Question

Ben uses a compass and a straightedge to bisect angle PQR, as shown below: The figure shows two rays QP and QR having a common end point Q. An arc drawn from Q cuts the ray QP at A and the ray QR at B. Two small arcs intersect between the lines QP and QR. The point of intersection of the arcs is labeled S. The points Q, S, and T are joined with a line. Which statement best explains why Ben uses the same width to draw arcs from A and B which intersect at S? (5 points) a BQ = BS when ∠AQS ≅ BQS. b AQ = AS when ∠AQS ≅ BQS. c ∠AQS ≅ BQS when AS = BS and AQ = BQ. d ∠AQS ≅ BQS when AS = BQ and BS = AQ.

Answer 1

The angle bisector QS is constructed using arcs of the same width

intersecting above the segment joining equidistant point from Q.

Correct Response;

• c. ∠AQS ≅ ∠BQS when AS = BS and AQ = BQ

Reasons why the selected option is correct;

The steps to construct an angle bisector are as follows;

• Draw an arc from the vertex of the angle, Q, intersecting the rays forming the angle, QP and QR, at points A and B respectively.

• From points A, and B, draw arcs having same radii to intersect between the rays QP and QR at point S.

• Join the point of intersection of the small arcs at S to Q to bisect the angle PQR.

The reason why Ben uses the same width to draw arcs from A and B is as follows;

The point A and B are equidistant from point Q, therefore, point Q is point of intersection of arcs of radius AQ = BQ drawn from A and B.

Similarly point S is the point of intersection of arcs AS = BS from points A and B.

Which gives that the line QS is the perpendicular bisector of the segment AB, where ΔABQ is an isosceles triangle, therefore, QS bisects vertex angle ∠PQR.

Therefore, the correct option is the option c.;

• c. ∠AQS ≅ ∠BQS when AS = BS and AQ = BQ

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