Geometry: Shapes & Angles,polygons) GL ASSESSMENT QUESTIONS 11 plus exam part -4

 Geometry: Shapes & Angles,polygons) GL ASSESSMENT QUESTIONS 11 plus exam part -4

Part 2: 100+ Practice Questions (GL Assessment Style)

Section 1: 2D Shapes & Polygons (30 Questions)

1. What is the name of a triangle with all sides different lengths?
   A) Equilateral B) Isosceles C) Scalene D) Right-angled

2. How many lines of symmetry does a regular hexagon have?
   A) 3 B) 4 C) 6 D) 8

3. A quadrilateral has exactly one pair of parallel sides. What is it called?
   A) Parallelogram B) Rhombus C) Trapezium D) Kite

4. What is the sum of interior angles in a pentagon?
   A) 360° B) 540° C) 720° D) 900°

5. A regular polygon has interior angles of 150°. How many sides does it have?
   A) 8 B) 10 C) 12 D) 15

6. Which quadrilateral always has diagonals that bisect each other at right angles?
   A) Rectangle B) Rhombus C) Parallelogram D) Trapezium

7. What is the size of each exterior angle of a regular nonagon?
   A) 30° B) 40° C) 45° D) 50°

8. True or False: A square is a special type of rhombus.

9. An isosceles triangle has a vertex angle of 80°. What is the size of each base angle?
   A) 40° B) 50° C) 60° D) 70°

10. How many diagonals does an octagon have?
    A) 16 B) 20 C) 24 D) 28

11. Which shape has rotational symmetry of order 2 but only one line of symmetry?
    A) Square B) Rectangle C) Rhombus D) Parallelogram

12. What type of triangle has angles 45°, 45°, and 90°?
    A) Scalene right-angled B) Isosceles right-angled C) Equilateral D) Obtuse

13. The interior angles of a quadrilateral are in the ratio 1:2:3:4. What is the size of the largest angle?
    A) 72° B) 96° C) 120° D) 144°

14. A regular polygon has an exterior angle of 24°. How many sides does it have?
    A) 12 B) 15 C) 18 D) 20

15. Which statement is true for all parallelograms?
    A) All angles are 90° B) All sides are equal C) Opposite sides are equal D) Diagonals are equal

16. What is the order of rotational symmetry of an equilateral triangle?
    A) 1 B) 2 C) 3 D) 4

17. A heptagon has how many sides?
    A) 5 B) 6 C) 7 D) 8

18. Three angles in a quadrilateral are 85°, 95°, and 110°. What is the fourth angle?
    A) 60° B) 70° C) 80° D) 90°

19. True or False: Every rectangle is a parallelogram.

20. A kite has one angle of 120° and another of 50°. What could the other two angles be?
    A) 120° and 50° B) 120° and 70° C) 95° and 95° D) 85° and 85°

21. What is the sum of exterior angles of any polygon?
    A) 90° × n B) 180° C) 360° D) 540°

22. A polygon has 10 sides. What is the sum of its interior angles?
    A) 1080° B) 1260° C) 1440° D) 1620°

23. Which shape has exactly two lines of symmetry?
    A) Square B) Rectangle C) Rhombus D) Kite

24. An equilateral triangle has perimeter 24cm. What is its area?
    A) 16√3 cm² B) 24 cm² C) 16 cm² D) 27.7 cm²

25. True or False: A trapezium can have two right angles.

26. What is the specific name for a quadrilateral with all sides equal and all angles 90°?

27. A regular polygon has interior angles of 162°. How many sides does it have?

28. The diagonals of which quadrilateral are always equal in length?
    A) Rhombus B) Square C) Rectangle D) Both B and C

29. What is the size of one interior angle of a regular dodecagon (12 sides)?

30. A triangle has sides of 6cm, 8cm, and 10cm. What type of triangle is it?

Section 2: 3D Shapes (20 Questions)

31. How many faces does a triangular prism have?
    A) 4 B) 5 C) 6 D) 8

32. A cube has:
    A) 6 faces, 8 edges, 12 vertices
    B) 6 faces, 12 edges, 8 vertices
    C) 8 faces, 12 edges, 6 vertices
    D) 12 faces, 8 edges, 6 vertices

33. What is the name of a 3D shape with 2 circular faces and 1 curved surface?
    A) Cone B) Cylinder C) Sphere D) Prism

34. How many vertices does a square-based pyramid have?
    A) 4 B) 5 C) 6 D) 8

35. A tetrahedron has how many triangular faces?
    A) 3 B) 4 C) 5 D) 6

36. Which net will make a cube?
    [Diagram showing different arrangements of 6 squares]

37. A cuboid has dimensions 5cm × 4cm × 3cm. What is its volume?
    A) 12 cm³ B) 48 cm³ C) 60 cm³ D) 120 cm³

38. How many edges does a hexagonal prism have?
    A) 12 B) 15 C) 18 D) 24

39. What shape are the faces of a dodecahedron?
    A) Squares B) Pentagons C) Hexagons D) Triangles

40. True or False: A cone has 2 faces and 1 edge.

41. If you slice a cube parallel to one face, what shape is the cross-section?
    A) Triangle B) Square C) Rectangle D) Hexagon

42. How many more edges does a cube have than a tetrahedron?
    A) 4 B) 6 C) 8 D) 12

43. A cylinder has radius 7cm and height 10cm. What is its volume? (Use π = 22/7)
    A) 1540 cm³ B) 154 cm³ C) 440 cm³ D) 15400 cm³

44. What is the surface area of a cube with side 5cm?
    A) 25 cm² B) 100 cm² C) 125 cm² D) 150 cm²

45. Which shape has Euler's formula F + V = E + 2?
    A) All polyhedra B) Only prisms C) Only pyramids D) Only cubes

46. How many rectangular faces does a triangular prism have?

47. What is the name of a prism with a pentagon as its cross-section?

48. A cube has volume 64 cm³. What is the length of one edge?
    A) 2cm B) 4cm C) 8cm D) 16cm

49. True or False: A sphere has faces, edges, and vertices.

50. Which 3D shape has exactly one vertex?
    A) Cube B) Cylinder C) Cone D) Sphere

Section 3: Angles & Angle Rules (30 Questions)

51. What type of angle is 125°?
    A) Acute B) Obtuse C) Reflex D) Right

52. Angles on a straight line add up to:
    A) 90° B) 180° C) 270° D) 360°

53. Two angles are supplementary. One is 75°. What is the other?
    A) 15° B) 105° C) 115° D) 285°

54. In a triangle, angles are 2x, 3x, and 4x. What is the value of x?
    A) 10° B) 20° C) 30° D) 40°

55. Vertically opposite angles are always:
    A) Supplementary B) Complementary C) Equal D) Different

56. Find angle a in this triangle with angles 40° and 60°:
    A) 60° B) 70° C) 80° D) 90°

57. Angles around a point add up to:
    A) 90° B) 180° C) 270° D) 360°

58. In a right-angled triangle, the two acute angles are:
    A) Supplementary B) Complementary C) Equal D) Both 45°

59. One angle of an isosceles triangle is 100°. What are the other two angles?
    A) 40° each B) 50° each C) 80° each D) 90° and 10°

