Lines & Angles - ACT Math

1. What is the definition of complementary angles?

   Two angles summing to $90^{\text{°}}$. Together they form a right angle.

2. When a transversal cuts parallel lines, what is true about alternate interior angles?

   Alternate interior angles are congruent. Interior angles on opposite sides are equal.

3. What is the property of angles on a straight line?

   They sum to $180^{\text{°}}$. Linear pair angles are supplementary.

4. Identify the angle that forms a linear pair with $130^{\text{°}}$.

   $50^{\text{°}}$. Linear pair angles are supplementary, so $180° - 130° = 50°$.

5. How many degrees are in each angle of an equilateral triangle?

   $60^{\text{°}}$. All angles in an equilateral triangle are equal.

6. Find the measure of an angle in a regular pentagon.

   $108^{\text{°}}$. Using formula: $\frac{(5-2)\times180°}{5} = 108°$.

7. What is the measure of each interior angle in a regular heptagon?

   $128.57^\circ$. Using formula: $\frac{(7-2)\times180^\circ}{7} ≈ 128.57^\circ$.

8. What is the sum of the interior angles of a pentagon ($n=5$)?

   $540^\circ$. Using $(n-2) \cdot 180^\circ$ with $n=5$.

9. What is the definition of supplementary angles?

   Two angles summing to $180^{\text{°}}$. Together they form a straight line.

10. What is the formula for the sum of interior angles in an $n$-sided polygon?

    $(n-2) \times 180^{\text{°}}$. Derived from dividing a polygon into triangles.

11. When a transversal cuts parallel lines, what is true about same-side interior angles?

    Same-side interior angles are supplementary. Interior angles on same side add to $180^\circ$.

12. Define corresponding angles in parallel lines.

    Angles in matching corners equal to each other. They're in the same relative position at each intersection.

13. What is the definition of supplementary angles in degrees?

    Two angles whose measures sum to $180^\circ$. Together they form a straight angle.

14. If two angles form a linear pair, what equation relates their measures?

    $m\angle 1 + m\angle 2 = 180^\circ$. Linear pairs always sum to a straight angle.

15. Identify the vertical angle to $60^{\text{°}}$ in an intersecting lines scenario.

    $60^{\text{°}}$. Vertical angles are always equal.

16. What is the definition of complementary angles in degrees?

    Two angles whose measures sum to $90^\circ$. Together they form a right angle.

17. What is the measure of a straight angle?

    $180^\circ$. Half a complete rotation.

18. Identify the type of angles if one is $110^{\text{°}}$ and another is $70^{\text{°}}$.

    Supplementary angles. Since $110° + 70° = 180°$.

19. If two angles are supplementary, what equation relates their measures?

    $m\angle 1 + m\angle 2 = 180^\circ$. Definition of supplementary angles.

20. What is the measure of a full rotation (one complete turn)?

    $360^\circ$. Complete circle rotation.

21. What is the definition of adjacent angles?

    Two angles with a common side and vertex. They share a vertex and one side between them.

22. Which condition proves two lines are parallel using corresponding angles?

    If corresponding angles are congruent, the lines are parallel. Converse of corresponding angles theorem.

23. In a triangle, an exterior angle is $120^\circ$ and one remote interior angle is $50^\circ$; what is the other?

    $70^\circ$. Exterior angle theorem: $120^\circ = 50^\circ + x$.

24. What is the measure of an angle that is supplementary to $125^{\text{°}}$?

    $55^{\text{°}}$. Supplementary angles sum to $180°$, so $180° - 125° = 55°$.

25. What is each interior angle of a regular $n$-gon?

    $\frac{(n-2)\cdot 180^\circ}{n}$. Total interior sum divided by number of sides.

26. What is the sum of the interior angles of a quadrilateral?

    $360^{\text{°}}$. Sum of interior angles increases by $180°$ for each additional side.

27. Find the measure of an angle in an equilateral triangle.

    $60^{\text{°}}$. All angles are equal in an equilateral triangle.

