Solve Problems With Angle Relationships - 7th Grade Math

1. What is a linear pair of angles?

   Adjacent angles whose noncommon sides form a line. Their outer rays form a straight line ($180^\circ$).

2. What is the definition of supplementary angles?

   Two angles whose measures add to $180^\circ$. Their sum equals a straight angle.

3. Find $x$ if adjacent angles form $90^\circ$ and are labeled $x$ and $x + 14$.

   $38$. Set $x + (x + 14) = 90$ and solve.

4. What is the value of $x$ if a linear pair has measures $(2x + 15)^\circ$ and $105^\circ$?

   $x = 30$. Linear pair sums to $180$: $(2x + 15) + 105 = 180$.

5. Identify the missing angle: angles $112^\circ$ and $x^\circ$ are supplementary. What is $x$?

   $x = 68^\circ$. Since $112 + x = 180$, subtract 112 from both sides.

6. What equation relates vertical angles with measures $x^\circ$ and $y^\circ$?

   $x = y$. Vertical angles are always congruent.

7. What is the definition of adjacent angles?

   Two angles that share a vertex and a common side, with no overlap. They touch at one point and share one ray.

8. Find $x$: two complementary angles are $(x + 18)^\circ$ and $(2x - 6)^\circ$.

   $x = 26$. Set sum to $90$: $(x + 18) + (2x - 6) = 90$.

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

   Two angles whose measures add to $90^\circ$. The sum equals a right angle.

10. Find $x$: vertical angles are labeled $(5x - 20)^\circ$ and $(3x + 16)^\circ$.

    $x = 18$. Set vertical angles equal: $5x - 20 = 3x + 16$.

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

    Two angles whose measures add to $180^\circ$. The sum equals a straight angle.

12. What equation represents supplementary angles with measures $x^\circ$ and $y^\circ$?

    $x + y = 180$. Supplementary angles sum to $180^\circ$.

13. What equation represents two supplementary angles with measures $x^\circ$ and $y^\circ$?

    $x + y = 180$. Supplementary angles sum to $180^\circ$.

14. What is $x$ if intersecting lines create vertical angles $x^\circ$ and $x^\circ$, and an adjacent angle is $110^\circ$?

    $x=70$. Adjacent to vertical angles forms linear pair: $x + 110 = 180$.

15. What is the definition of complementary angles?

    Two angles whose measures add to $90^\circ$. Their sum equals a right angle.

16. What is the definition of adjacent angles in terms of sides and vertex?

    They share a vertex and a common side, with no overlap. Adjacent angles are side-by-side without overlapping interior.

17. What is the definition of supplementary angles in terms of their measures?

    Two angles whose measures sum to $180^\circ$. The sum of their measures equals a straight angle.

18. Find $x$ if supplementary angles are labeled $5x - 10$ and $3x + 30$ (degrees).

    $20$. Set $(5x - 10) + (3x + 30) = 180$ and solve.

19. What is the definition of vertical angles?

    Opposite angles formed by intersecting lines; they are congruent. Formed when two lines cross.

20. Find $x$: adjacent angles on a straight line are $x^\circ$ and $(x + 40)^\circ$.

    $x = 70^\circ$. Adjacent angles on a line sum to $180$: $x + (x + 40) = 180$.

21. What is the relationship between angles that form a linear pair?

    They are supplementary, so their sum is $180^\circ$. Adjacent angles on a straight line form a linear pair.

22. What is the relationship between vertical angles in a figure?

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

23. Find $x$ if supplementary angles are labeled $2x + 15$ and $x + 30$ (degrees).

    $45$. Set $(2x + 15) + (x + 30) = 180$ and solve.

24. Find $x$ if vertical angles are labeled $3x + 10$ and $2x + 40$ (degrees).

    $30$. Set $3x + 10 = 2x + 40$ and solve for $x$.

25. What is the definition of vertical angles formed by intersecting lines?

    Opposite angles formed by intersecting lines. They share a vertex but not sides when lines cross.

26. What is $x$ if two complementary angles have measures $x^\circ$ and $35^\circ$?

    $x=55$. Since $x + 35 = 90$, subtract 35 from both sides.

27. Find $x$ if a linear pair is labeled $4x$ and $3x$ (degrees).

    $\frac{180}{7}$. Linear pair sums to $180^\circ$, so $4x + 3x = 180$.

28. What is a linear pair of angles?

    Adjacent angles whose noncommon sides form a line; sum is $180^\circ$. Adjacent angles on a straight line are supplementary.

29. What is always true about the measures of vertical angles?

    Vertical angles are congruent (equal in measure). This property makes them useful for finding unknown angles.

30. Find $x$ if complementary angles are labeled $4x$ and $2x + 18$ (degrees).

    $12$. Set $4x + (2x + 18) = 90$ and solve.

