Theorems about Triangles - Geometry

1. Identify whether $P$ lies on the perpendicular bisector of $\overline{AB}$ if $PA = PB$.

   Yes; $PA=PB$ implies $P$ is on the perpendicular bisector. Converse of perpendicular bisector theorem applies.

2. Find $m\angle 2$ if $m\angle 1 = 115^\circ$ and $\angle 1$ and $\angle 2$ are alternate interior with parallel lines.

   $m\angle 2 = 115^\circ$. Alternate interior angles are congruent with parallel lines.

3. What converse theorem uses corresponding angles to prove two lines are parallel?

   If corresponding angles are congruent, then lines are parallel. Converse used to prove lines are parallel.

4. What is the converse of the perpendicular bisector theorem?

   If $PA=PB$, then $P$ lies on the perpendicular bisector of $\overline{AB}$. Equal distances imply the point is on the bisector.

5. What is the definition of a perpendicular bisector of segment $\overline{AB}$?

   A line perpendicular to $\overline{AB}$ through its midpoint. Passes through midpoint at $90^\circ$ angle.

6. Find $m\angle 2$ if $m\angle 1 = 101^\circ$ and $\angle 1$ and $\angle 2$ are same-side interior with parallel lines.

   $m\angle 2 = 79^\circ$. Same-side interior angles are supplementary: $180-101=79$.

7. Find $m\angle B$ if $m\angle A = 47^\circ$ and $\angle A$ and $\angle B$ form a linear pair.

   $m\angle B = 133^\circ$. Linear pair angles sum to $180^\circ$.

8. What is the definition of perpendicular lines in terms of angle measure?

   They intersect to form a $90^\circ$ angle. Form right angles at their intersection.

9. What does it mean for a point $P$ to be equidistant from points $A$ and $B$?

   $PA = PB$. Equal distance means same length from both points.

10. What is the measure of each angle if two vertical angles together sum to $180^\circ$?

    Each angle is $90^\circ$. Vertical angles sum to $180^\circ$ only when both are $90^\circ$.

11. Find $m\angle 2$ if $m\angle 1 = 62^\circ$ and $\angle 1$ and $\angle 2$ are corresponding angles with parallel lines.

    $m\angle 2 = 62^\circ$. Corresponding angles are congruent with parallel lines.

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

    Coplanar lines that do not intersect. Never meet no matter how far extended.

13. Identify whether lines are parallel if alternate interior angles measure $48^\circ$ and $48^\circ$.

    Yes; congruent alternate interior angles imply parallel lines. Equal alternate interior angles prove lines are parallel.

14. What is the definition of alternate interior angles for two lines cut by a transversal?

    Interior angles on opposite sides of the transversal. Between the lines but on opposite sides.

15. Find $x$ if $m\angle 1 = 3x+10$ and $m\angle 2 = 5x-30$, and $\angle 1$ and $\angle 2$ are vertical.

    $x = 20$. Set vertical angles equal: $3x+10=5x-30$, solve for $x$.

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

    They are supplementary (sum to $180^\circ$). Adjacent angles forming a straight line sum to $180^\circ$.

17. When two parallel lines are cut by a transversal, what is true about alternate interior angles?

    Alternate interior angles are congruent. Key theorem for parallel line proofs.

18. Identify the angle pair: two adjacent angles whose noncommon sides are opposite rays.

    Linear pair. Adjacent angles forming a straight line.

19. Identify the correct conclusion if $PA = PB$ for point $P$ and segment $\overline{AB}$.

    $P$ lies on the perpendicular bisector of $\overline{AB}$. Converse of perpendicular bisector theorem.

20. When two parallel lines are cut by a transversal, what is true about corresponding angles?

    Corresponding angles are congruent. Key theorem for parallel line proofs.

21. What property justifies that if $a=b$ then $b=a$ in an angle proof?

    Symmetric Property of Equality. Allows reversing the order of an equality.

22. Find $x$ if $m\angle 1 = 9x+9$ and $m\angle 2 = 3x+15$, and $\angle 1$ and $\angle 2$ are same-side interior with parallel lines.

    $x = 13$. Set same-side interior sum to $180$: $9x+9+3x+15=180$.

23. What is the definition of alternate exterior angles for two lines cut by a transversal?

    Exterior angles on opposite sides of the transversal. Outside the lines but on opposite sides.

24. What is the definition of corresponding angles for two lines cut by a transversal?

    Angles in matching positions at the two intersections. Same relative position at both intersection points.

25. What is the measure relationship for complementary angles $m\angle A$ and $m\angle B$?

    $m\angle A + m\angle B = 90^\circ$. Two angles whose measures add to $90^\circ$.

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

    They are supplementary (sum to $180^\circ$). Interior angles on same side are supplementary.

27. What converse theorem uses alternate interior angles to prove two lines are parallel?

    If alternate interior angles are congruent, then lines are parallel. Converse used to prove lines are parallel.

28. What is the definition of a midpoint of segment $\overline{AB}$?

    A point $M$ on $\overline{AB}$ with $AM = MB$. Divides segment into two equal parts.

