Question

c) the construction of copying ∠qpr is started below. the next step is to set the width of the compass to the length of ab. how does this step ensure that the new angle will be congruent to the original angle?

Explanation:

Step1: Recall SSS Congruence Criterion

When constructing an angle, if we can show that the three sides of two triangles are equal (SSS - Side - Side - Side congruence criterion), the triangles are congruent.

Step2: Relate to Angle Construction

In the process of copying an angle, we use the compass to create arcs. By setting the compass width to the length of \(\overline{AB}\), we are ensuring that one of the sides of the new - formed triangle (in the angle - copying construction) is equal to the corresponding side of the triangle formed by the original angle. Along with the radii of the arcs (which are equal by the nature of compass usage in construction), this helps in satisfying the SSS congruence criterion for the two triangles (one with the original angle and the other with the copied angle).

Answer:

By using the SSS (Side - Side - Side) congruence criterion. When we set the compass width to the length of \(\overline{AB}\) (a side - length in the triangle associated with the original angle \(\angle QPR\)), and considering the equal radii of the arcs (another two sides in the triangles formed during construction), the two triangles (one with the original angle and the other with the copied angle) will be congruent. Since congruent triangles have congruent corresponding angles, the new angle will be congruent to the original angle.