28. When a transversal cuts parallel lines, what is true about alternate exterior angles?

    Alternate exterior angles are congruent. Exterior angles on opposite sides are equal.

29. What is each interior angle of a regular hexagon $(n=6)$?

    $120^\circ$. Using $\frac{(n-2) \cdot 180^\circ}{n}$ with $n=6$.

30. Find the measure of the missing angle if two angles are $90^{\text{°}}$ and $30^{\text{°}}$ in a triangle.

    $60^{\text{°}}$. Triangle angles sum to $180°$, so $180° - 90° - 30° = 60°$.

31. State the formula to calculate the measure of each interior angle of a regular polygon.

    $\frac{(n-2)\times180}{n}$. Uses the sum of interior angles divided by number of sides.

32. If two angles are supplementary and one measures $145^\circ$, what is the other?

    $35^\circ$. Supplementary angles sum to $180^\circ$.

33. Which condition proves two lines are parallel using alternate interior angles?

    If alternate interior angles are congruent, the lines are parallel. Converse of alternate interior angles theorem.

34. What is the definition of parallel lines in a plane?

    Coplanar lines that never intersect. Same plane, same direction, never meet.

35. Define a linear pair of angles.

    Adjacent angles that form a straight line. They share a side and their sum is $180°$.

36. If two angles form a linear pair and one measures $112^\circ$, what is the other?

    $68^\circ$. Linear pairs sum to $180^\circ$.

37. What is the property of angles around a point?

    They sum to $360^{\text{°}}$. Full rotation around any point equals $360°$.

38. What is the measure of each angle in a regular decagon?

    $144^{\text{°}}$. Using formula: $\frac{(10-2)\times180°}{10} = 144°$.

39. What is the definition of vertical angles?

    Opposite angles formed by intersecting lines; they are congruent. They have equal measures when lines cross.

40. If a regular polygon has each exterior angle $24^\circ$, how many sides does it have?

    $15$. Using $n = \frac{360^\circ}{24^\circ}$.

41. Find the measure of each angle in a regular triangle.

    $60^{\text{°}}$. Regular triangle means equilateral triangle.

42. If corresponding angles measure $(3x+10)^\circ$ and $(5x-30)^\circ$, what is $x$?

    $20$. Set corresponding angles equal and solve.

43. What is the sum of the exterior angles of any polygon?

    $360^{\text{°}}$. This holds for any polygon regardless of number of sides.

44. Define an angle bisector.

    A ray dividing an angle into two equal parts. It creates two congruent angles.

45. What is the exterior angle theorem for a triangle?

    An exterior angle equals the sum of the two remote interior angles. Exterior equals sum of non-adjacent interior angles.

46. Find the measure of an angle in a regular nonagon.

    $140^{\text{°}}$. Using formula: $\frac{(9-2)\times180°}{9} = 140°$.

47. What is a transversal?

    A line that intersects two or more other lines. Cuts across multiple lines.

48. What is each interior angle of a regular hexagon ($n=6$)?

    $120^\circ$. Using $\frac{(n-2) \cdot 180^\circ}{n}$ with $n=6$.

49. What is the sum of the interior angles of a triangle?

    $180^{\text{°}}$. This is a fundamental property of all triangles.

50. What is the measure of an angle that is complementary to $65^{\text{°}}$?

    $25^{\text{°}}$. Complementary angles sum to $90°$, so $90° - 65° = 25°$.

51. What is the relationship between alternate interior angles?

    They are equal. When parallel lines are cut by a transversal.

52. What is the definition of adjacent angles?

    Two angles sharing a vertex and a side with nonoverlapping interiors. Next to each other but don't overlap inside.

53. What is the measure of a right angle?

    $90^\circ$. Quarter of a complete rotation.

54. Identify the angle type if two angles measure $45^{\text{°}}$ and $45^{\text{°}}$.

    Complementary angles. Since $45° + 45° = 90°$.

