Geometry test questions on the SAS postulate and triangle congruence

📚 Quick Study Guide

• 📐 SAS Postulate: The Side-Angle-Side (SAS) postulate states that 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 two triangles are congruent.
• 📏 Congruent Sides: Congruent sides are sides that have the same length. We often mark congruent sides with small tick marks.
• 🧮 Included Angle: The included angle is the angle formed by two specific sides of a triangle. Make sure it's the angle *between* the two sides you're comparing!
• ✅ Triangle Congruence: If the SAS postulate conditions are met, we can confidently say that the two triangles are congruent.
• 💡 Important Note: The order matters! It must be Side-Angle-Side in both triangles.

Practice Quiz

1. Which of the following statements correctly describes the SAS postulate?

   1. Two angles and the included side of one triangle are congruent to the corresponding parts of another triangle.
   2. Two sides and a non-included angle of one triangle are congruent to the corresponding parts of another triangle.
   3. Two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle.
   4. All three sides of one triangle are congruent to the corresponding sides of another triangle.

2. In $\triangle ABC$ and $\triangle XYZ$, $AB = XY$, $BC = YZ$, and $\angle B = \angle Y$. Which postulate proves that $\triangle ABC \cong \triangle XYZ$?

   1. ASA
   2. SSS
   3. SAS
   4. AAS

3. Given: $PQ = RS$, $QR = SP$, and $\angle PQR = 50^{\circ}$. What additional information is needed to prove $\triangle PQR \cong \triangle RSP$ using SAS?

   1. $\angle QRS = 50^{\circ}$
   2. $\angle RSP = 50^{\circ}$
   3. $PR = PR$
   4. $\angle RPQ = \angle RQS$

4. In $\triangle DEF$, $DE = 5$, $EF = 7$, and $\angle E = 60^{\circ}$. In $\triangle LMN$, $LM = 5$, $MN = 7$, and $\angle M = 50^{\circ}$. Are the triangles congruent based on the SAS postulate?

   1. Yes
   2. No
   3. Cannot be determined
   4. Only if the third side is also equal

5. If two triangles share a common side, is that side congruent to itself? Which property justifies this?

   1. Yes, Reflexive Property
   2. No, Symmetric Property
   3. Yes, Transitive Property
   4. No, Associative Property

6. Which of the following sets of information is sufficient to prove that $\triangle ABC \cong \triangle DEF$ using the SAS Postulate?

   1. $AB = DE$, $AC = DF$, $\angle B = \angle E$
   2. $AB = DE$, $BC = EF$, $\angle B = \angle E$
   3. $AC = DF$, $BC = EF$, $\angle A = \angle D$
   4. $AB = DE$, $BC = EF$, $\angle C = \angle F$

7. If $\triangle PQR$ and $\triangle STU$ have $PQ \cong ST$ and $QR \cong TU$, what additional congruence is needed to prove $\triangle PQR \cong \triangle STU$ using the SAS Postulate?

   1. $\angle P \cong \angle S$
   2. $\angle R \cong \angle U$
   3. $\angle Q \cong \angle T$
   4. $\angle P \cong \angle U$