31. What is $x$ if a linear pair has measures $(2x+10)^\circ$ and $90^\circ$?

    $x=40$. Since $(2x + 10) + 90 = 180$, solve: $2x = 80$, so $x = 40$.

32. Identify the missing angle: angles $35^\circ$ and $x^\circ$ are complementary. What is $x$?

    $x = 55^\circ$. Since $35 + x = 90$, subtract 35 from both sides.

33. What is the missing angle if one angle in a complementary pair is $62^\circ$?

    $28^\circ$. Subtract from $90^\circ$: $90 - 62 = 28$.

34. What is $x$ if supplementary angles measure $(x+15)^\circ$ and $(2x+45)^\circ$?

    $x=40$. Set $(x + 15) + (2x + 45) = 180$, then $3x + 60 = 180$.

35. What equation represents two complementary angles with measures $x^\circ$ and $y^\circ$?

    $x + y = 90$. Complementary angles sum to $90^\circ$.

36. Find $x$: $\angle A$ and $\angle B$ are vertical; $m\angle A = (x + 7)^\circ$, $m\angle B = (2x - 13)^\circ$.

    $x = 20$. Set vertical angles equal: $x + 7 = 2x - 13$.

37. Find $x$: two angles form a linear pair; measures are $(x - 9)^\circ$ and $(3x + 21)^\circ$.

    $x = 42$. Linear pair sums to $180$: $(x - 9) + (3x + 21) = 180$.

38. What is the value of $x$ if $x$ and $35^\circ$ are complementary?

    $55$. Since $x + 35 = 90$, subtract $35$ from both sides.

39. What is $x$ if a linear pair measures $(4x)^\circ$ and $(2x+30)^\circ$?

    $x=25$. Set $4x + (2x + 30) = 180$, then $6x + 30 = 180$.

40. Find $x$ if vertical angles are labeled $7x$ and $140$ (degrees).

    $20$. Vertical angles are equal, so $7x = 140$.

41. Find $x$ if a straight angle is split into $3$ equal angles, each labeled $x^\circ$.

    $60$. Three equal angles sum to $180^\circ$, so $3x = 180$.

42. What is the missing angle if one angle in a supplementary pair is $147^\circ$?

    $33^\circ$. Subtract from $180^\circ$: $180 - 147 = 33$.

43. Find $x$ if a linear pair is labeled $x + 25$ and $2x + 5$ (degrees).

    $50$. Set $(x + 25) + (2x + 5) = 180$ and solve.

44. What is the definition of adjacent angles?

    Two angles that share a common vertex and a common side. They are next to each other with a shared ray between them.

45. What is always true about the measures of angles in a linear pair?

    They are supplementary: sum is $180^\circ$. Linear pairs always form a straight angle together.

46. What is $x$ if vertical angles measure $(5x-10)^\circ$ and $(3x+30)^\circ$?

    $x=20$. Set $5x - 10 = 3x + 30$, then $2x = 40$, so $x = 20$.

47. What is $x$ if a straight angle is split into three adjacent angles: $x^\circ$, $(x+20)^\circ$, and $40^\circ$?

    $x=60$. Set $x + (x + 20) + 40 = 180$, then $2x + 60 = 180$.

48. What is the value of $x$ if vertical angles have measures $(3x + 10)^\circ$ and $85^\circ$?

    $x = 25$. Set vertical angles equal: $3x + 10 = 85$, then solve for $x$.

49. What is the definition of complementary angles in terms of their measures?

    Two angles whose measures sum to $90^\circ$. The sum of their measures equals a right angle.

50. What is the angle-sum equation for two complementary angles with measures $a$ and $b$?

    $a+b=90$. Complementary angles sum to $90°$.

51. What is $x$ if vertical angles have measures $(3x+5)^\circ$ and $50^\circ$?

    $x=15$. Set $3x + 5 = 50$, then solve: $3x = 45$, so $x = 15$.

52. What is the value of $x$ if $x$ and $128^\circ$ are supplementary?

    $52$. Since $x + 128 = 180$, subtract $128$ from both sides.

53. What is the angle-sum equation for two supplementary angles with measures $a$ and $b$?

    $a+b=180$. Supplementary angles sum to $180°$.

54. What is $x$ if adjacent angles add to a right angle and measure $x^\circ$ and $(x+20)^\circ$?

    $x=35$. Set $x + (x + 20) = 90$, then $2x + 20 = 90$, so $x = 35$.

55. Identify the relationship: two adjacent angles form a straight line.

    They are supplementary (a linear pair), so the sum is $180^\circ$. Adjacent angles on a line form $180^\circ$.

56. What equation represents complementary angles with measures $x^\circ$ and $y^\circ$?

    $x + y = 90$. Complementary angles sum to $90^\circ$.

57. What is $x$ if two supplementary angles have measures $x^\circ$ and $128^\circ$?

    $x=52$. Since $x + 128 = 180$, subtract 128 from both sides.

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