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? bq = bs when ∠aqs ≅ ∠bqs. aq = as when ∠aqs ≅ ∠bqs. ∠aqs ≅ ∠bqs when as = bs and aq = bq. ∠aqs ≅ ∠bqs when as = bq and bs = aq

Explanation:

Step1: Recall angle - bisection construction principle

In an angle - bisection construction using a compass and straightedge, when we draw arcs of the same radius from the two intersection points (A and B) of the initial arc with the sides of the angle, we are creating two triangles $\triangle AQS$ and $\triangle BQS$.

Step2: Analyze congruent - triangle conditions

If $AS = BS$ (by drawing arcs of the same width from A and B) and $AQ=BQ$ (by the first step of the angle - bisection construction where we draw an arc from the vertex Q that intersects the sides of the angle), and $QS$ is common to both triangles. Then, by the Side - Side - Side (SSS) congruence criterion, $\triangle AQS\cong\triangle BQS$.

Step3: Determine the relationship between angles

When $\triangle AQS\cong\triangle BQS$, corresponding parts of congruent triangles are congruent. So, $\angle AQS\cong\angle BQS$.

Answer:

$\angle AQS\cong\angle BQS$ when $AS = BS$ and $AQ = BQ$.