Question

given $\triangle abc$, describe how to construct $\triangle pqr$ congruent to $\triangle abc$ using each of the following combinations.
a. two sides of $\triangle abc$ and an angle included between these two sides
b. the three sides of $\triangle abc$
c. two angles and a side included between these angles
b. choose the correct answer below
a. copy $overline{ab}$ from $\triangle abc$ to form $overline{pq}$. draw an arc with the center at $p$ and the length of $overline{ac}$ as the radius and an arc with the center at $q$ and the length of $overline{bc}$ as the radius. the intersection of these two arcs is point $r$. draw segments $overline{pr}$ and $overline{qr}$ to complete the triangle.
b. copy $overline{ab}$ from $\triangle abc$ to form $overline{qr}$. draw an arc with the center at $r$ and the length of $overline{ac}$ as the radius and an arc with the center at $q$ and the length of $overline{bc}$ as the radius. the intersection of these two arcs is point $p$. draw segments $overline{pr}$ and $overline{pq}$ to complete the triangle.
c. copy $overline{ab}$ from $\triangle abc$ to form $overline{pr}$. draw an arc with the center at $p$ and the length of $overline{bc}$ as the radius and an arc with the center at $r$ and the length of $overline{ac}$ as the radius. the intersection of these two arcs is point $q$. draw segments $overline{pq}$ and $overline{qr}$ to complete the triangle.

Explanation:

Step1: Recall SSS congruence criterion

To construct a triangle congruent to another using three - side lengths (SSS - Side - Side - Side congruence), we first copy one side. Then, using the endpoints of the copied side as centers and the lengths of the other two sides of the original triangle as radii, we draw arcs. The intersection of these arcs gives the third vertex of the new triangle.

Step2: Analyze option A

In option A, we copy $\overline{AB}$ from $\triangle ABC$ to form $\overline{PQ}$. Then we draw an arc with center at $P$ and length of $\overline{AC}$ as the radius and an arc with center at $Q$ and length of $\overline{BC}$ as the radius. The intersection of these two arcs is point $R$. Then we draw segments $\overline{PR}$ and $\overline{QR}$ to complete the triangle. This is the correct SSS construction method.

Step3: Analyze option B

In option B, the side - naming and construction steps are incorrect. We should start by copying a side and then using its endpoints as centers for the arcs, not naming the sides in the way presented here.

Step4: Analyze option C

In option C, the side - naming and construction steps are incorrect. The initial side - copying and subsequent arc - drawing with centers and radii are not in the correct order as per the SSS construction.

Answer:

A. Copy $\overline{AB}$ from $\triangle ABC$ to form $\overline{PQ}$. Draw an arc with the center at $P$ and the length of $\overline{AC}$ as the radius and an arc with the center at $Q$ and the length of $\overline{BC}$ as the radius. The intersection of these two arcs is point $R$. Draw segments $\overline{PR}$ and $\overline{QR}$ to complete the triangle.