Establish Angle Facts Using Arguments - 8th Grade Math

1. If $\ell_1\parallel \ell_2$ and a same-side interior angle is $47^\circ$, what is the other same-side interior angle?

   $133^\circ$. Same-side interior angles are supplementary: $180^\circ - 47^\circ = 133^\circ$.

2. If $\triangle ABC\sim\triangle DEF$ and $m\angle A=42^\circ$, what is $m\angle D$?

   $42^\circ$. Similar triangles have corresponding angles congruent.

3. What is the sum of the interior angles of any triangle?

   $180^\circ$. All triangles have interior angles that sum to exactly $180^\circ$.

4. What is the sum of the interior angles of any triangle?

   $180^\circ$. The three angles of any triangle always add up to a straight angle.

5. What is the angle-angle similarity criterion for triangles?

   If two angles are congruent, the triangles are similar. Two pairs of equal angles guarantee the third pair is also equal.

6. Find the third angle of a triangle if two angles are $x^\circ$ and $x^\circ$ and the third is $40^\circ$; find $x$.

   $x=70^\circ$. $x + x + 40 = 180°$, so $2x = 140°$.

7. Find the adjacent interior angle if an exterior angle of a triangle is $118^\circ$.

   $62^\circ$. $180^\circ - 118^\circ = 62^\circ$ since they form a linear pair.

8. Identify the relationship: vertical angles have what equality relationship?

   Vertical angles are congruent. Opposite angles formed by intersecting lines are equal.

9. What is the alternate interior angles theorem for parallel lines cut by a transversal?

   Alternate interior angles are congruent. Interior angles on opposite sides of the transversal are equal.

10. Two lines are parallel. If a same-side interior angle is $112^\circ$, what is the other same-side interior angle?

    $68^\circ$. $180^\circ - 112^\circ = 68^\circ$ since they're supplementary.

11. Find the exterior angle measure if the two remote interior angles are $35^\circ$ and $75^\circ$.

    $110^\circ$. Exterior angle equals the sum of remote interior angles.

12. Find the remote interior angle if an exterior angle is $130^\circ$ and the other remote interior angle is $55^\circ$.

    $75^\circ$. $130^\circ - 55^\circ = 75^\circ$ using the exterior angle theorem.

13. What is the relationship between an exterior angle of a triangle and its two remote interior angles?

    Exterior angle $=$ sum of the two remote interior angles. The exterior angle equals the sum of the two non-adjacent interior angles.

14. Find $x$ if alternate interior angles are $5x-15$ and $3x+25$ degrees for parallel lines.

    $x=20$. Alternate interior angles are equal: $5x - 15 = 3x + 25$.

15. What is the Angle-Angle (AA) similarity criterion for triangles?

    If two angles are congruent, the triangles are similar. Two pairs of congruent angles guarantee similarity.

16. If two triangles have angle measures $30^\circ$ and $70^\circ$ matching, are the triangles similar by AA?

    Yes, similar by AA. Two angle pairs match, so the third angles must also match.

17. Find the missing angle in Triangle $2$ if Triangle $1$ is similar by $AA$ with angles $35^\circ,65^\circ$ and Triangle $2$ has $35^\circ$ and $x$.

    $x=65^\circ$. Similar triangles have all corresponding angles congruent.

18. Find the third angle of a triangle if two angles are $65^\circ$ and $45^\circ$.

    $70^\circ$. $180^\circ - 65^\circ - 45^\circ = 70^\circ$ using the triangle angle sum.

19. What is the measure of each interior angle in an equilateral triangle?

    Each interior angle is $60^\circ$. All three equal angles must sum to $180°$, so each is $180° ÷ 3$.

20. Find $x$ if an exterior angle is $140^\circ$ and its adjacent interior angle is $x^\circ$.

    $x=40^\circ$. $140° + x = 180°$ since they're supplementary.

21. If $\ell_1\parallel \ell_2$ and a corresponding angle is $118^\circ$, what is its corresponding angle measure?

    $118^\circ$. Corresponding angles are congruent when lines are parallel.

22. When two parallel lines are cut by a transversal, what is true about same-side (consecutive) interior angles?

    They are supplementary: sum is $180^\circ$. They're on the same side of the transversal between parallels.

23. Find $x$ if an exterior angle of a triangle is $110^\circ$ and one remote interior angle is $45^\circ$.

    $x=65^\circ$. $110° = 45° + x$, so $x = 110° - 45°$.

24. What is the exterior angle theorem for a triangle (in words, as an equation)?

    $m\angle \text{ext}=m\angle \text{remote }1+m\angle \text{remote }2$. An exterior angle equals the sum of its two non-adjacent interior angles.

25. When two parallel lines are cut by a transversal, which angle pair is always congruent: corresponding angles or adjacent angles?

    Corresponding angles are congruent. They occupy the same position relative to the transversal.

26. What is the exterior angle of a triangle equal to in terms of the two remote interior angles?

    Exterior angle $=$ sum of the two remote interior angles. Forms a straight line with the adjacent interior angle.

27. Find the exterior angle of a triangle if the two remote interior angles are $50^\circ$ and $60^\circ$.

    $110^\circ$. $50^\circ + 60^\circ = 110^\circ$ using the exterior angle theorem.

28. Find the third angle: a triangle has angles $40^\circ$ and $65^\circ$. What is the third angle?

    $75^\circ$. Use $180^\circ - 40^\circ - 65^\circ = 75^\circ$.

29. Identify whether triangles are similar by AA if one has angles $40^\circ$ and $70^\circ$ and the other has $40^\circ$ and $70^\circ$.

    Yes, they are similar by $AA$. Two pairs of equal angles satisfy the AA criterion.

