Question

ben uses a compass and a straightedge to bisect angle pqr, as shown below. which statement best explains why ben uses the same width to draw arcs from a and b that intersect at s? bo = bs when ∠aqs = ∠bqs. ao = as when ∠aqs = ∠bqs. ∠aqs = ∠bqs when as = bs and ao = bo. ∠aqs = ∠bqs when as = bo and bs = ao.

Explanation:

Step1: Recall angle - bisection principle

In the construction of an angle - bisector, we use the SSS (Side - Side - Side) congruence criterion for triangles. When we draw arcs of the same width from points A and B (which are equidistant from the vertex Q of the angle ∠PQR) and they intersect at S, we create two triangles △AQS and △BQS.
We have AQ = BQ (by the initial construction of arcs from the vertex to the sides of the angle to get points A and B), AS = BS (because the arcs are of the same width), and QS is common to both triangles.

Step2: Apply SSS congruence

By the SSS congruence criterion, △AQS≅△BQS. When two triangles are congruent, their corresponding angles are equal. So, ∠AQS = ∠BQS.

Answer:

∠AQS = ∠BQS when AS = BS and AQ = BQ