60. Two parallel lines are crossed by a transversal. Corresponding angles are:
    A) Equal B) Supplementary C) Complementary D) None of these

61. Calculate the reflex angle if the acute angle is 35°:
    A) 35° B) 145° C) 215° D) 325°

62. In quadrilateral PQRS, angle P = 80°, Q = 100°, R = 120°. Find angle S:
    A) 40° B) 50° C) 60° D) 70°

63. Angles x and 3x are on a straight line. What is x?
    A) 30° B) 45° C) 60° D) 90°

64. True or False: An obtuse angle is between 90° and 180°.

65. One angle of a parallelogram is 70°. What are the other angles?
    A) 70°, 110°, 110° B) 70°, 70°, 110° C) 70°, 120°, 110° D) 70°, 100°, 100°

66. Find the missing angle in this triangle: 35°, 45°, ?
    A) 80° B) 90° C) 100° D) 110°

67. Two angles are complementary. Their difference is 20°. What is the larger angle?
    A) 35° B) 45° C) 55° D) 65°

68. In a regular pentagon, what is each interior angle?
    A) 72° B) 108° C) 120° D) 135°

69. Angles a, b, c are around a point. If a = 110°, b = 95°, find c:
    A) 115° B) 125° C) 135° D) 155°

70. Alternate angles formed by parallel lines are:
    A) Equal B) Supplementary C) Complementary D) Vertically opposite

71. A triangle cannot have:
    A) Two acute angles B) Two obtuse angles C) One right angle D) Three acute angles

72. Calculate the third angle in a triangle with angles 55° and 65°:
    A) 50° B) 60° C) 70° D) 80°

73. True or False: A quadrilateral can have all four angles obtuse.

74. The exterior angle of a triangle equals:
    A) Sum of two interior angles B) Sum of all interior angles C) Sum of two opposite interior angles D) 180° minus one interior angle

75. Find angle x if angles are 2x, 3x, and 5x in a triangle:
    A) 18° B) 30° C) 36° D) 45°

76. Co-interior angles between parallel lines sum to:
    A) 90° B) 180° C) 270° D) 360°

77. One angle of a rhombus is 50°. What is the adjacent angle?
    A) 40° B) 50° C) 130° D) 140°

78. How many degrees are there in a full turn?
    A) 90° B) 180° C) 270° D) 360°

79. The supplement of 45° is:
    A) 45° B) 90° C) 135° D) 180°

80. In a trapezium with one pair of parallel sides, two angles on the same side are:
    A) Equal B) Supplementary C) Complementary D) Vertically opposite

Section 4: Symmetry & Coordinates (20 Questions)

81. What is the order of rotational symmetry of a square?
    A) 1 B) 2 C) 3 D) 4

82. Point A has coordinates (3, 4). What are the coordinates of its reflection in the y-axis?
    A) (-3, 4) B) (3, -4) C) (-3, -4) D) (4, 3)

83. Which shape has infinite lines of symmetry?
    A) Square B) Circle C) Equilateral triangle D) Regular hexagon

84. The point (0, 0) is called:
    A) Origin B) Axis C) Vertex D) Midpoint

85. What is the line symmetry of the letter H?
    A) 1 B) 2 C) 3 D) 4

86. Rotate point (2, 3) 90° clockwise about the origin. What are the new coordinates?
    A) (3, -2) B) (-3, 2) C) (3, 2) D) (-2, 3)

87. Which quadrilateral has no lines of symmetry?
    A) Square B) Rectangle C) Parallelogram (general) D) Rhombus

88. Point P is (5, 2). Point Q is (9, 2). What is the length PQ?
    A) 2 B) 4 C) 6 D) 8

89. True or False: A scalene triangle has no rotational symmetry.

90. Midpoint of (2, 4) and (6, 8) is:
    A) (3, 4) B) (4, 6) C) (4, 4) D) (6, 4)

91. Which shape has rotational symmetry order 3?
    A) Square B) Equilateral triangle C) Rectangle D) Regular pentagon

92. Point in Quadrant III has coordinates:
    A) (+, +) B) (-, +) C) (-, -) D) (+, -)

93. How many lines of symmetry does a regular octagon have?
    A) 4 B) 6 C) 8 D) 10

94. Reflect point (4, -1) in the x-axis:
    A) (4, 1) B) (-4, 1) C) (-4, -1) D) (4, -1)

95. Which letter has rotational symmetry order 2?
    A) A B) H C) N D) S

96. Distance between (1, 1) and (4, 5) is:
    A) 3 B) 4 C) 5 D) 6

97. True or False: A kite always has exactly one line of symmetry.

98. What is the order of rotational symmetry of the letter Z?
    A) 0 B) 1 C) 2 D) 4

99. Point R is 3 units right and 2 units down from point (1, 5). What are its coordinates?
    A) (4, 3) B) (4, 7) C) (-2, 3) D) (-2, 7)

100. Which transformation changes (x, y) to (x, -y)?
     A) Reflection in x-axis B) Reflection in y-axis C) 90° rotation D) Translation

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Part 3: 10 Previous Year GL Assessment Questions

Question 1: The diagram shows two parallel lines crossed by a transversal. One angle is marked 65°. What is the size of angle x?
[Diagram showing parallel lines with 65° angle and angle x marked as corresponding]
A) 65° B) 115° C) 125° D) 180°

Question 2: Which 3D shape has exactly 5 faces, 8 edges, and 5 vertices?
A) Cube B) Cuboid C) Square-based pyramid D) Triangular prism

Question 3: A regular polygon has interior angles of 135°. How many sides does it have?
A) 6 B) 8 C) 10 D) 12

Question 4: *Point P has coordinates (3, -2). In which quadrant is it located?*
A) Quadrant I B) Quadrant II C) Quadrant III D) Quadrant IV

Question 5: The net of a 3D shape is shown. When folded, which shape does it make?
[Net showing 6 squares in a cross shape]
A) Cube B) Cuboid C) Square-based pyramid D) Tetrahedron

Question 6: In an isosceles triangle, the vertex angle is 40°. What is the size of each base angle?
A) 40° B) 60° C) 70° D) 80°

Question 7: Which quadrilateral always has diagonals that are equal in length?
A) Rhombus B) Kite C) Parallelogram D) Rectangle

Question 8: Angles around a point are 85°, 95°, 120°, and x. What is the value of x?
A) 50° B) 60° C) 70° D) 80°

Question 9: How many lines of symmetry does a regular pentagon have?
A) 3 B) 4 C) 5 D) 6

Question 10: A cube has volume 125 cm³. What is the surface area of one face?
A) 5 cm² B) 25 cm² C) 100 cm² D) 150 cm²

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Part 4: Answer Key & Detailed Solutions

Section 1 Solutions:

