Congruent and Similar Triangles Examples

📚 Quick Study Guide

• 📐 Congruent Triangles: Two triangles are congruent if they have the same size and shape. This means all corresponding sides and angles are equal. We can prove congruence using criteria like SSS, SAS, ASA, AAS, and HL.
• 📏 SSS (Side-Side-Side): If all three sides of one triangle are congruent to the corresponding three sides of another triangle, then the triangles are congruent.
• 📐 SAS (Side-Angle-Side): If two sides and the included angle of one triangle are congruent to the corresponding two sides and included angle of another triangle, then the triangles are congruent.
• ✨ ASA (Angle-Side-Angle): If two angles and the included side of one triangle are congruent to the corresponding two angles and included side of another triangle, then the triangles are congruent.
• 💫 AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are congruent to the corresponding two angles and non-included side of another triangle, then the triangles are congruent.
• 💪 HL (Hypotenuse-Leg): Applicable only to right triangles. If the hypotenuse and one leg of one right triangle are congruent to the corresponding hypotenuse and leg of another right triangle, then the triangles are congruent.
• 💡 Similar Triangles: Two triangles are similar if they have the same shape but not necessarily the same size. This means all corresponding angles are equal, and corresponding sides are in proportion. We can prove similarity using criteria like AA, SSS, and SAS.
• 👯 AA (Angle-Angle): If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
• ⚖️ SSS (Side-Side-Side): If all three sides of one triangle are proportional to the corresponding three sides of another triangle, then the triangles are similar.
• 🌱 SAS (Side-Angle-Side): If two sides of one triangle are proportional to the corresponding two sides of another triangle, and the included angles are congruent, then the triangles are similar.

🧪 Practice Quiz

1. Which of the following congruence criteria requires knowing all three sides of two triangles?

   1. A) SAS
   2. B) ASA
   3. C) SSS
   4. D) AA

2. If $\triangle ABC \sim \triangle XYZ$, and $AB = 5$, $XY = 10$, and $BC = 7$, what is the length of $YZ$?

   1. A) 3.5
   2. B) 14
   3. C) 12
   4. D) 10

3. In $\triangle PQR$, $\angle P = 50^\circ$ and $\angle Q = 60^\circ$. In $\triangle LMN$, $\angle L = 50^\circ$ and $\angle M = 60^\circ$. Are the triangles similar? If so, by which criterion?

   1. A) Yes, by SAS
   2. B) Yes, by AA
   3. C) Yes, by SSS
   4. D) No, they are not similar

4. Which congruence criterion applies *only* to right triangles?

   1. A) SAS
   2. B) ASA
   3. C) SSS
   4. D) HL

5. If $\triangle DEF \cong \triangle GHI$, $DE = 8$, $EF = 6$, and $GH = 8$, what is the length of $HI$?

   1. A) 10
   2. B) 6
   3. C) 8
   4. D) Cannot be determined

6. Two triangles have proportional sides with a scale factor of 3. Are the triangles similar?

   1. A) Yes, by AA
   2. B) Yes, by SSS
   3. C) Yes, by SAS
   4. D) No, they are not similar

7. Given $\triangle ABC$ and $\triangle A'B'C'$ where $\angle A = \angle A'$ and $\frac{AB}{A'B'} = \frac{AC}{A'C'}$. Which similarity criterion can be used to prove $\triangle ABC \sim \triangle A'B'C'$?

   1. A) AA
   2. B) SSS
   3. C) SAS
   4. D) ASA