55. What is the measure of an angle in a regular dodecagon?

    $150^{\text{°}}$. Using formula: $\frac{(12-2)\times180^\circ}{12} = 150^\circ$.

56. What is the measure of an angle that is supplementary to $45^{\text{°}}$?

    $135^{\text{°}}$. Supplementary angles sum to $180°$, so $180° - 45° = 135°$.

57. What is the angle sum of a triangle in degrees?

    $180^\circ$. Fundamental triangle angle property.

58. If alternate interior angles measure $(2x+15)^\circ$ and $(5x-45)^\circ$, what is $x$?

    $20$. Set alternate interior angles equal and solve.

59. What is the measure of the exterior angle of a regular nonagon?

    $40^{\text{°}}$. Exterior angle = $\frac{360°}{9} = 40°$ for a nonagon.

60. If two angles are vertical and one measures $37^\circ$, what is the other?

    $37^\circ$. Vertical angles are always congruent.

61. What is the measure of each exterior angle of a regular octagon?

    $45^\circ$. Exterior angle = $\frac{360^\circ}{8} = 45^\circ$ for an octagon.

62. What is the measure of an angle that is complementary to $30^{\text{°}}$?

    $60^{\text{°}}$. Complementary angles sum to $90°$, so $90° - 30° = 60°$.

63. What is each exterior angle of a regular polygon with $n$ sides?

    $\frac{360^\circ}{n}$. Full rotation divided by number of sides.

64. What is the relationship between vertical angles formed by two intersecting lines?

    Vertical angles are congruent (equal in measure). Fundamental property of intersecting lines.

65. What is the sum of the interior angles of an $n$-gon?

    $(n-2)\cdot 180^\circ$. General polygon angle sum formula.

66. What is a linear pair of angles?

    Adjacent angles whose noncommon sides form a straight line ($180^\circ$). Adjacent angles that together form a line.

67. What is the property of vertically opposite angles?

    They are equal. Formed by intersecting lines, they're always equal.

68. If two lines are perpendicular, what is the measure of the angles formed?

    All four angles measure $90^\circ$. Perpendicular lines form right angles.

69. If two angles are complementary and one measures $29^\circ$, what is the other?

    $61^\circ$. Complementary angles sum to $90^\circ$.

70. If two angles are complementary, what equation relates their measures?

    $m\angle 1 + m\angle 2 = 90^\circ$. Definition of complementary angles.

71. What is the measure of each exterior angle of a regular decagon?

    $36^{\text{°}}$. Exterior angle = $\frac{360°}{10} = 36°$ for a decagon.

72. What is the measure of an exterior angle of a regular hexagon?

    $60^{\text{°}}$. Exterior angle = $\frac{360°}{6} = 60°$ for a hexagon.

73. When a transversal cuts parallel lines, what is true about corresponding angles?

    Corresponding angles are congruent. Same position angles are equal.

74. Find the missing angle if two angles are $70^{\text{°}}$ and $20^{\text{°}}$ in a triangle.

    $90^{\text{°}}$. Triangle angles sum to $180°$, so $180° - 70° - 20° = 90°$.

75. Which condition proves two lines are parallel using same-side interior angles?

    If same-side interior angles are supplementary, the lines are parallel. Converse of same-side interior theorem.

76. In a triangle, angles are $50^\circ$ and $65^\circ$; what is the third angle?

    $65^\circ$. Triangle angles sum to $180^\circ$.

77. Define alternate exterior angles in parallel lines.

    Non-adjacent and equal angles outside parallel lines. They're on opposite sides of the transversal.

78. What is each exterior angle of a regular octagon ($n=8$)?

    $45^\circ$. Using $\frac{360^\circ}{n}$ with $n=8$.

79. If same-side interior angles measure $(4x+10)^\circ$ and $(2x+50)^\circ$, what is $x$?

    $20$. Same-side interior angles sum to $180^\circ$.

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