30. What is the measure of an exterior angle if the adjacent interior angle is $128^\circ$?

    $52^\circ$. Linear pair: $180^\circ - 128^\circ = 52^\circ$.

31. What is the relationship between an exterior angle and its adjacent interior angle in a triangle?

    They are supplementary: sum $=180^\circ$. They form a linear pair on a straight line.

32. When two parallel lines are cut by a transversal, which angle pair is always congruent: alternate interior angles or same-side interior angles?

    Alternate interior angles are congruent. They form a Z-pattern and are equal when lines are parallel.

33. What is the same-side (consecutive) interior angles theorem for parallel lines cut by a transversal?

    Same-side interior angles are supplementary (sum $180^\circ$). Interior angles on the same side of the transversal sum to $180^\circ$.

34. Two lines are parallel. If a corresponding angle is $72^\circ$, what is the measure of its corresponding partner?

    $72^\circ$. Corresponding angles are congruent when lines are parallel.

35. What is always true about vertical angles formed by two intersecting lines?

    Vertical angles are congruent. Opposite angles formed by intersecting lines are always equal.

36. What is the measure of each exterior angle of an equilateral triangle?

    Each exterior angle is $120^\circ$. Each supplements its adjacent $60°$ interior angle.

37. Find $x$ if the interior angles of a triangle are $50^\circ$, $60^\circ$, and $x^\circ$.

    $x=70^\circ$. $50° + 60° + x = 180°$, so $x = 180° - 110°$.

38. Identify whether triangles are similar by AA: Triangle $1$ has angles $40^\circ,70^\circ$; Triangle $2$ has $40^\circ,70^\circ$.

    Yes, similar by $AA$. Two pairs of congruent angles satisfy the AA criterion.

39. Find the measure of each angle in an equilateral triangle.

    Each angle is $60^\circ$. $180^\circ \div 3 = 60^\circ$ since all angles are equal.

40. Find $x$ if corresponding angles formed by a transversal are labeled $3x+10$ and $7x-30$ degrees.

    $x=10$. Corresponding angles are equal: $3x + 10 = 7x - 30$.

41. What is the AA similarity criterion for triangles?

    If two angles are congruent, the triangles are similar. Two pairs of congruent angles guarantee triangle similarity.

42. Identify the angle pair that is congruent when parallel lines are cut by a transversal and angles are in matching corners.

    Corresponding angles are congruent. These angles are in the same position relative to their vertices.

43. What conclusion about a triangle's angles is supported by drawing a line parallel to one side through the opposite vertex?

    The interior angles sum to $180^\circ$. Creates alternate interior angles that prove the sum is $180^\circ$.

44. What is the alternate exterior angles theorem for parallel lines cut by a transversal?

    Alternate exterior angles are congruent. Exterior angles on opposite sides of the transversal are equal.

45. Find the exterior angle: the two remote interior angles are $35^\circ$ and $80^\circ$. What is $m\angle \text{ext}$?

    $115^\circ$. Apply exterior angle theorem: $35^\circ + 80^\circ = 115^\circ$.

46. What is the corresponding angles postulate for parallel lines cut by a transversal?

    Corresponding angles are congruent. Angles in matching positions have equal measures.

47. If an exterior angle of a triangle is $140^\circ$ and one remote interior angle is $65^\circ$, what is the other remote interior angle?

    $75^\circ$. Use exterior angle theorem: $140^\circ - 65^\circ = 75^\circ$.

48. Identify the relationship: an exterior angle and its adjacent interior angle form what type of angle pair?

    Linear pair (supplementary, sum $180^\circ$). They share a vertex and form a straight line.

49. Two lines are parallel. If an alternate interior angle is $105^\circ$, what is its alternate interior partner?

    $105^\circ$. Alternate interior angles are congruent when lines are parallel.

50. What does it mean for two triangles to be similar in terms of angles and side lengths?

    Corresponding angles congruent; sides proportional. Similar triangles have equal angles and proportional side lengths.

51. Identify the angle pair that is congruent when parallel lines are cut by a transversal and angles are outside on opposite sides.

    Alternate exterior angles are congruent. These angles are outside the parallels on opposite sides of the transversal.

52. What is always true about a linear pair of angles?

    They are supplementary: sum is $180^\circ$. Adjacent angles that form a straight line.

53. If $\ell_1\parallel \ell_2$ and an alternate interior angle is $63^\circ$, what is the other alternate interior angle?

    $63^\circ$. Alternate interior angles are congruent when lines are parallel.

54. What is the relationship of vertical angles formed by intersecting lines?

    Vertical angles are congruent. Opposite angles formed by two intersecting lines are always equal.

55. Identify the relationship between an exterior angle and its adjacent interior angle in a triangle.

    They are supplementary: sum is $180^\circ$. They form a straight angle on the same line.

56. What is the relationship of same-side interior angles when two parallel lines are cut by a transversal?

    Same-side interior angles are supplementary: sum $=180^\circ$. These angles are between the parallels on the same side of the transversal.

57. Find $x$ if same-side interior angles are $2x+20$ and $3x+10$ degrees for parallel lines.

    $x=30$. Same-side angles sum to $180°$: $(2x + 20) + (3x + 10) = 180°$.

58. A triangle has angles $x^\circ$, $50^\circ$, and $60^\circ$. What is $x$?

    $70^\circ$. Use angle sum: $180^\circ - 50^\circ - 60^\circ = 70^\circ$.

59. Identify the angle pair that is congruent when parallel lines are cut by a transversal and angles are inside on opposite sides.

    Alternate interior angles are congruent. These angles are between the parallels on opposite sides of the transversal.

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