Question

consider point q which is the same distance from points a and b. ray qp bisects ∠aqb. use the diagram below to answer the remaining questions. line segment ac is congruent to line segment. the distance from point p to point a is the same as the distance from point p to point. the measure of ∠acq is °.

Explanation:

Step1: Recall congruence property for angle - bisector

Since ray $QP$ bisects $\angle AQB$ and $QA = QB$, by the Angle - Bisector Theorem and congruence of triangles ($\triangle AQC\cong\triangle BQC$ by Side - Angle - Side (SAS) criterion where $QA = QB$, $\angle AQC=\angle BQC$, and $QC = QC$), line segment $AC$ is congruent to line segment $BC$.

Step2: Recall equidistance property

The set of points equidistant from two points $A$ and $B$ is the perpendicular bisector of the line segment $\overline{AB}$. Since ray $QP$ is the angle - bisector of $\angle AQB$ and $QA = QB$, point $P$ lies on the perpendicular bisector of $\overline{AB}$. So the distance from point $P$ to point $A$ is the same as the distance from point $P$ to point $B$.

Step3: Recall angle property

Since $\triangle AQC\cong\triangle BQC$, $\angle ACQ=\angle BCQ$. And since $QP$ is the angle - bisector of $\angle AQB$ and $QA = QB$, $QP$ is also the perpendicular bisector of $\overline{AB}$. So $\angle ACQ = 90^{\circ}$.

Answer:

Line segment $AC$ is congruent to line segment $BC$.
The distance from point $P$ to point $A$ is the same as the distance from point $P$ to point $B$.
The measure of $\angle ACQ$ is $90^{\circ}$.