1. C) Scalene

2. C) 6

3. C) Trapezium

4. B) 540° ((5-2)×180 = 3×180)

5. C) 12 (Interior 150°, exterior = 30°, sides = 360÷30 = 12)

6. B) Rhombus

7. B) 40° (360÷9 = 40°)

8. True (A square has all properties of a rhombus plus right angles)

9. B) 50° ((180-80)÷2 = 50°)

10. B) 20 (n(n-3)/2 = 8×5/2 = 20)

11. B) Rectangle

12. B) Isosceles right-angled

13. D) 144° (1+2+3+4=10 parts, 360÷10=36 per part, largest=4×36=144°)

14. B) 15 (360÷24 = 15)

15. C) Opposite sides are equal

16. C) 3

17. C) 7

18. B) 70° (360-(85+95+110)=70°)

19. True

20. C) 95° and 95° (Sum of angles=360, remaining sum=190°, in kite two angles are equal)

21. C) 360°

22. C) 1440° ((10-2)×180=8×180)

23. C) Rhombus

24. D) 27.7 cm² (side=8cm, area=(√3/4)×64=16√3≈27.7)

25. True

26. Square

27. 20 (Interior=162°, exterior=18°, sides=360÷18=20)

28. D) Both B and C

29. 150° ((12-2)×180÷12=10×180÷12=150)

30. Right-angled (6²+8²=10²)

Section 2 Solutions:

31. B) 5

32. B) 6 faces, 12 edges, 8 vertices

33. B) Cylinder

34. B) 5

35. B) 4

36. [The "T" shape net cannot make a cube; cross shape can]

37. C) 60 cm³ (5×4×3)

38. C) 18

39. B) Pentagons

40. True (1 circular face, 1 curved face, 1 circular edge)

41. B) Square

42. B) 6 (Cube:12 edges, Tetrahedron:6 edges, difference=6)

43. A) 1540 cm³ (πr²h=(22/7)×49×10=1540)

44. D) 150 cm² (6×5²=6×25)

45. A) All polyhedra

46. 3

47. Pentagonal prism

48. B) 4cm (∛64=4)

49. False (Sphere has none)

50. C) Cone

Section 3 Solutions:

51. B) Obtuse

52. B) 180°

53. B) 105° (180-75)

54. B) 20° (2x+3x+4x=180, 9x=180, x=20)

55. C) Equal

56. C) 80° (180-(40+60)=80)

57. D) 360°

58. B) Complementary (sum to 90°)

59. A) 40° each (180-100=80, 80÷2=40)

60. A) Equal

61. D) 325° (360-35)

62. C) 60° (360-(80+100+120)=60)

63. B) 45° (x+3x=180, 4x=180, x=45)

64. True

65. A) 70°, 110°, 110° (Adjacent angles supplementary)

66. C) 100° (180-(35+45)=100)

67. C) 55° (x+y=90, x-y=20, solving: 2x=110, x=55)

68. B) 108° ((5-2)×180÷5=3×180÷5=108)

69. D) 155° (360-(110+95)=155)

70. A) Equal

71. B) Two obtuse angles (sum would exceed 180°)

72. B) 60° (180-(55+65)=60)

73. False (Sum would exceed 360°)

74. C) Sum of two opposite interior angles

75. A) 18° (2x+3x+5x=180, 10x=180, x=18)

76. B) 180°

77. C) 130° (Adjacent angles supplementary in rhombus)

78. D) 360°

79. C) 135° (180-45)

80. B) Supplementary

Section 4 Solutions:

81. D) 4

82. A) (-3, 4)

83. B) Circle

84. A) Origin

85. B) 2 (horizontal and vertical)

86. A) (3, -2)

87. C) Parallelogram (general)

88. B) 4 (9-5=4)

89. True

90. B) (4, 6) ((2+6)/2=4, (4+8)/2=6)

91. B) Equilateral triangle

92. C) (-, -)

93. C) 8

94. A) (4, 1)

95. C) N (also H, I, O, S, X, Z)

96. C) 5 (√[(4-1)²+(5-1)²]=√[9+16]=√25=5)

97. False (A kite can have 2 lines if it's a rhombus)

98. C) 2 (rotational symmetry of 180°)

99. A) (4, 3) (1+3=4, 5-2=3)

100. A) Reflection in x-axis

10 GL Assessment Questions Solutions:

1. A) 65° (Corresponding angles are equal with parallel lines)

2. C) Square-based pyramid (5 faces: 1 square + 4 triangles, 8 edges, 5 vertices)

3. B) 8 (Interior=135°, exterior=45°, sides=360÷45=8)

4. D) Quadrant IV (x positive, y negative)

5. A) Cube (The cross-shaped net of 6 squares folds to make a cube)

6. C) 70° ((180-40)÷2=140÷2=70)

7. D) Rectangle (Both square and rectangle have equal diagonals)

8. B) 60° (360-(85+95+120)=60)

9. C) 5 (Regular pentagon has 5 lines of symmetry)

10. B) 25 cm² (Side = ∛125=5cm, face area=5×5=25 cm²)

Emoji Compound Words Quiz + Math Logic Puzzles (Brain Test)

 🧠 Welcome to the Emoji & Math Brain Challenge!

This fun post includes:

• Emoji Compound Words Quiz

• Math & Logic Puzzles

• Answer Key at the bottom

Rules:

• Try all questions before checking answers

• No cheating ❌

• Comment your final score 💬

• Share with friends 📤

👇 Scroll down to start the challenge!

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🐞

Hi! I’m Quiztle 🤓

Try all questions first! The answer key is at the bottom. Don’t scroll too fast 😉

🧩 Emoji Compound Words Quiz

1. 🌞 + 🌻 = ?
2. 📚 + 🐛 = ?
3. ❄️ + ⚽ = ?
4. 🌧️ + ☔ = ?
5. 🌙 + ✨ = ?
6. 🐦 + 🏠 = ?
7. 🏠 + 🛏️ = ?
8. 🧊 + 🍦 = ?
9. 🍎 + 🥧 = ?
10. 🧠 + 💡 = ?

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🧮 Math & Logic Puzzle Game

1. 8 + 7 × 2 = ?
2. 50 − (5 × 6) = ?
3. If 2 → 6 and 3 → 12, what is 4 → ?
4. Which number comes next: 2, 6, 12, 20, ___
5. I am an odd number. Remove one letter and I become even. What number am I?

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🎯 Score Guide

Correct Answers	Your Level
0 – 5	Beginner 🌱
6 – 10	Good 👍
11 – 14	Smart 🧠
15	Genius 🏆

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✅ Answer Key

Emoji Quiz Answers

1. Sunflower
2. Bookworm
3. Snowball
4. Raincoat
5. Moonlight
6. Birdhouse
7. Bedroom
8. Ice cream
9. Apple pie
10. Brainstorm

Math & Logic Answers

1. 22
2. 20
3. 20
4. 30
5. Seven

🎉 Done! Comment your score and challenge your friends.

