Question

given △abc, describe how to construct △pqr congruent to △abc using each of the following combinations
a. two sides of △abc and an angle included between these two sides.
b. the three sides of △abc
c. two angles and a side included between these angles

copy ab from △abc to form pr. draw an arc with the center at p and the length of bc as the radius and an arc with the center at r and the length of ac as the radius. the intersection of these two arcs is point q. draw segments pq and qr to complete the triangle.
follow the steps to construct △pqr. choose the correct answer below

Explanation:

Step1: Analyze Side - Angle - Side (SAS) construction

When two sides and the included angle of one triangle are given to construct a congruent triangle, we first draw one of the sides. Then we construct the included - angle at one of the endpoints of the drawn side and mark off the length of the second side along the ray of the constructed angle.

Step2: Analyze Side - Side - Side (SSS) construction

For SSS construction, we copy one side. Then, using the lengths of the other two sides as radii, we draw arcs from the endpoints of the copied side. The intersection of these arcs gives the third vertex of the congruent triangle.

Step3: Analyze Angle - Side - Angle (ASA) construction

When two angles and the included side are given, we first draw the included side. Then we construct the two angles at the two endpoints of the drawn side. The intersection of the rays of the constructed angles gives the third vertex of the congruent triangle.

Step4: Match with the given description

The description "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" is an SSS construction.

Answer:

We are using the SSS (Side - Side - Side) congruence criterion to construct $\triangle PQR$ congruent to $\triangle ABC$. Looking at the options, we need to find the one that represents the correct correspondence of sides for an SSS - constructed congruent triangle. Without seeing the specific markings on the options clearly, if the sides are marked correctly according to the SSS construction where the lengths of the sides of $\triangle PQR$ match the lengths of the sides of $\triangle ABC$ in the described construction, we can choose the appropriate option. Since the construction is based on copying sides and using them to form a new triangle, we are looking for a pair of triangles where the side - lengths are marked as congruent in the same order as the construction steps. If we assume that the markings on the triangles in the options represent side - length congruence, we need to find the option where the sides of $\triangle PQR$ are congruent to the sides of $\triangle ABC$ as per the SSS construction. Without the ability to see the markings on the options precisely, we can't give a definite letter answer, but the correct option should show that $PR = AB$, $PQ=BC$, and $QR = AC$ based on the construction steps.