Theorems about Parallelograms - Geometry

1. What must be true about points $D$ and $E$ to conclude $DE$ is a midsegment in $\triangle ABC$?

   $D$ and $E$ are midpoints of two sides of $\triangle ABC$. Midsegments require endpoints at midpoints of two triangle sides.

2. Find $BC$ if $DE$ is a midsegment parallel to $BC$ and $DE=\frac{3}{2}$.

   $BC=3$. $BC = 2 \times DE = 2 \times \frac{3}{2} = 3$.

3. What is the third angle if a triangle has angles $90^\circ$ and $15^\circ$?

   $75^\circ$. $180° - 90° - 15° = 75°$ using angle sum theorem.

4. What is the measure of the vertex angle in an isosceles triangle with base angles $32^\circ$?

   $116^\circ$. Vertex angle: $180° - 32° - 32° = 116°$ using angle sum.

5. What is the required conclusion if $D$ and $E$ are midpoints of $AB$ and $AC$?

   $DE\parallel BC$ and $DE=\frac{1}{2}BC$. The Triangle Midsegment Theorem's required conclusions.

6. Find $AG$ if $G$ is the centroid on median $AM$ and $AM=12$.

   $AG=8$. $AG = \frac{2}{3} \times 12 = 8$ using the centroid ratio.

7. What is each base angle in an isosceles triangle if the vertex angle is $100^\circ$?

   $40^\circ$. $(180° - 100°) ÷ 2 = 40°$ for each base angle.

8. Identify the congruent sides if $\angle B\cong\angle C$ in $\triangle ABC$.

   $AB\cong AC$. Congruent angles indicate the opposite sides are congruent.

9. What ratio does the centroid create on each median, measured from vertex to centroid to midpoint?

   $2:1$. The centroid divides each median in a 2:1 ratio.

10. Identify the line parallel to $BC$ if $D$ is midpoint of $AB$ and $E$ is midpoint of $AC$.

    $DE\parallel BC$. The midsegment $DE$ is parallel to the third side $BC$.

11. What is a median of a triangle?

    A segment from a vertex to the midpoint of the opposite side. A median connects a vertex to the opposite side's midpoint.

12. Find $BC$ if $D,E$ are midpoints and the midsegment $DE=11$.

    $BC=22$. Third side is twice the midsegment: $11 × 2 = 22$.

13. Identify the congruent angles if $AB\cong AC$ in $\triangle ABC$.

    $\angle B\cong\angle C$. Congruent sides $AB \cong AC$ create congruent base angles.

14. What is the measure of the third angle if two angles are $95^\circ$ and $40^\circ$?

    $45^\circ$. Use $180° - 95° - 40° = 45°$ from the angle sum theorem.

15. What theorem states that the medians of a triangle meet at a single point?

    Concurrency of Medians Theorem. States that the three medians intersect at one point.

16. What is the point of concurrency of the three medians of a triangle called?

    Centroid. The intersection point of all three medians in a triangle.

17. If $G$ is the centroid on median $AM$, what is the relationship between $AG$ and $GM$?

    $AG=2GM$. The centroid creates a 2:1 division on each median.

18. Find $x$ if the angles of a triangle are $2x^\circ$, $x^\circ$, and $60^\circ$.

    $x=40$. $2x + x + 60 = 180$, so $3x = 120$ and $x = 40$.

19. Find $x$ if the angles of a triangle are $3x^\circ$, $x^\circ$, and $20^\circ$.

    $x=40$. $3x + x + 20 = 180$, so $4x = 160$ and $x = 40$.

20. What is the relationship between a midsegment $DE$ and the third side $BC$ in $\triangle ABC$?

    $DE\parallel BC$. Midsegments are always parallel to the third side.

21. Find $DE$ if $D,E$ are midpoints of two sides and $BC=18$.

    $DE=9$. Midsegment length is half of the third side: $18 ÷ 2 = 9$.

22. Find $GM$ if $G$ is the centroid on median $AM$ and $AM=15$.

    $GM=5$. $GM = \frac{1}{3} \times AM = \frac{1}{3} \times 15 = 5$.

23. What is the name of the segment joining the midpoints of two sides of a triangle?

    Midsegment. The segment connecting midpoints of two triangle sides.

24. What is the sum of the interior angle measures of any triangle?

    $m\angle A+m\angle B+m\angle C=180^\circ$. The Triangle Angle Sum Theorem states all triangles have this property.

25. What is the length relationship in $\triangle ABC$ if $D,E$ are midpoints of $AB,AC$?

    $DE=\frac{1}{2}BC$. Midsegments are exactly half the length of the third side.