Geometry: Shapes & Angles,polygons) GL ASSESSMENT QUESTIONS 11 plus exam part -3

 

Geometry: Shapes & Angles,polygons) GL ASSESSMENT QUESTIONS 11 plus exam part -3

Section A: Angles & Types of Angles (10 Questions)

1. What is the correct name for an angle that is greater than 180° but less than 360°?

2. Look at the clock. What type of angle is between the hands at 2 o'clock?

3. An angle is measured at 91°. What type of angle is it?

4. How many right angles are there in a full turn?

5. Two angles are complementary. One is 37°. What is the other?

6. An angle is half the size of its supplement. What is the size of the smaller angle?

7. In a triangle, one angle is a right angle and another is 30°. What is the third angle?

8. A reflex angle is 245°. What is the size of the corresponding acute/obtuse angle inside the shape?

9. Which of these is not possible for the angles of a triangle? A) 45°, 55°, 80° B) 60°, 60°, 60° C) 100°, 40°, 40° D) 30°, 70°, 100°

10. A full rotation is divided into 5 equal angles. How big is each angle?

Section B: Polygons (10 Questions)

11. What is the sum of the interior angles of a heptagon?

12. A regular polygon has an exterior angle of 24°. How many sides does it have?

13. The interior angles of a pentagon are 100°, 110°, 115°, 105° and *x*. Find the value of *x*.

14. What is the name of a nine-sided polygon?

15. Is it possible for a regular polygon to have an interior angle of 170°? Explain your answer.

16. How many diagonals can you draw from one vertex of a hexagon?

17. The sum of the interior angles of a polygon is 1260°. How many sides does it have?

18. A regular octagon has how many lines of symmetry?

19. Each interior angle of a regular polygon is 144°. What is the name of the polygon?

20. True or False: A circle is a polygon.

Section C: Properties of 2D Shapes (50 Questions)

This section covers Triangles and Quadrilaterals (sides, angles, symmetry).

21. What type of triangle has all sides different lengths?

22. A triangle has angles of 70° and 70°. What is the size of the third angle and what type of triangle is it?

23. How many lines of symmetry does an equilateral triangle have?

24. What is the order of rotational symmetry of a square?

25. Which quadrilateral has only one pair of parallel sides?

26. All squares are rectangles. Are all rectangles squares?

27. A rhombus has all sides equal. Does it always have all angles equal?

28. What is the specific name for a quadrilateral with both pairs of opposite sides parallel and equal?

29. A parallelogram has one angle of 75°. What are the sizes of its other three angles?

30. How many right angles does a kite have?

31. What is the difference between a scalene and an isosceles triangle?

32. Draw a sketch of a trapezium.

33. A triangle has one line of symmetry and no rotational symmetry. What type of triangle is it?

34. True or False: A rhombus is a regular polygon.

35. What is the size of each angle in an equilateral triangle?

36. A quadrilateral has rotational symmetry of order 4. What could it be?

37. How many pairs of equal sides does an isosceles triangle have?

38. What is the sum of the angles in any quadrilateral?

39. A shape has four lines of symmetry and rotational symmetry of order 4. What is it?

40. True or False: Every square is a rhombus.

41. In an isosceles triangle, the angle at the apex (top) is 40°. What is the size of each base angle?

42. A rectangle has a length of 8cm and a width of 5cm. What is the perimeter?

43. A square has a perimeter of 36cm. What is its area?

44. A parallelogram has an area of 20cm² and a base of 5cm. What is its height?

45. A triangle is drawn on a centimetre grid. Its vertices are at (1,1), (1,5), and (4,1). What is its area?

46. What is the name given to the longest side of a right-angled triangle?

47. If two angles in a triangle are 45° and 55°, what is the third angle and what type of triangle is it?

48. A quadrilateral has exactly two lines of symmetry. It is not a rectangle. What could it be?

49. How many sides does a heptagon have?

50. True or False: A trapezium can have a right angle.

51. A regular polygon has an interior angle of 135°. How many sides does it have?

52. What is the exterior angle of a regular nonagon?

53. A triangle has sides of length 5cm, 5cm, and 8cm. What type of triangle is it?

54. A triangle has sides of length 3cm, 4cm, and 5cm. What type of triangle is it?

55. A quadrilateral has all sides equal and one angle of 90°. What is its name?

56. A quadrilateral has opposite angles equal and all sides equal, but it is not a square. What is it?

57. How many lines of symmetry does a regular hexagon have?

58. What is the order of rotational symmetry of an equilateral triangle?

59. True or False: A circle has infinite lines of symmetry.

60. A shape has rotational symmetry of order 2 and no line symmetry. What could it be?

61. A triangle has an area of 15cm² and a base of 6cm. What is its height?

62. A square has an area of 49cm². What is its perimeter?

63. A rectangle has a perimeter of 24cm and a length of 8cm. What is its width?

64. A parallelogram has a base of 10cm and a height of 3cm. What is its area?

65. A trapezium has parallel sides of 6cm and 10cm, and a height of 4cm. What is its area?

66. What is the name of a triangle with all angles less than 90°?

67. What is the name of a triangle with one angle greater than 90°?

68. True or False: A rhombus has diagonals that bisect each other at right angles.

69. How many vertices does a pentagon have?

70. What is the name of a quadrilateral where diagonals are equal and bisect each other at right angles?

Section D: Angle Rules (30 Questions)

*Apply these rules: Straight line=180°, Around a point=360°, Vertically opposite are equal, Triangle sum=180°, Quadrilateral sum=360°.*

71. Two angles on a straight line are 125° and *x*. Find *x*.

72. Three angles around a point are 100°, 150° and *y*. Find *y*.

73. In the diagram, two straight lines cross. One angle is 40°. What are the sizes of the other three angles?

74. In an isosceles triangle, the vertex angle is 50°. What is the size of each base angle?

75. A quadrilateral has angles of 80°, 95°, and 110°. What is the fourth angle?

76. Find the size of angle *a* in a triangle with angles 60° and 70°.

77. Angles *p* and *q* are vertically opposite. If *p* is 110°, what is *q*?

78. In a right-angled triangle, one acute angle is twice the other. What are the sizes of the angles?

79. A straight line has three angles on it, *x*, 2*x*, and 90°. Find the value of *x*.

80. In a parallelogram, one angle is 110°. What are the sizes of the other three angles?

81. The angles of a triangle are in the ratio 2:3:4. Find the size of the largest angle.

82. In a rhombus, one angle is 65°. What is the size of the angle adjacent to it?

83. Angles *a* and *b* are on a straight line. Angle *a* is 3 times angle *b*. Find angle *b*.

84. Four angles around a point are 2*x*, 3*x*, 4*x*, and 5*x*. Find the value of *x*.

85. In a pentagon, four of the angles are 100°, 110°, 120°, and 130°. What is the fifth angle?

86. An isosceles triangle has a base angle of 40°. What is the vertex angle?

87. Two angles in a triangle are 35° and 45°. What is the third angle?

88. In a quadrilateral, three angles are equal and the fourth is 90°. What is the size of each of the equal angles?

89. Angles *c* and *d* are complementary. Angle *c* is 15° more than angle *d*. Find angle *c*.

90. The exterior angle of a triangle is 120° and one of its interior opposite angles is 50°. What is the other interior opposite angle?

