The sum of angles in any triangle is 180^(°). Given A = 80^(°) and B = 60^(°) for ABC, we can find C by subtracting the sum of A and B from 180^(°).

Subtract 140^(°) from 180^(°) to calculate the measure of C in ABC.

Since the ratios of all corresponding sides are equal, the triangles ABC and RQP are similar by the SSS (Side-Side-Side) similarity criterion. Note the correspondence of vertices: A to R, B to Q, C to P.

Since the ratios of all corresponding sides are equal, the triangles ABC and RQP are similar by the SSS (Side-Side-Side) similarity criterion. Note the correspondence of vertices: A to R, B to Q, C to P.

When two triangles are similar, their corresponding angles are equal. From the similarity ABC RQP, we can establish the correspondence between their angles.

Given that C = 40^(°) and knowing that P corresponds to C due to the similarity ABC RQP, we can conclude that P is also 40^(°).

Observe Fig. 6.30 and then find $\angle P$ . [Figure: $\triangle ABC$ with AB = 3.8, BC = 6, CA = $3\sqrt{3}$ , $\angle A = 80°$ , $\angle B = 60°$ ; and $\triangle PQR$ with PQ = 12, QR = 7.6, RP = $6\sqrt{3}$ .]

Answer: $\angle P = 40^{\circ}$

Step-by-step solution

Step 1: Calculate the third angle of triangle ABC

The sum of angles in any triangle is $180^{\circ}$ . Given $\angle A = 80^{\circ}$ and $\angle B = 60^{\circ}$ for $\triangle ABC$ , we can find $\angle C$ by subtracting the sum of $\angle A$ and $\angle B$ from $180^{\circ}$ .

Step 2: Substitute and compute $\angle C$

Substitute the given values of $\angle A = 80^{\circ}$ and $\angle B = 60^{\circ}$ into the formula for the sum of angles in a triangle to find $\angle C$ . The sum of these two angles is $140^{\circ}$ .

Step 3: Simplify to find $\angle C$

Subtract $140^{\circ}$ from $180^{\circ}$ to calculate the measure of $\angle C$ in $\triangle ABC$ .

Step 4: Compare Side Ratios of Triangles

To determine if the triangles are similar, we compare the ratios of their corresponding sides. We calculate the ratios of $AB$ to $RQ$ , $BC$ to $QP$ , and $CA$ to $PR$ . We observe that all three ratios are equal to $\frac{1}{2}$ .

Step 5: Establish Triangle Similarity by SSS criterion

Since the ratios of all corresponding sides are equal, the triangles $\triangle ABC$ and $\triangle RQP$ are similar by the SSS (Side-Side-Side) similarity criterion. Note the correspondence of vertices: A to R, B to Q, C to P.

Step 6: Determine Corresponding Angles

When two triangles are similar, their corresponding angles are equal. From the similarity $\triangle ABC \sim \triangle RQP$ , we can establish the correspondence between their angles.

Step 7: Find the measure of Angle P

Given that $\angle C = 40^{\circ}$ and knowing that $\angle P$ corresponds to $\angle C$ due to the similarity $\triangle ABC \sim \triangle RQP$ , we can conclude that $\angle P$ is also $40^{\circ}$ .

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Observe Fig. 6.30 and then find ∠P\angle P∠P. [Figure: △ABC\triangle ABC△ABC with AB = 3.8, BC = 6, CA = 333\sqrt{3}33​, ∠A=80°\angle A = 80°∠A=80°, ∠B=60°\angle B = 60°∠B=60°; and △PQR\triangle PQR△PQR with PQ = 12, QR = 7.6, RP = 636\sqrt{3}63​.]