Question

these steps were followed to complete the construction.

• an arc was drawn, with the compass, from point y through the sides of the angle to create
• two arcs were drawn, each with the same compass setting, one from point x and one from
• a ray was drawn, with the straightedge, from point y through point p.

based on the construction, which of the following is not true?
a point p is equidistant from points x and z
b (mangle pyx = mangle pyz)
c (mangle xyz=\frac{1}{2}(mangle pyz))
d ray (yp) is an angle bisector of (angle xyz)

Explanation:

Step1: Recall angle - bisector construction properties

The given construction is of an angle - bisector of $\angle XYZ$. When constructing an angle - bisector, the ray $YP$ divides $\angle XYZ$ into two equal angles. So, $m\angle PYX=m\angle PYZ$ and ray $YP$ is the angle - bisector of $\angle XYZ$. Also, by the construction, point $P$ is equidistant from the sides of the angle (points $X$ and $Z$ on the sides of the angle).

Step2: Analyze each option

• Option A: In the construction of an angle - bisector, the point $P$ on the angle - bisector is equidistant from the points $X$ and $Z$ on the sides of the angle. This is true.
• Option B: Since $YP$ is the angle - bisector of $\angle XYZ$, $m\angle PYX = m\angle PYZ$. This is true.
• Option C: Since $YP$ is the angle - bisector of $\angle XYZ$, $m\angle PYZ=\frac{1}{2}(m\angle XYZ)$, not $m\angle XYZ=\frac{1}{2}(m\angle PYZ)$. This is false.
• Option D: By the construction steps, ray $YP$ is an angle - bisector of $\angle XYZ$. This is true.

Answer:

C. $m\angle XYZ=\frac{1}{2}(m\angle PYZ)$