91. In a trapezium with one pair of parallel sides, one angle is 85°. What is the angle on the same side of the transversal?

92. A triangle has angles of (x+10)°, (2x-20)°, and 60°. Find the value of x.

93. A quadrilateral has angles of x, 2x, 3x, and 4x. Find the value of x.

94. In a regular hexagon, what is the size of each interior angle?

95. The angles of a triangle are (2y)°, (3y)°, and (4y)°. Find the value of y.

96. On a straight line, angles are 2a, 3a, and 4a. Find the smallest angle.

97. Around a point, angles are a, 2a, 3a, 4a, and 5a. Find the largest angle.

98. In an isosceles triangle, the vertex angle is 4 times a base angle. Find the vertex angle.

99. Two vertically opposite angles are (3x+10)° and (5x-20)°. Find the value of x.

100. In a right-angled isosceles triangle, what are the sizes of the two acute angles?

ANSWER KEY

Section A: Angles & Types of Angles

1. What is the correct name for an angle that is greater than 180° but less than 360°?
Answer: Reflex angle
Explanation: Angles are classified as acute (< 90°), right (90°), obtuse (between 90° and 180°), reflex (between 180° and 360°), and full turn (360°). Therefore, an angle between 180° and 360° is a reflex angle.

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2. Look at the clock. What type of angle is between the hands at 2 o’clock?
Answer: Acute
Explanation: At 2 o’clock, the hour hand is at 2 and the minute hand is at 12. The angle between each hour mark on a clock is 30° (360° ÷ 12 = 30°). From 12 to 2 is 2 × 30° = 60°, which is less than 90°, so it is acute.

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3. An angle is measured at 91°. What type of angle is it?
Answer: Obtuse
Explanation: An obtuse angle is greater than 90° and less than 180°. Since 91° lies in that range, it is obtuse.

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4. How many right angles are there in a full turn?
Answer: 4
Explanation: A full turn is 360°. A right angle is 90°. 360° ÷ 90° = 4.

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5. Two angles are complementary. One is 37°. What is the other?
Answer: 53°
Explanation: Complementary angles add up to 90°. So, 90° − 37° = 53°.

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6. An angle is half the size of its supplement. What is the size of the smaller angle?
Answer: 60°
Explanation: Let the smaller angle be $x$. Its supplement is $2x$. Supplementary angles add to 180°, so $x+2x=180\mathrm{°}$, giving $3x=180\mathrm{°}$ and $x=60\mathrm{°}$.

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7. In a triangle, one angle is a right angle and another is 30°. What is the third angle?
Answer: 60°
Explanation: The sum of angles in a triangle is 180°. 180° − (90° + 30°) = 60°.

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8. A reflex angle is 245°. What is the size of the corresponding acute/obtuse angle inside the shape?
Answer: 115°
Explanation: The corresponding angle inside a full rotation is 360° − reflex angle. 360° − 245° = 115°, which is obtuse.

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9. Which of these is not possible for the angles of a triangle?
A) 45°, 55°, 80°
B) 60°, 60°, 60°
C) 100°, 40°, 40°
D) 30°, 70°, 100°

Answer: D) 30°, 70°, 100°
Explanation: The angles in a triangle must sum to 180°. Option D sums to 200°, so it is not possible.

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10. A full rotation is divided into 5 equal angles. How big is each angle?
Answer: 72°

Explanation: Full rotation = 360°. Divided into 5 equal parts: 360° ÷ 5 = 72°.

Section B: Polygons

11. What is the sum of the interior angles of a heptagon?
Answer: 900°
Explanation: A heptagon has 7 sides. The formula for the sum of interior angles is $(n-2)\times {180}^{\circ }$, where $n$ is the number of sides. So, $(7-2)\times {180}^{\circ }=5\times {180}^{\circ }={900}^{\circ }$.

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12. A regular polygon has an exterior angle of 24°. How many sides does it have?
Answer: 15 sides
Explanation: For any regular polygon, the sum of exterior angles is 360°. The number of sides is ${360}^{\circ }÷\text{exterior angle}$. So, ${360}^{\circ }÷{24}^{\circ }=15$.

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13. The interior angles of a pentagon are 100°, 110°, 115°, 105° and *x*. Find the value of *x*.
Answer: 110°
Explanation: A pentagon has 5 sides, so the sum of interior angles = $(5-2)\times {180}^{\circ }=3\times {180}^{\circ }={540}^{\circ }$. The given angles sum to $100+110+115+105={430}^{\circ }$. Therefore, $x={540}^{\circ }-{430}^{\circ }={110}^{\circ }$.

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14. What is the name of a nine-sided polygon?
Answer: Nonagon
Explanation: Standard polygon names: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10), etc.

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15. Is it possible for a regular polygon to have an interior angle of 170°? Explain your answer.
Answer: Yes
Explanation: If interior angle = 170°, then exterior angle = 180° - 170° = 10°. The number of sides would be ${360}^{\circ }÷{10}^{\circ }=36$. Since 36 is a whole number, a regular 36-sided polygon (triacontakaihexagon) is possible.

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16. How many diagonals can you draw from one vertex of a hexagon?
Answer: 3
Explanation: A hexagon has 6 vertices. From one vertex, you cannot draw a diagonal to itself or to the two adjacent vertices (they are already connected by sides). So diagonals = $6-3=3$.

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17. The sum of the interior angles of a polygon is 1260°. How many sides does it have?
Answer: 9 sides
Explanation: Using $(n-2)\times {180}^{\circ }={1260}^{\circ }$.
$n-2=1260÷180=7$
$n=7+2=9$.

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18. A regular octagon has how many lines of symmetry?
Answer: 8
Explanation: A regular octagon has 8 equal sides and 8 equal angles, so it has 8 lines of symmetry (through opposite vertices and through midpoints of opposite sides).

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19. Each interior angle of a regular polygon is 144°. What is the name of the polygon?
Answer: Decagon
Explanation: If interior = 144°, exterior = 180° - 144° = 36°. Number of sides = ${360}^{\circ }÷{36}^{\circ }=10$. A 10-sided polygon is a decagon.

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20. True or False: A circle is a polygon.
Answer: False
Explanation: A polygon is a closed shape with straight sides. A circle has a curved boundary, so it is not a polygon

Section C: Properties of 2D Shapes (First 20 of the 50 questions)

21. What type of triangle has all sides different lengths?
Answer: Scalene triangle
Explanation: Triangles are classified by side lengths:

• Equilateral: all sides equal

• Isosceles: two sides equal

• Scalene: all sides different

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22. A triangle has angles of 70° and 70°. What is the size of the third angle and what type of triangle is it?
Answer: 40°, isosceles triangle
Explanation: Sum of angles = 180°. Third angle = 180° - (70° + 70°) = 40°. Since two angles are equal, the sides opposite them are equal → isosceles triangle.

