1. **Problem 5:** Given in triangles $\triangle ABC$ and $\triangle PQR$, the sides satisfy $AB = QR$, $BC = PR$, and $CA = PQ$. We need to find which triangle congruence statement is true. 2. **Formula and Rule:** Two triangles are congruent if their corresponding sides and angles are equal. Here, the given equalities are side equalities. By the SSS (Side-Side-Side) congruence rule, if all three sides of one triangle are equal to the corresponding three sides of another triangle, the triangles are congruent. 3. **Work:** Given $AB = QR$, $BC = PR$, and $CA = PQ$, the order of vertices in $\triangle PQR$ must correspond to $\triangle ABC$ for congruence. The matching order is $A \to P$, $B \to Q$, $C \to R$. 4. **Conclusion:** Therefore, $\triangle ABC \cong \triangle PQR$. --- 5. **Problem 6:** In $\triangle ACB$ and $\triangle PQR$, given $AC = PQ$ and $BC = RQ$, to prove congruence by SAS (Side-Angle-Side) rule, which angle equality is needed? 6. **Formula and Rule:** SAS rule states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent. 7. **Work:** The two sides given are $AC = PQ$ and $BC = RQ$. The included angle between these sides in $\triangle ACB$ is $\angle C$, and in $\triangle PQR$ it is $\angle Q$. 8. **Conclusion:** So, the needed angle equality is $\angle C = \angle Q$. --- 9. **Problem 7:** In $\triangle ABC$, $BC = AB$ and $\angle B = 80^\circ$. Find $\angle A$. 10. **Formula and Rule:** In an isosceles triangle, angles opposite equal sides are equal. Since $BC = AB$, angles opposite these sides are equal, so $\angle A = \angle C$. 11. **Work:** Sum of angles in a triangle is $180^\circ$. $$\angle A + \angle B + \angle C = 180^\circ$$ Since $\angle A = \angle C$, let $\angle A = \angle C = x$. $$x + 80^\circ + x = 180^\circ$$ $$2x = 100^\circ$$ $$x = 50^\circ$$ 12. **Conclusion:** $\angle A = 50^\circ$. --- 13. **Problem 8:** Given $\triangle ABC \cong \triangle PQR$ and $\triangle ABC$ is not congruent to $\triangle RPQ$, which statement is not true? 14. **Work:** Since $\triangle ABC \cong \triangle PQR$, corresponding sides are equal: $AB = PQ$, $BC = QR$, $AC = PR$. If $\triangle ABC$ is not congruent to $\triangle RPQ$, then the order of vertices is different, so corresponding sides change. Check options: - (a) $BC = PQ$ (False, because $BC$ corresponds to $QR$ in $PQR$, not $PQ$) - (b) $AC = PR$ (True) - (c) $QR = BC$ (True) - (d) $AB = PQ$ (True) 15. **Conclusion:** Option (a) is not true. --- 16. **Problem 9:** Assertion (A): Two triangles with angles $30^\circ$, $70^\circ$, and $80^\circ$ but different side lengths are not congruent. Reason (R): Two triangles with all corresponding angles equal but unequal corresponding sides cannot be congruent. 17. **Explanation:** Congruence requires both equal angles and equal sides. Equal angles alone imply similarity, not congruence. 18. **Conclusion:** Both A and R are true, and R correctly explains A. --- **Final answers:** 5. (a) $\triangle ABC \cong \triangle PQR$ 6. (d) $\angle C = \angle Q$ 7. (c) $50^\circ$ 8. (a) $BC = PQ$ is not true 9. (a) Both A and R are true and R is the correct explanation of A.