29. Find $x$ if $m\angle 1 = 4x+8$ and $m\angle 2 = 6x-22$, and $\angle 1$ and $\angle 2$ are alternate interior with parallel lines.

    $x = 15$. Set alternate interior angles equal: $4x+8=6x-22$.

30. What is the definition of congruent angles in terms of measures?

    Their measures are equal. Congruent angles have identical degree measures.

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

    Opposite (nonadjacent) angles formed by intersecting lines. These angles share only a vertex and don't overlap.

32. What theorem states the relationship between vertical angles?

    Vertical angles are congruent. Proven using properties of supplementary angles.

33. What property justifies adding the same value to both sides of an equation in proofs?

    Addition Property of Equality. Maintains equality when adding same values.

34. Which postulate justifies splitting an angle into two parts to add measures?

    Angle Addition Postulate. States that $m\angle ABC = m\angle ABD + m\angle DBC$.

35. Which angle pair is always supplementary when two lines intersect?

    A linear pair. Adjacent angles with opposite rays always sum to $180^\circ$.

36. Find $PB$ if $P$ lies on the perpendicular bisector of $\overline{AB}$ and $PA = 9$.

    $PB = 9$. Perpendicular bisector theorem: equal distances to endpoints.

37. Find $m\angle 2$ if $m\angle 1 = 128^\circ$ and $\angle 1$ and $\angle 2$ are vertical.

    $m\angle 2 = 128^\circ$. Vertical angles are congruent by theorem.

38. Identify whether lines are parallel if same-side interior angles measure $120^\circ$ and $60^\circ$.

    Yes; supplementary same-side interior angles imply parallel lines. Same-side interior angles sum to $180^\circ$.

39. Identify the missing reason: vertical angles are congruent because they are both supplementary to the same angle.

    Congruent Supplements Theorem. If supplements of same angle, then angles are congruent.

40. What is the definition of same-side (consecutive) interior angles?

    Interior angles on the same side of the transversal. Between the lines on the same side.

41. What theorem connects a point on a perpendicular bisector to distances from endpoints?

    A point on it is equidistant from the segment endpoints. Every point on it has equal distances to endpoints.

42. When two parallel lines are cut by a transversal, what is true about alternate exterior angles?

    Alternate exterior angles are congruent. Key theorem for parallel line proofs.

43. What property justifies that if $a=b$ and $b=c$ then $a=c$ in an angle proof?

    Transitive Property of Equality. Allows chaining equalities together.

44. Identify the correct conclusion if $P$ is on the perpendicular bisector of $\overline{AB}$.

    $PA = PB$. Direct application of perpendicular bisector theorem.

45. What property justifies subtracting the same value from both sides of an equation in proofs?

    Subtraction Property of Equality. Maintains equality when subtracting same values.

46. What is the definition of a transversal in geometry?

    A line that intersects two or more lines at distinct points. Creates angle pairs at each intersection point.

47. Find $m\angle 1$ if $\angle 1$ and $\angle 2$ are vertical and $m\angle 2 = 3m\angle 1$.

    $m\angle 1 = 45^\circ$. Set $m\angle 1 + 3m\angle 1 = 180$, solve for $m\angle 1$.

48. What converse theorem uses same-side interior angles to prove two lines are parallel?

    If same-side interior angles are supplementary, then lines are parallel. Converse used to prove lines are parallel.

49. Identify whether lines are parallel if corresponding angles measure $70^\circ$ and $70^\circ$.

    Yes; congruent corresponding angles imply parallel lines. Equal corresponding angles prove lines are parallel.

50. What is the measure relationship for supplementary angles $m\angle A$ and $m\angle B$?

    $m\angle A + m\angle B = 180^\circ$. Two angles whose measures add to $180^\circ$.

51. Find $x$ if $PA = 2x+1$, $PB = 5x-11$, and $P$ lies on the perpendicular bisector of $\overline{AB}$.

    $x = 4$. Set equal distances: $2x+1=5x-11$, solve for $x$.

52. What is the locus of points equidistant from endpoints $A$ and $B$?

    The perpendicular bisector of $\overline{AB}$. Set of all points equidistant from two endpoints.

53. Find $m\angle 2$ if $\angle 1$ and $\angle 2$ are a linear pair and $m\angle 2 = 2m\angle 1$.

    $m\angle 2 = 120^\circ$. Set $m\angle 1 + 2m\angle 1 = 180$, then $m\angle 2 = 2(60) = 120$.

54. Find $x$ if $m\angle A = 2x+15$ and $m\angle B = 5x-9$, and $\angle A$ and $\angle B$ are supplementary.

    $x = 34$. Set supplementary angles equal: $2x+15+5x-9=180$.

55. Find $x$ if $m\angle 1 = 7x-4$ and $m\angle 2 = 3x+20$, and $\angle 1$ and $\angle 2$ are corresponding with parallel lines.

    $x = 6$. Set corresponding angles equal: $7x-4=3x+20$.

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