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 (overline{ab}). how does this step ensure that the new angle will be congruent to the original angle?

Explanation:

Step1: Recall the SSS (Side - Side - Side) Congruence Criterion

When constructing an angle, if we can show that the three sides of two triangles (formed by the original angle and the constructed angle) are equal, then the triangles are congruent.

Step2: Analyze the role of setting the compass width to \(AB\)

In the angle - copying construction, we have arcs drawn. Let's assume in the original angle \(\angle QPR\), we have an arc with a certain radius (say \(r\)) intersecting the sides of the angle at two points (forming a chord of length \(AB\)). When we set the compass width to \(AB\) for the new angle construction, we are ensuring that one of the sides (the chord of the arc) of the triangle formed by the new angle (with the same radius \(r\) for the arc) is equal to the corresponding side (the chord \(AB\)) of the triangle formed by the original angle \(\angle QPR\)). Along with the fact that the radius of the arcs (the other two'sides' of the two - triangle system) are equal (since we use the same compass radius for drawing arcs in both the original and constructed angle cases), by the SSS congruence criterion (\(SSS: \text{If }a = a',b = b',c=c'\text{ then }\triangle ABC\cong\triangle A'B'C'\)), the two angles (the angles of the two congruent triangles) are congruent.

Answer:

By the SSS (Side - Side - Side) congruence criterion. When we set the compass width to the length of \(AB\) (a chord of the arc in the original angle), and since the radius of the arcs (used in the construction of both the original and new angle) is the same, the three'sides' (two radii of the arcs and the chord \(AB\) and its corresponding chord in the new - angle construction) of the two triangles (formed by the original and new angles) are equal. So, the two angles (the original \(\angle QPR\) and the new angle) are congruent.