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23. How many lines of symmetry does an equilateral triangle have?
Answer: 3
Explanation: An equilateral triangle has three equal sides and three equal angles. Each line from a vertex to the midpoint of the opposite side is a line of symmetry → 3 lines.

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24. What is the order of rotational symmetry of a square?
Answer: 4
Explanation: A square fits onto itself 4 times during a full 360° rotation (at 90°, 180°, 270°, and 360°).

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25. Which quadrilateral has only one pair of parallel sides?
Answer: Trapezium (UK definition)
Explanation: In UK geometry:

• Trapezium: 1 pair of parallel sides

• Trapezoid (US): 1 pair of parallel sides

• Parallelogram: 2 pairs of parallel sides

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26. All squares are rectangles. Are all rectangles squares?
Answer: No
Explanation: A square is a special rectangle with all sides equal. Rectangles have opposite sides equal but not necessarily all four sides equal.

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27. A rhombus has all sides equal. Does it always have all angles equal?
Answer: No
Explanation: A rhombus has all sides equal, but angles are only equal if it's a square. A "slanted" rhombus has unequal opposite angles.

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28. What is the specific name for a quadrilateral with both pairs of opposite sides parallel and equal?
Answer: Parallelogram
Explanation: Definition of a parallelogram: both pairs of opposite sides are parallel (and therefore equal in length).

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29. A parallelogram has one angle of 75°. What are the sizes of its other three angles?
Answer: 75°, 105°, 105°
Explanation: In a parallelogram:

• Opposite angles are equal → another 75°

• Adjacent angles are supplementary (add to 180°) → 180° - 75° = 105° for the other two angles

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30. How many right angles does a kite have?
Answer: Typically 0, but can have 2 in a special case (right kite)
Explanation: A standard kite has no right angles unless specified as a "right kite," which has two opposite right angles.

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31. What is the difference between a scalene and an isosceles triangle?
Answer: Scalene has all sides different; isosceles has at least two equal sides.
Explanation: Classification by sides:

• Scalene: no equal sides

• Isosceles: two equal sides

• Equilateral: three equal sides

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32. Draw a sketch of a trapezium.
Answer: [Visual description] A quadrilateral with one pair of parallel sides. Often drawn with horizontal parallel lines, the other two sides non-parallel.

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33. A triangle has one line of symmetry and no rotational symmetry. What type of triangle is it?
Answer: Isosceles triangle
Explanation: An isosceles triangle has exactly one line of symmetry (through the vertex angle to the midpoint of the base) and rotational symmetry of order 1 (none).

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34. True or False: A rhombus is a regular polygon.
Answer: False
Explanation: A regular polygon must have all sides equal AND all angles equal. A rhombus has all sides equal but not necessarily all angles equal (unless it's a square).

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35. What is the size of each angle in an equilateral triangle?
Answer: 60°
Explanation: All angles equal in equilateral triangle → 180° ÷ 3 = 60°.

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36. A quadrilateral has rotational symmetry of order 4. What could it be?
Answer: Square
Explanation: Among quadrilaterals, only a square has rotational symmetry of order 4 (90° rotations). A rhombus has order 2.

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37. How many pairs of equal sides does an isosceles triangle have?
Answer: 1 pair
Explanation: Exactly two sides are equal in an isosceles triangle, forming one pair of equal sides.

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38. What is the sum of the angles in any quadrilateral?
Answer: 360°
Explanation: Formula: $(n-2)\times 180\mathrm{°}$ with n=4 → $2\times 180\mathrm{°}=360\mathrm{°}$.

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39. A shape has four lines of symmetry and rotational symmetry of order 4. What is it?
Answer: Square
Explanation: A square has 4 lines of symmetry (through opposite vertices and midpoints) and rotational symmetry of order 4.

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40. True or False: Every square is a rhombus.
Answer: True
Explanation: A square meets the definition of a rhombus (all sides equal) with the additional property of all angles being 90°.

Section C: Properties of 2D Shapes

41. In an isosceles triangle, the angle at the apex (top) is 40°. What is the size of each base angle?
Answer: 70°
Explanation: Sum of angles = 180°. Let each base angle = $x$. Then $40\mathrm{°}+x+x=180\mathrm{°}$ → $2x=140\mathrm{°}$ → $x=70\mathrm{°}$.

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42. A rectangle has a length of 8cm and a width of 5cm. What is the perimeter?
Answer: 26 cm
Explanation: Perimeter of rectangle = $2\times (\text{length}+\text{width})=2\times (8+5)=2\times 13=26$ cm.

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43. A square has a perimeter of 36cm. What is its area?
Answer: 81 cm²
Explanation: Perimeter = $4\times \text{side}$. Side = $36÷4=9$ cm. Area = side² = $9\times 9=81$ cm².

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44. A parallelogram has an area of 20 cm² and a base of 5 cm. What is its height?
Answer: 4 cm
Explanation: Area of parallelogram = base × height. So $20=5\times h$ → $h=20÷5=4$ cm.

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45. A triangle is drawn on a centimetre grid. Its vertices are at (1,1), (1,5), and (4,1). What is its area?
Answer: 6 cm²
Explanation: This is a right-angled triangle with:

• Base along x-axis from (1,1) to (4,1) → length = 3 cm

• Height along y-axis from (1,1) to (1,5) → length = 4 cm
  Area = $\text{½}\times \text{base}\times \text{height}=\text{½}\times 3\times 4=6$ cm².

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46. What is the name given to the longest side of a right-angled triangle?
Answer: Hypotenuse
Explanation: In a right-angled triangle, the side opposite the right angle is the hypotenuse, and it is always the longest side (by Pythagoras).

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47. If two angles in a triangle are 45° and 55°, what is the third angle and what type of triangle is it?
Answer: 80°, acute-angled triangle
Explanation: Third angle = $180\mathrm{°}-(45\mathrm{°}+55\mathrm{°})=80\mathrm{°}$. All angles < 90°, so acute-angled.

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48. A quadrilateral has exactly two lines of symmetry. It is not a rectangle. What could it be?
Answer: Rhombus
Explanation: A rectangle has 2 lines of symmetry, but if "not a rectangle," another possibility is a rhombus (lines through opposite vertices).

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49. How many sides does a heptagon have?
Answer: 7
Explanation: Greek prefixes: hepta- = 7.

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50. True or False: A trapezium can have a right angle.
Answer: True
Explanation: A right-angled trapezium has two right angles adjacent to each other on one of the parallel sides.

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51. A regular polygon has an interior angle of 135°. How many sides does it have?
Answer: 8 sides (octagon)
Explanation: If interior = 135°, exterior = 180° - 135° = 45°. Sides = $360\mathrm{°}÷45\mathrm{°}=8$.

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52. What is the exterior angle of a regular nonagon?
Answer: 40°
Explanation: Nonagon = 9 sides. Exterior angle = $360\mathrm{°}÷9=40\mathrm{°}$.

