Congruent Triangles Worksheet - Identify and justify congruence between triangle pairs using geometric markings.

To determine if the pairs of triangles are congruent, we need to use the criteria for triangle congruence: Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Angle-Angle-Side (AAS), and Hypotenuse-Leg (HL) for right triangles. Let's analyze each pair step by step.

Pair 1:

- Triangles: \( \triangle PQR \) and \( \triangle ABC \)
- Analysis:
- Both triangles are right triangles.
- They have one pair of corresponding legs marked as equal (PQ = AB).
- They have another pair of corresponding legs marked as equal (QR = BC).
- The hypotenuses are not marked as equal.
- Conclusion: By the Leg-Leg (LL) criterion for right triangles, these triangles are congruent.
- Statement: \( \triangle PQR \cong \triangle ABC \) by LL.

Pair 2:

- Triangles: \( \triangle LPM \) and \( \triangle TQS \)
- Analysis:
- One angle is marked as equal (\(\angle LPM = \angle TQS\)).
- One side is marked as equal (PM = QS).
- Another angle is marked as equal (\(\angle LMP = \angle TQS\)).
- Conclusion: By the Angle-Side-Angle (ASA) criterion, these triangles are congruent.
- Statement: \( \triangle LPM \cong \triangle TQS \) by ASA.

Pair 3:

- Triangles: \( \triangle ABC \) and \( \triangle PQR \)
- Analysis:
- One angle is marked as equal (\(\angle BAC = \angle QPR\)).
- One side is marked as equal (AB = PQ).
- Another angle is marked as equal (\(\angle ABC = \angle PQR\)).
- Conclusion: By the Angle-Side-Angle (ASA) criterion, these triangles are congruent.
- Statement: \( \triangle ABC \cong \triangle PQR \) by ASA.

Pair 4:

- Triangles: \( \triangle STW \) and \( \triangle TVW \)
- Analysis:
- One side is marked as equal (ST = TV).
- One angle is marked as equal (\(\angle STW = \angle TVW\)).
- Another side is marked as equal (TW = VW).
- Conclusion: By the Side-Angle-Side (SAS) criterion, these triangles are congruent.
- Statement: \( \triangle STW \cong \triangle TVW \) by SAS.

Pair 5:

- Triangles: \( \triangle STV \) and \( \triangle ADE \)
- Analysis:
- Both triangles are right triangles.
- One leg is marked as equal (ST = AD).
- The hypotenuses are marked as equal (SV = AE).
- Conclusion: By the Hypotenuse-Leg (HL) criterion for right triangles, these triangles are congruent.
- Statement: \( \triangle STV \cong \triangle ADE \) by HL.

Pair 6:

- Triangles: \( \triangle EFG \) and \( \triangle CAB \)
- Analysis:
- One angle is marked as equal (\(\angle EFG = \angle CAB\)).
- One side is marked as equal (EF = CA).
- Another angle is marked as equal (\(\angle FEG = \angle ACB\)).
- Conclusion: By the Angle-Angle-Side (AAS) criterion, these triangles are congruent.
- Statement: \( \triangle EFG \cong \triangle CAB \) by AAS.

Pair 7:

- Triangles: \( \triangle LMN \) and \( \triangle PQR \)
- Analysis:
- All three sides are marked as equal (LM = PQ, MN = QR, LN = PR).
- Conclusion: By the Side-Side-Side (SSS) criterion, these triangles are congruent.
- Statement: \( \triangle LMN \cong \triangle PQR \) by SSS.

Pair 8:

- Triangles: \( \triangle ABC \) and \( \triangle CDA \)
- Analysis:
- Both triangles share a common side (AC).
- Two pairs of corresponding sides are marked as equal (AB = CD, BC = DA).
- The included angles are right angles, which are equal.
- Conclusion: By the Side-Angle-Side (SAS) criterion, these triangles are congruent.
- Statement: \( \triangle ABC \cong \triangle CDA \) by SAS.

Final Answer:

\[
\boxed{
\begin{array}{ll}
1. & \triangle PQR \cong \triangle ABC \text{ by LL} \\
2. & \triangle LPM \cong \triangle TQS \text{ by ASA} \\
3. & \triangle ABC \cong \triangle PQR \text{ by ASA} \\
4. & \triangle STW \cong \triangle TVW \text{ by SAS} \\
5. & \triangle STV \cong \triangle ADE \text{ by HL} \\
6. & \triangle EFG \cong \triangle CAB \text{ by AAS} \\
7. & \triangle LMN \cong \triangle PQR \text{ by SSS} \\
8. & \triangle ABC \cong \triangle CDA \text{ by SAS} \\
\end{array}
}
\]