26. Find $GM$ if $G$ is the centroid on median $AM$ and $AM=12$.

    $GM=4$. $GM = \frac{1}{3} \times 12 = 4$ using the centroid ratio.

27. What theorem states that if two angles of a triangle are congruent, then the opposite sides are congruent?

    Converse of the Isosceles Triangle Theorem. The reverse of the Isosceles Triangle Theorem.

28. Find $x$ if an isosceles triangle has base angles $x^\circ$ and vertex angle $50^\circ$.

    $x=65$. $x + x + 50 = 180$, so $2x = 130$ and $x = 65$.

29. Identify the base angles in $\triangle ABC$ if it is isosceles with base $BC$.

    $\angle B$ and $\angle C$. Base angles are opposite the congruent sides in isosceles triangles.

30. Identify the equal sides in $\triangle ABC$ if it is isosceles with base $BC$.

    $AB\cong AC$. In an isosceles triangle, the two non-base sides are equal.

31. What does the Triangle Midsegment Theorem conclude about the midsegment and the third side?

    Midsegment is parallel and half as long as the third side. The Triangle Midsegment Theorem's two main conclusions.

32. What is each base angle in an isosceles triangle if the vertex angle is $18^\circ$?

    $81^\circ$. $(180° - 18°) ÷ 2 = 81°$ for each base angle.

33. Find $GM$ if $G$ is the centroid on median $AM$ and $AG=10$.

    $GM=5$. $GM = \frac{1}{2} \times AG = \frac{1}{2} \times 10 = 5$.

34. What theorem states that if two sides of a triangle are congruent, then the base angles are congruent?

    Isosceles Triangle Theorem. This theorem connects congruent sides to congruent angles.

35. What is the third angle if a triangle has angles $x^\circ$, $x^\circ$, and $110^\circ$?

    $x=35$. $x + x + 110 = 180$, so $2x = 70$ and $x = 35$.

36. Find $AM$ if $G$ is the centroid on median $AM$ and $GM=7$.

    $AM=21$. $AM = 3 \times GM = 3 \times 7 = 21$ from centroid ratio.

37. What is the measure of the third angle if two angles are $50^\circ$ and $60^\circ$?

    $70^\circ$. Use $180° - 50° - 60° = 70°$ from the angle sum theorem.

38. What is the measure of each base angle in an isosceles triangle with vertex angle $40^\circ$?

    $70^\circ$. Base angles are equal: $(180° - 40°) ÷ 2 = 70°$ each.

39. What is the vertex angle in an isosceles triangle if each base angle is $48^\circ$?

    $84^\circ$. $180° - 48° - 48° = 84°$ for the vertex angle.

40. Which segment is a median if $M$ is the midpoint of $BC$ in $\triangle ABC$?

    $AM$. A median connects a vertex to the opposite side's midpoint.

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

    $60^\circ$. All angles in an equilateral triangle are equal.

42. What is the third angle if a triangle has angles $120^\circ$ and $25^\circ$?

    $35^\circ$. $180° - 120° - 25° = 35°$ using angle sum theorem.

43. Find $x$ if an isosceles triangle has vertex angle $2x^\circ$ and base angles $40^\circ$.

    $x=50$. $2x + 40 + 40 = 180$, so $2x = 100$ and $x = 50$.

44. Find $AG$ if $G$ is the centroid and $GM=6$ on median $AM$.

    $AG=12$. $AG = 2 \times GM = 2 \times 6 = 12$ from centroid ratio.

45. Find $AG$ if $G$ is the centroid on median $AM$ and $AM=15$.

    $AG=10$. $AG = \frac{2}{3} \times AM = \frac{2}{3} \times 15 = 10$.

46. Find $AM$ if $G$ is the centroid on median $AM$ and $AG=14$.

    $AM=21$. If $AG = 14$, then $AM = \frac{3}{2} \times 14 = 21$.

47. Identify the length of midsegment $DE$ if it is parallel to $BC$ and $BC=2k$.

    $DE=k$. Midsegment length is half the parallel side: $\frac{1}{2}(2k) = k$.

48. Find $x$ if an isosceles triangle has base angles $3x^\circ$ and vertex angle $36^\circ$.

    $x=24$. $3x + 3x + 36 = 180$, so $6x = 144$ and $x = 24$.

49. What is the point of concurrency of the medians, and what is its standard symbol?

    Centroid, often labeled $G$. The medians' intersection point, commonly labeled $G$.

50. Find $x$ if the angles of a triangle are $x^\circ$, $60^\circ$, and $80^\circ$.

    $x=40$. $x + 60 + 80 = 180$, so $x = 40$.

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