Question

What are the missing parts that correctly complete the proof?

Given: Point A is on the bisector of angle P Q R.
Prove: Point A is equidistant from the sides of angle P Q R.
Art: Three rays Q P, Q A, and Q R point in rightward direction sharing a common endpoint at Q. Ray Q A is horizontal and bisects angle P Q R equally.
A. The perpendicular from point A to ray Q P is made by a dotted line.
B. The perpendicular cuts the ray Q P at X. Angle Q X A is labeled a right angle.
C. The perpendicular from point A to ray Q R is made by another dotted line.
D. The perpendicular cuts ray Q R at Y. Angle Q Y A is labeled a right angle.

Answer 1

The missing parts involve proving that perpendicular segments from point A to the sides of angle PQR are congruent, which is achieved using the Hypotenuse-Leg (HL) congruence theorem to show triangles formed are congruent, and hence the segments are equal.

Step-by-step explanation:

The missing parts to correctly complete the proof that point A is equidistant from the sides of angle PQR involve showing that the perpendicular segments from point A to each side of the angle are congruent.

Since ray QA bisects angle PQR, and segments AX and AY are perpendicular to QP and QR respectively, triangles QAX and QAY are right triangles sharing a hypotenuse QA and have congruent angles at Q (because QA bisects angle PQR).

Therefore, by the Hypotenuse-Leg (HL) congruence theorem, ΔQAX ≅ ΔQAY, implying that the perpendiculars AX and AY are congruent, so point A is equidistant from the sides of angle PQR.