2 $$ riangle A B C $$ and $$ riangle P Q R $$ in which $$ A B=P Q $$ and the angles at $$ A, B, P $$ and $$ R $$ are respectively $$ 40^{\circ}, 60^{\circ} $$, 4h and $$ 80^{\circ} $$. Pove that the two triangle are congruent

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2 $$ 	riangle A B C $$ and $$ 	riangle P Q R $$ in which $$ A B=P Q $$ and the angles at $$ A, B, P $$ and $$ R $$ are respectively $$ 40^{\circ}, 60^{\circ} $$, 4h and $$ 80^{\circ} $$. Pove that the two triangle are congruent

UpStudy AI · Accepted Answer

To prove that triangles $$ \triangle ABC $$ and $$ \triangle PQR $$ are congruent, we'll use the **Angle-Side-Angle (ASA)** congruence criterion. Here's a step-by-step explanation:

### Given:
1. **Side Equality:** $$ AB = PQ $$
2. **Angle Measures:**
   - $$ \angle A = 40^\circ $$
   - $$ \angle B = 60^\circ $$
   - $$ \angle P = 40^\circ $$
   - $$ \angle R = 80^\circ $$

### Steps to Prove Congruence:

1. **Determine Remaining Angles:**
   - In $$ \triangle ABC $$:
     $$
     \angle C = 180^\circ - \angle A - \angle B = 180^\circ - 40^\circ - 60^\circ = 80^\circ
     $$
   - In $$ \triangle PQR $$:
     $$
     \angle Q = 180^\circ - \angle P - \angle R = 180^\circ - 40^\circ - 80^\circ = 60^\circ
     $$

2. **Compare Corresponding Angles:**
   - $$ \angle A = \angle P = 40^\circ $$
   - $$ \angle B = \angle Q = 60^\circ $$
   - $$ \angle C = \angle R = 80^\circ $$

3. **Apply ASA Congruence Criterion:**
   - **First Angle:** $$ \angle A = \angle P $$
   - **Included Side:** $$ AB = PQ $$
   - **Second Angle:** $$ \angle B = \angle Q $$
   
   Since two angles and the included side of $$ \triangle ABC $$ are equal to two angles and the included side of $$ \triangle PQR $$, by the **ASA criterion**, the triangles are congruent.

### Conclusion:
By the **Angle-Side-Angle (ASA)** congruence theorem, triangles $$ \triangle ABC $$ and $$ \triangle PQR $$ are congruent.

**Final Answer:**
Since AB = PQ and two pairs of corresponding angles are equal, triangles ABC and PQR satisfy ASA and are congruent.
Triangles $$ \triangle ABC $$ and $$ \triangle PQR $$ are congruent because they have two pairs of equal corresponding angles and the included side $$ AB = PQ $$.

2 and in which and the
angles at and are respectively , 4h
and . Pove that the two triangle are congruent

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2 and in which and the angles at and are respectively , 4h and . Pove that the two triangle are congruent