Congruence and Similarity to Solve Problems - Geometry

1. In right $\triangle ABC$ with hypotenuse $AB=13$ and leg $AC=5$, what is the other leg $BC$?

   $12$. $\sqrt{13^2 - 5^2} = \sqrt{169 - 25} = 12$.

2. What theorem states that a line parallel to one side of a triangle divides the other two sides proportionally?

   Triangle proportionality theorem. States parallel line divides other sides proportionally.

3. What is the Pythagorean Theorem statement for a right triangle with legs $a,b$ and hypotenuse $c$?

   $a^2+b^2=c^2$. Sum of squares of legs equals square of hypotenuse.

4. In $\triangle ABC$, $DE\parallel BC$, $AD=4$, $AB=10$, and $BC=15$. What is $DE$?

   $6$. $DE = BC \cdot \frac{AD}{AB} = 15 \cdot \frac{2}{5} = 6$.

5. Using $AC^2=AB\cdot AD$ and $BC^2=AB\cdot DB$, what identity follows after adding?

   $AC^2+BC^2=AB\cdot(AD+DB)$. Adding the two leg-segment relations.

6. In right $\triangle ABC$, $AD=4$ and $DB=5$ on hypotenuse $AB$. What is $AB$?

   $9$. $AD + DB = 4 + 5 = 9$.

7. In right $\triangle ABC$ with $\angle C=90^\circ$, what does the similarity proof conclude about $AC^2+BC^2$?

   $AC^2+BC^2=AB^2$. Pythagorean theorem derived from similarity relations.

8. In right $\triangle ABC$ with $\angle C=90^\circ$, if $CD$ is the altitude to $AB$, what triangles are similar?

   $\triangle ABC\sim\triangle ACD\sim\triangle BCD$. Altitude to hypotenuse creates three similar triangles.

9. If $AD+DB=AB$ in the altitude-to-hypotenuse setup, what does $AB\cdot(AD+DB)$ simplify to?

   $AB^2$. Since $AD + DB = AB$, this equals $AB^2$.

10. In a triangle, if a segment is parallel to one side, what angle relationship justifies AA similarity?

    Corresponding angles are congruent. Parallel lines create congruent corresponding angles.

11. In right $\triangle ABC$, $AB=20$ and $DB=5$. Using similarity, what is $BC$?

    $10$. $BC = \sqrt{AB \cdot DB} = \sqrt{20 \cdot 5} = 10$.

12. In right $\triangle ABC$ with altitude $CD$ to $AB$, what relation links leg $AC$ to $AB$ and $AD$?

    $AC^2=AB\cdot AD$. Leg squared equals hypotenuse times adjacent segment.

13. What similarity criterion is commonly used to prove the Pythagorean Theorem via an altitude in a right triangle?

    AA similarity. AA similarity establishes similar triangles in altitude proof.

14. In $\triangle ABC$, $DE\parallel BC$, $AE=9$, $AC=12$. What is $\frac{AE}{AC}$?

    $\frac{3}{4}$. $\frac{9}{12} = \frac{3}{4}$.

15. In $\triangle ABC$, with $DE\parallel BC$, what proportionality relates the divided sides $AB$ and $AC$?

    $\frac{AD}{DB}=\frac{AE}{EC}$. Parallel line divides sides proportionally (segments).

16. In $\triangle ABC$, $AD=3$, $DB=6$, $AE=4$, $EC=8$. Which conclusion follows about $DE$?

    $DE\parallel BC$. $\frac{3}{6} = \frac{4}{8} = \frac{1}{2}$, so proportional division.

17. What is the immediate algebraic consequence of $\frac{BC}{AB}=\frac{DB}{BC}$?

    $BC^2=AB\cdot DB$. Cross-multiplying the proportion gives this relation.

18. What is the similarity-based relationship between corresponding sides of similar triangles?

    Corresponding side lengths are proportional. Fundamental property of similar triangles.

19. Identify the proportionality used in the altitude-to-hypotenuse proof: $\frac{AC}{AB}=$ what?

    $\frac{AD}{AC}$. From similarity $\triangle ABC \sim \triangle ACD$.

20. In $\triangle ABC$, if $DE\parallel BC$ with $D\in AB$ and $E\in AC$, what similarity must hold?

    $\triangle ADE\sim\triangle ABC$. Parallel line creates similar triangles with same vertex order.

21. What is the immediate algebraic consequence of $\frac{CD}{AD}=\frac{DB}{CD}$ in the altitude setup?

    $CD^2=AD\cdot DB$. Cross-multiplying the altitude proportion gives this relation.

22. In $\triangle ABC$, with $DE\parallel BC$, what ratio equality compares the smaller to the whole triangle on side $AB$ and base $BC$?

    $\frac{AD}{AB}=\frac{DE}{BC}$. Similar triangles have proportional corresponding sides.

23. In right $\triangle ABC$ with $AC=6$ and $BC=8$, what is the hypotenuse $AB$?

    $10$. $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = 10$.

24. In right $\triangle ABC$, $AB=18$ and $DB=2$. Using $BC^2=AB\cdot DB$, what is $BC$?

    $6$. $BC = \sqrt{18 \cdot 2} = \sqrt{36} = 6$.

25. In $\triangle ABC$, $AD=3$, $DB=6$, $AE=4$, $EC=10$. Which conclusion follows about $DE$?

    Cannot conclude $DE\parallel BC$. $\frac{3}{6} = \frac{1}{2}$ but $\frac{4}{10} = \frac{2}{5}$, not proportional.

26. In $\triangle ABC$, with $DE\parallel BC$, what ratio equality compares the smaller to the whole triangle on side $AC$ and base $BC$?

