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 = $90\mathrm{°}÷2=45\mathrm{°}$.