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53. A triangle has sides of length 5 cm, 5 cm, and 8 cm. What type of triangle is it?
Answer: Isosceles triangle
Explanation: Two sides equal (5 cm) → isosceles.

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54. A triangle has sides of length 3 cm, 4 cm, and 5 cm. What type of triangle is it?
Answer: Right-angled triangle
Explanation: $3^2+4^2=9+16=25=5^2$, so it satisfies Pythagoras' theorem → right-angled.

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55. A quadrilateral has all sides equal and one angle of 90°. What is its name?
Answer: Square
Explanation: All sides equal → rhombus. One angle 90° → all angles 90° (because adjacent angles supplementary, opposite angles equal) → square.

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56. A quadrilateral has opposite angles equal and all sides equal, but it is not a square. What is it?
Answer: Rhombus
Explanation: All sides equal = rhombus. If not a square, then angles are not 90°.

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57. How many lines of symmetry does a regular hexagon have?
Answer: 6
Explanation: A regular hexagon has 6 lines of symmetry (through opposite vertices and through midpoints of opposite sides).

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58. What is the order of rotational symmetry of an equilateral triangle?
Answer: 3
Explanation: Fits onto itself 3 times in a 360° rotation (at 120°, 240°, 360°).

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59. True or False: A circle has infinite lines of symmetry.
Answer: True
Explanation: Any line through the center of a circle is a line of symmetry → infinite such lines.

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60. A shape has rotational symmetry of order 2 and no line symmetry. What could it be?
Answer: Parallelogram (non-rectangle, non-rhombus)
Explanation: Example: a slanted parallelogram (not rectangle, not rhombus) has order 2 rotational symmetry but no lines of symmetry.

61. A triangle has an area of 15 cm² and a base of 6 cm. What is its height?
Answer: 5 cm
Explanation: Area of triangle = $\text{½}\times \text{base}\times \text{height}$.
$15=\text{½}\times 6\times h$
$15=3h$
$h=5$ cm.

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62. A square has an area of 49 cm². What is its perimeter?
Answer: 28 cm
Explanation: Area = side² = 49 → side = √49 = 7 cm.
Perimeter = $4\times \text{side}=4\times 7=28$ cm.

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63. A rectangle has a perimeter of 24 cm and a length of 8 cm. What is its width?
Answer: 4 cm
Explanation: Perimeter = $2(\text{length}+\text{width})$.
$24=2(8+w)$
$12=8+w$
$w=4$ cm.

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64. A parallelogram has a base of 10 cm and a height of 3 cm. What is its area?
Answer: 30 cm²
Explanation: Area of parallelogram = base × height = $10\times 3=30$ cm².

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65. A trapezium has parallel sides of 6 cm and 10 cm, and a height of 4 cm. What is its area?
Answer: 32 cm²
Explanation: Area of trapezium = $\text{½}\times (\text{sum of parallel sides})\times \text{height}$
= $\text{½}\times (6+10)\times 4$
= $\text{½}\times 16\times 4$
= $8\times 4=32$ cm².

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66. What is the name of a triangle with all angles less than 90°?
Answer: Acute-angled triangle

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67. What is the name of a triangle with one angle greater than 90°?
Answer: Obtuse-angled triangle

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68. True or False: A rhombus has diagonals that bisect each other at right angles.
Answer: True
Explanation: This is a key property of a rhombus (and also of a square, which is a special rhombus).

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69. How many vertices does a pentagon have?
Answer: 5

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70. What is the name of a quadrilateral where diagonals are equal and bisect each other at right angles?
Answer: Square
Explanation: Only a square has all these properties among quadrilaterals.

• Diagonals equal ✓ (rectangle also has this, but doesn't bisect at right angles)

• Diagonals bisect each other ✓ (all parallelograms have this)

• Diagonals bisect at right angles ✓ (rhombus has this, but diagonals aren't equal)
  Only a square satisfies all three.

Section D: Angle Rules (First 15 of 30 questions)

71. Two angles on a straight line are 125° and *x*. Find *x*.
Answer: 55°
Explanation: Angles on a straight line sum to 180°.
$125\mathrm{°}+x=180\mathrm{°}$
$x=180\mathrm{°}-125\mathrm{°}=55\mathrm{°}$

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72. Three angles around a point are 100°, 150° and *y*. Find *y*.
Answer: 110°
Explanation: Angles around a point sum to 360°.
$100\mathrm{°}+150\mathrm{°}+y=360\mathrm{°}$
$250\mathrm{°}+y=360\mathrm{°}$
$y=360\mathrm{°}-250\mathrm{°}=110\mathrm{°}$

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73. In the diagram, two straight lines cross. One angle is 40°. What are the sizes of the other three angles?
Answer: 40°, 140°, 140°
Explanation: Vertically opposite angles are equal → one opposite angle is 40°. Adjacent angles on a straight line are supplementary → $180\mathrm{°}-40\mathrm{°}=140\mathrm{°}$. So the four angles are: 40°, 140°, 40°, 140°.

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74. In an isosceles triangle, the vertex angle is 50°. What is the size of each base angle?
Answer: 65°
Explanation: Sum of angles = 180°. Let each base angle = $b$.
$50\mathrm{°}+b+b=180\mathrm{°}$
$50\mathrm{°}+2b=180\mathrm{°}$
$2b=130\mathrm{°}$
$b=65\mathrm{°}$

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75. A quadrilateral has angles of 80°, 95°, and 110°. What is the fourth angle?
Answer: 75°
Explanation: Sum of angles in quadrilateral = 360°.
$80\mathrm{°}+95\mathrm{°}+110\mathrm{°}+x=360\mathrm{°}$
$285\mathrm{°}+x=360\mathrm{°}$
$x=360\mathrm{°}-285\mathrm{°}=75\mathrm{°}$

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76. Find the size of angle *a* in a triangle with angles 60° and 70°.
Answer: 50°
Explanation: Triangle sum = 180°.
$a=180\mathrm{°}-(60\mathrm{°}+70\mathrm{°})=180\mathrm{°}-130\mathrm{°}=50\mathrm{°}$

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77. Angles *p* and *q* are vertically opposite. If *p* is 110°, what is *q*?
Answer: 110°
Explanation: Vertically opposite angles are equal.

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78. In a right-angled triangle, one acute angle is twice the other. What are the sizes of the angles?
Answer: 30°, 60°, 90°
Explanation: Let smaller acute angle = $x$, then larger acute = $2x$.
Right angle = 90°.
$x+2x+90\mathrm{°}=180\mathrm{°}$
$3x=90\mathrm{°}$
$x=30\mathrm{°}$, so angles are 30°, 60°, 90°.

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79. A straight line has three angles on it, *x*, 2*x*, and 90°. Find the value of *x*.
Answer: 30°
Explanation: Angles on a straight line sum to 180°.
$x+2x+90\mathrm{°}=180\mathrm{°}$
$3x=90\mathrm{°}$
$x=30\mathrm{°}$

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80. In a parallelogram, one angle is 110°. What are the sizes of the other three angles?
Answer: 110°, 70°, 70°
Explanation: Opposite angles equal → another 110°. Adjacent angles supplementary → $180\mathrm{°}-110\mathrm{°}=70\mathrm{°}$. So angles: 110°, 70°, 110°, 70°.

