1 In A ABC, AB =5 cm, AC =6 cm, Z A = 50". 
In APQR, PQ = 5 cm, PR =6 cm and ZP=50. 
(a) Are the triangles ABC and PQR equal? 
(b) If equal, write pairs of equal angles.

To determine if triangles \( \triangle ABC \) and \( \triangle PQR \) are congruent or similar, we need to check if they satisfy any of the standard criteria for triangle congruence or similarity.

### Given Data:
- In \( \triangle ABC \):
  - \( AB = 5 \) cm, \( AC = 6 \) cm, and \( \angle A = 50^\circ \).

- In \( \triangle PQR \):
  - \( PQ = 5 \) cm, \( PR = 6 \) cm, and \( \angle P = 50^\circ \).

### (a) Are \( \triangle ABC \) and \( \triangle PQR \) congruent?

To check if the triangles are congruent, we can use the **SAS (Side-Angle-Side) Congruence Rule**, which states that if two sides and the included angle of one triangle are equal to the corresponding two sides and included angle of another triangle, the triangles are congruent.

For \( \triangle ABC \) and \( \triangle PQR \):
- \( AB = PQ = 5 \) cm.
- \( AC = PR = 6 \) cm.
- \( \angle A = \angle P = 50^\circ \).

Since two sides and the included angle are the same in both triangles, **the triangles are congruent by SAS congruence**.

### (b) If congruent, what are the pairs of equal angles?

Since \( \triangle ABC \cong \triangle PQR \), all corresponding angles of the triangles are equal. The pairs of equal angles are:
1. \( \angle A = \angle P \) (given as \( 50^\circ \)).
2. \( \angle B = \angle Q \).
3. \( \angle C = \angle R \).

### Summary:
- \( \triangle ABC \cong \triangle PQR \) by the **SAS congruence rule**.
- The pairs of equal angles are \( \angle A = \angle P \), \( \angle B = \angle Q \), and \( \angle C = \angle R \).

Would you like a more detailed explanation, or do you have any further questions?

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Here are 5 related questions to consider:
1. What are the other criteria for triangle congruence?
2. Can triangles be similar but not congruent? How?
3. How would you prove two triangles are similar using angles?
4. What happens if only one side length is the same between two triangles?
5. Can we determine the other angles in \( \triangle ABC \) and \( \triangle PQR \)?

**Tip:** Remember that the SAS rule requires the included angle between the two known sides to be the same for congruence.

In a triangle ABC, AB = 5 cm, AC = 6 cm, ∠A = 50°. In triangle PQR, PQ = 5 cm, PR = 6 cm, and ∠P = 50°. (a) Are the triangles ABC and PQR equal? (b) If equal, write pairs of equal angles.

Learn how to determine if triangles ABC and PQR are congruent using the SAS (Side-Angle-Side) rule. This problem involves checking equal sides and angles in the two triangles. Suitable for students in Grades 8-10.

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