Geometry Exam Prep: Mastering Triangle Side-Angle Comparisons

📚 Quick Study Guide

• 📐 Angle-Side Relationship: In any triangle, the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle.
• 📏 Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For sides $a$, $b$, and $c$: $a + b > c$, $a + c > b$, and $b + c > a$.
• 💡 Hinge Theorem (SAS Inequality): If two sides of one triangle are congruent to two sides of another triangle, then the triangle with the larger included angle has the longer third side.
• 🧭 Converse of the Hinge Theorem (SSS Inequality): If two sides of one triangle are congruent to two sides of another triangle, then the triangle with the longer third side has the larger included angle.
• 📝 Exterior Angle Theorem: The measure of an exterior angle of a triangle is greater than the measure of either of the two non-adjacent interior angles.

🧪 Practice Quiz

1. In $\triangle ABC$, $AB = 5$, $BC = 8$, and $AC = 10$. Which angle is the largest?
   1. $\angle A$
   2. $\angle B$
   3. $\angle C$
   4. Cannot be determined
2. In $\triangle PQR$, $\angle P = 50^\circ$, $\angle Q = 60^\circ$, and $\angle R = 70^\circ$. Which side is the longest?
   1. $PQ$
   2. $QR$
   3. $PR$
   4. Cannot be determined
3. Which of the following sets of side lengths can form a triangle?
   1. 2, 3, 6
   2. 4, 5, 9
   3. 5, 8, 12
   4. 1, 2, 3
4. Given $\triangle ABC$ and $\triangle DEF$ with $AB = DE$, $BC = EF$, and $\angle B > \angle E$, which of the following is true?
   1. $AC > DF$
   2. $AC < DF$
   3. $AC = DF$
   4. Cannot be determined
5. Given $\triangle XYZ$ and $\triangle UVW$ with $XY = UV$, $YZ = VW$, and $XZ > UW$, which of the following is true?
   1. $\angle Y > \angle V$
   2. $\angle Y < \angle V$
   3. $\angle Y = \angle V$
   4. Cannot be determined
6. In $\triangle ABC$, if $AB = 7$ and $BC = 4$, which of the following could be a possible length for $AC$?
   1. 2
   2. 3
   3. 11
   4. 5
7. In $\triangle ABC$, $\angle A = 80^\circ$ and $\angle B = 40^\circ$. Which side is the shortest?
   1. $AB$
   2. $BC$
   3. $AC$
   4. Cannot be determined