Practice worksheet for identifying congruent triangles using SSS and SAS theorems.

To determine which triangles are congruent, we need to look at the markings on the sides and angles. We are looking for pairs that match either the SSS (Side-Side-Side) rule or the SAS (Side-Angle-Side) rule.

* SSS Rule: All three sides of one triangle are equal to all three sides of another triangle.
* SAS Rule: Two sides and the angle *between* them in one triangle are equal to two sides and the angle *between* them in another triangle.

Let's look at each problem step-by-step:

1. Analyze Problem 1:
* Triangle XYZ: Has a right angle. The legs have 1 tick and 2 ticks. The hypotenuse has 3 ticks.
* Triangle UVW: Has a right angle. The legs have 1 tick and 2 ticks. The hypotenuse has 2 ticks.
* *Check:* The hypotenuses do not match (3 ticks vs 2 ticks). So, this is not a match.
* Triangle EFG: Has a right angle. The legs have 1 tick and 2 ticks. The hypotenuse has 3 ticks.
* *Check:* Triangle XYZ and Triangle EFG both have legs with 1 and 2 ticks, and hypotenuses with 3 ticks. Since all three sides match, they are congruent by SSS. Also, since the angle between the legs (the right angle) is the same, they are also congruent by SAS.

2. Analyze Problem 2:
* Triangle DEF: Right angle at F. Legs DF (1 tick) and EF (2 ticks).
* Triangle GHI: Right angle at H. Legs IH (1 tick) and GH (2 ticks).
* *Check:* Both have legs of length 1 and 2, and the angle between them is a right angle ($90^\circ$). This matches the SAS rule.
* Triangle JKL: Right angle at L. Legs JL (1 tick) and KL (2 ticks), but the hypotenuse JK has 2 ticks.
* *Check:* In triangle GHI, the side with 2 ticks is a leg. In triangle JKL, a side with 2 ticks is the hypotenuse. They are not arranged the same way.

Conclusion for 2: Triangles DEF and GHI are congruent.

3. Analyze Problem 3:
* Triangle XYZ: Angle X is marked. Sides XZ (1 tick) and XY (2 ticks) surround the angle. Side ZY has 1 tick.
* Triangle ABC: Angle A is marked. Sides AC (1 tick) and AB (2 ticks) surround the angle. Side BC has 2 ticks.
* *Check:* Let's compare XYZ and ABC.
* Side XZ (1 tick) matches Side AC (1 tick).
* Side XY (2 ticks) matches Side AB (2 ticks).
* Angle X matches Angle A.
* Since two sides and the included angle match, they are congruent by SAS.
* Triangle DEF: Angle D is marked. Sides DE (1 tick) and DF (2 ticks) surround the angle. Side EF has 1 tick.
* *Check:* Compare XYZ and DEF.
* Side XZ (1 tick) matches Side DE (1 tick).
* Side XY (2 ticks) matches Side DF (2 ticks).
* Angle X matches Angle D.
* They are congruent by SAS.

Conclusion for 3: All three triangles (XYZ, ABC, and DEF) are congruent to each other.

4. Analyze Problem 4:
* Triangle JKL: Right angle at J. Legs KJ (1 tick) and JL (2 ticks). Hypotenuse KL has 1 tick.
* Triangle MNO: Right angle at M. Legs NM (1 tick) and MO (2 ticks). Hypotenuse NO has 1 tick.
* *Check:* Legs match (1 and 2 ticks), and the included right angle matches. Congruent by SAS.
* Triangle PQR: Right angle at P. Legs QP (1 tick) and PR (2 ticks). Hypotenuse QR has 1 tick.
* *Check:* Legs match (1 and 2 ticks), and the included right angle matches. Congruent by SAS.

Conclusion for 4: All three triangles (JKL, MNO, and PQR) are congruent to each other.

5. Analyze Problem 5:
* Triangle STU: No right angle shown. Sides SU (2 ticks), UT (1 tick), TS (2 ticks).
* Triangle PQR: No right angle shown. Sides PQ (1 tick), QR (2 ticks), RP (3 ticks).
* *Check:* The side lengths (ticks) are completely different from triangle STU. Not congruent.
* Triangle ABC: No right angle shown. Sides AB (1 tick), BC (2 ticks), CA (3 ticks).
* *Check:* Compare PQR and ABC.
* Side PQ (1 tick) matches Side AB (1 tick).
* Side QR (2 ticks) matches Side BC (2 ticks).
* Side RP (3 ticks) matches Side CA (3 ticks).
* All three sides match. Congruent by SSS.

Conclusion for 5: Triangles PQR and ABC are congruent.

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Final Answer:
1. $\triangle XYZ \cong \triangle EFG$ (by SSS or SAS)
2. $\triangle DEF \cong \triangle GHI$ (by SAS)
3. $\triangle XYZ \cong \triangle ABC \cong \triangle DEF$ (by SAS)
4. $\triangle JKL \cong \triangle MNO \cong \triangle PQR$ (by SAS)
5. $\triangle PQR \cong \triangle ABC$ (by SSS)