    $\frac{AE}{AC}=\frac{DE}{BC}$. Similar triangles have proportional corresponding sides.

27. In right $\triangle ABC$ with altitude $CD$ to $AB$, what relation links leg $BC$ to $AB$ and $DB$?

    $BC^2=AB\cdot DB$. Leg squared equals hypotenuse times adjacent segment.

28. What is the converse theorem called: if a line divides two sides proportionally, then it is parallel to the third side?

    Converse of the triangle proportionality theorem. If division is proportional, then line is parallel to third side.

29. In right $\triangle ABC$, $AB=20$ and $AD=5$. Using similarity, what is $AC$?

    $10$. $AC = \sqrt{AB \cdot AD} = \sqrt{20 \cdot 5} = 10$.

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

    Midsegment. Connects midpoints of two triangle sides.

31. In $\triangle ABC$, $DE\parallel BC$, $AD=6$, $DB=3$, and $AE=10$. What is $EC$?

    $5$. Since $\frac{AE}{EC} = \frac{AD}{DB} = 2$, so $EC = \frac{10}{2} = 5$.

32. In $\triangle ABC$, $DE\parallel BC$, $AD=6$, $DB=3$. What is $\frac{AD}{DB}$?

    $2$. $\frac{6}{3} = 2$.

33. In right $\triangle ABC$ with altitude $CD$ to hypotenuse $AB$, what is the geometric mean relation for $CD$?

    $CD^2=AD\cdot DB$. Altitude is geometric mean of hypotenuse segments.

34. What is the key conclusion of the converse: if $\frac{AD}{DB}=\frac{AE}{EC}$, then what must be true?

    $DE\parallel BC$. Converse: proportional division implies parallel line.

35. In right $\triangle ABC$ with $AB=25$ and $AC=7$, what is $BC$?

    $24$. $\sqrt{25^2 - 7^2} = \sqrt{625 - 49} = 24$.

36. Identify the proportionality used in the altitude-to-hypotenuse proof: $\frac{BC}{AB}=$ what?

    $\frac{DB}{BC}$. From similarity $\triangle ABC \sim \triangle BCD$.

37. In right $\triangle ABC$, altitude $CD$ meets $AB$ at $D$. If $AD=9$ and $DB=16$, what is $CD$?

    $12$. $CD = \sqrt{AD \cdot DB} = \sqrt{9 \cdot 16} = 12$.

38. In a right triangle, what is the name of the side opposite the right angle?

    Hypotenuse. Longest side in right triangle, opposite 90° angle.

39. What does the midsegment theorem state about a midsegment and the third side of a triangle?

    It is parallel and has half the third side's length. Midsegment is parallel and half the length of third side.

40. What is the standard similarity criterion used to prove the triangle proportionality theorem?

    AA similarity. Angle-Angle similarity proves triangles are similar.

41. In $\triangle ABC$, $DE\parallel BC$, $AD=5$, $DB=15$, and $AE=4$. What is $EC$?

    $12$. $\frac{AD}{DB} = \frac{5}{15} = \frac{1}{3}$, so $EC = 4 \cdot 3 = 12$.

42. In $\triangle ABC$, $D\in AB$, $E\in AC$, and $\frac{AD}{DB}=\frac{AE}{EC}$. What must be true about $DE$?

    $DE\parallel BC$. Converse of triangle proportionality theorem.

43. What is the immediate algebraic consequence of $\frac{AC}{AB}=\frac{AD}{AC}$?

    $AC^2=AB\cdot AD$. Cross-multiplying the proportion gives this relation.

44. In $\triangle ABC$, $DE\parallel BC$, $AE=9$, $AC=12$, and $BC=20$. What is $DE$?

    $15$. $DE = BC \cdot \frac{AE}{AC} = 20 \cdot \frac{3}{4} = 15$.

45. Find the missing value: if $AC^2=AB\cdot AD$, $AB=15$, and $AD=6$, what is $AC$?

    $3\sqrt{10}$. $AC = \sqrt{15 \cdot 6} = \sqrt{90} = 3\sqrt{10}$.

46. Find the missing value: if $BC^2=AB\cdot DB$, $AB=12$, and $DB=3$, what is $BC$?

    $6$. $BC = \sqrt{12 \cdot 3} = \sqrt{36} = 6$.

47. In right $\triangle ABC$, $AB=18$ and $AD=8$. Using $AC^2=AB\cdot AD$, what is $AC$?

    $12$. $AC = \sqrt{18 \cdot 8} = \sqrt{144} = 12$.

48. In right $\triangle ABC$, $AD=1$ and $DB=9$. Using $CD^2=AD\cdot DB$, what is $CD$?

    $3$. $CD = \sqrt{1 \cdot 9} = \sqrt{9} = 3$.

49. Identify the name of the segment from a vertex perpendicular to the opposite side of a triangle.

    Altitude. Perpendicular segment from vertex to opposite side.

50. Identify the correct conclusion: if $DE\parallel BC$, which ratio equals $\frac{DE}{BC}$?

    $\frac{AD}{AB}$. From similar triangles, corresponding sides are proportional.

51. In $\triangle ABC$, with $DE\parallel BC$, what ratio equality compares the smaller to the whole triangle on side $AB$ and $AC$?

    $\frac{AD}{AB}=\frac{AE}{AC}$. Parallel line creates proportional ratios (part to whole).

52. In $\triangle ABC$, $DE\parallel BC$, $AD=4$, $AB=10$. What is $\frac{AD}{AB}$?

    $\frac{2}{5}$. $\frac{4}{10} = \frac{2}{5}$.

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