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81. The angles of a triangle are in the ratio 2:3:4. Find the size of the largest angle.
Answer: 80°
Explanation: Let angles = $2x,3x,4x$.
Sum: $2x+3x+4x=180\mathrm{°}$
$9x=180\mathrm{°}$
$x=20\mathrm{°}$
Largest = $4x=4\times 20\mathrm{°}=80\mathrm{°}$

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82. In a rhombus, one angle is 65°. What is the size of the angle adjacent to it?
Answer: 115°
Explanation: Adjacent angles in a rhombus (parallelogram property) are supplementary.
$180\mathrm{°}-65\mathrm{°}=115\mathrm{°}$

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83. Angles *a* and *b* are on a straight line. Angle *a* is 3 times angle *b*. Find angle *b*.
Answer: 45°
Explanation: $a+b=180\mathrm{°}$ and $a=3b$
$3b+b=180\mathrm{°}$
$4b=180\mathrm{°}$
$b=45\mathrm{°}$

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84. Four angles around a point are 2*x*, 3*x*, 4*x*, and 5*x*. Find the value of *x*.
Answer: $x={\frac{180}{7}}^{\circ }$ or ≈ 25.71°
Explanation: Sum around a point = 360°.
$2x+3x+4x+5x=360\mathrm{°}$
$14x=360\mathrm{°}$
$x=\frac{360\mathrm{°}}{14}=\frac{180\mathrm{°}}{7}$

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85. In a pentagon, four of the angles are 100°, 110°, 120°, and 130°. What is the fifth angle?
Answer: 80°
Explanation: Sum of interior angles of pentagon = $(5-2)\times 180\mathrm{°}=540\mathrm{°}$.
Given angles sum: $100+110+120+130=460\mathrm{°}$.
Fifth angle = $540\mathrm{°}-460\mathrm{°}=80\mathrm{°}$

Section D: Angle Rules (Questions 86–100)

86. An isosceles triangle has a base angle of 40°. What is the vertex angle?
Answer: 100°
Explanation: Base angles in an isosceles triangle are equal → both base angles = 40°.
Vertex angle = $180\mathrm{°}-(40\mathrm{°}+40\mathrm{°})=180\mathrm{°}-80\mathrm{°}=100\mathrm{°}$.

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87. Two angles in a triangle are 35° and 45°. What is the third angle?
Answer: 100°
Explanation: $180\mathrm{°}-(35\mathrm{°}+45\mathrm{°})=180\mathrm{°}-80\mathrm{°}=100\mathrm{°}$.

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88. In a quadrilateral, three angles are equal and the fourth is 90°. What is the size of each of the equal angles?
Answer: 90°
Explanation: Let each equal angle = $x$.
Sum: $3x+90\mathrm{°}=360\mathrm{°}$
$3x=270\mathrm{°}$
$x=90\mathrm{°}$.

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89. Angles *c* and *d* are complementary. Angle *c* is 15° more than angle *d*. Find angle *c*.
Answer: 52.5°
Explanation: Complementary → $c+d=90\mathrm{°}$
Given: $c=d+15\mathrm{°}$
Substitute: $(d+15\mathrm{°})+d=90\mathrm{°}$
$2d+15\mathrm{°}=90\mathrm{°}$
$2d=75\mathrm{°}$
$d=37.5\mathrm{°}$
$c=37.5\mathrm{°}+15\mathrm{°}=52.5\mathrm{°}$.

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90. The exterior angle of a triangle is 120° and one of its interior opposite angles is 50°. What is the other interior opposite angle?
Answer: 70°
Explanation: Exterior angle = sum of the two opposite interior angles.
$120\mathrm{°}=50\mathrm{°}+x$
$x=120\mathrm{°}-50\mathrm{°}=70\mathrm{°}$.

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91. In a trapezium with one pair of parallel sides, one angle is 85°. What is the angle on the same side of the transversal?
Answer: 95°
Explanation: In a trapezium, angles on the same side of the transversal (between a parallel line and the non-parallel side) are supplementary if the lines are parallel.
$180\mathrm{°}-85\mathrm{°}=95\mathrm{°}$.

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92. A triangle has angles of (x+10)°, (2x-20)°, and 60°. Find the value of x.
Answer: $x=\frac{130}{3}$ ≈ 43.33°
Explanation: Sum: $(x+10)+(2x-20)+60=180$
$3x+50=180$
$3x=130$
$x=\frac{130}{3}$.

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93. A quadrilateral has angles of x, 2x, 3x, and 4x. Find the value of x.
Answer: 36°
Explanation: Sum: $x+2x+3x+4x=10x=360\mathrm{°}$
$x=36\mathrm{°}$.

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94. In a regular hexagon, what is the size of each interior angle?
Answer: 120°
Explanation: Sum of interior angles = $(6-2)\times 180\mathrm{°}=720\mathrm{°}$.
Each interior angle = $720\mathrm{°}÷6=120\mathrm{°}$.

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95. The angles of a triangle are (2y)°, (3y)°, and (4y)°. Find the value of y.
Answer: 20°
Explanation: Sum: $2y+3y+4y=9y=180\mathrm{°}$
$y=20\mathrm{°}$.

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96. On a straight line, angles are 2a, 3a, and 4a. Find the smallest angle.
Answer: 40°
Explanation: Sum: $2a+3a+4a=9a=180\mathrm{°}$
$a=20\mathrm{°}$.
Smallest angle = $2a=2\times 20\mathrm{°}=40\mathrm{°}$.

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97. Around a point, angles are a, 2a, 3a, 4a, and 5a. Find the largest angle.
Answer: 120°
Explanation: Sum: $a+2a+3a+4a+5a=15a=360\mathrm{°}$
$a=24\mathrm{°}$.
Largest = $5a=5\times 24\mathrm{°}=120\mathrm{°}$.

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98. In an isosceles triangle, the vertex angle is 4 times a base angle. Find the vertex angle.
Answer: 120°
Explanation: Let base angle = $b$, vertex = $4b$.
Sum: $b+b+4b=180\mathrm{°}$
$6b=180\mathrm{°}$
$b=30\mathrm{°}$.
Vertex = $4\times 30\mathrm{°}=120\mathrm{°}$.

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99. Two vertically opposite angles are (3x+10)° and (5x-20)°. Find the value of x.
Answer: 15
Explanation: Vertically opposite angles are equal:
$3x+10=5x-20$
$10+20=5x-3x$
$30=2x$
$x=15$.

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100. In a right-angled isosceles triangle, what are the sizes of the two acute angles?
Answer: 45° each
Explanation: Right angle = 90°, remaining two acute angles are equal and sum to 90°. Each = $90\mathrm{°}÷2=45\mathrm{°}$.