Question

given δabc, describe how to construct δpqr congruent to δabc using each of the following combinations
a. two sides of δabc and an angle included between these two sides
b. the three sides of δabc
c. two angles and a side included between these angles
a. choose the correct answer below
a. copy ∠a from δabc to form ∠p. then copy sides ab and ac to form pq and pr, respectively. draw the third side, qr, by connecting the ends of pq and pr
b. copy ∠b from δabc to form ∠p. then copy sides ab and bc to form pq and pr, respectively. draw the third side, qr, by connecting the ends of pq and pr
c. copy ∠c from δabc to form ∠p. then copy sides ac and bc to form pq and pr, respectively. draw the third side, qr, by connecting the ends of pq and pr

Explanation:

Step1: Recall the Side - Angle - Side (SAS) congruence criterion

To construct a triangle congruent to $\triangle ABC$ using two sides and the included angle, we need to copy the angle and the two sides that form that angle.

Step2: Analyze option A

In option A, we copy $\angle A$ from $\triangle ABC$ to form $\angle P$. Then we copy sides $\overline{AB}$ and $\overline{AC}$ to form $\overline{PQ}$ and $\overline{PR}$ respectively. This follows the SAS congruence criterion as $\angle A$ is the included - angle between $\overline{AB}$ and $\overline{AC}$ in $\triangle ABC$, and $\angle P$ is the included - angle between $\overline{PQ}$ and $\overline{PR}$ in the new triangle $\triangle PQR$.

Step3: Analyze option B

In option B, we copy $\angle B$ from $\triangle ABC$ to form $\angle P$, but $\angle B$ is not the included angle between $\overline{AB}$ and $\overline{BC}$ in the way required for constructing a congruent triangle using SAS.

Step4: Analyze option C

In option C, we copy $\angle C$ from $\triangle ABC$ to form $\angle P$, but $\angle C$ is not the included angle between $\overline{AC}$ and $\overline{BC}$ in the way required for constructing a congruent triangle using SAS.

Answer:

A. Copy $\angle A$ from $\triangle ABC$ to form $\angle P$. Then copy sides $\overline{AB}$ and $\overline{AC}$ to form $\overline{PQ}$ and $\overline{PR}$ respectively. Draw the third side, $\overline{QR}$, by connecting the ends of $\overline{PQ}$ and $\overline{PR}$