Question

1. (12 pts) in the diagram below, $overrightarrow{yx}$ and $overrightarrow{yz}$ are opposite rays, $overrightarrow{ya}$ is the angle bisector of $angle byx$, and $overrightarrow{yc}$ is the angle bisector of $angle byz$. prove that m$angle ayc = 90$. hint: lots of equations to write down here. first off, there are angle bisectors given in the diagram, so you can immediately write down that m$angle ayb$ =m$angle ayx$ and m$angle cyb$ = m$angle cyz$ using the definition of angle bisector. you will also need some equations coming from the angle theorems we talked about. make substitutions!

Explanation:

Step1: Recall angle - bisector property

Since $\overrightarrow{YA}$ is the angle - bisector of $\angle BYX$, we have $m\angle AYB=m\angle AYX$. Let $m\angle AYB = m\angle AYX=x$.

Step2: Recall another angle - bisector property

Since $\overrightarrow{YC}$ is the angle - bisector of $\angle BYZ$, we have $m\angle CYB=m\angle CYZ$. Let $m\angle CYB = m\angle CYZ = y$.

Step3: Use the fact that $\overrightarrow{YX}$ and $\overrightarrow{YZ}$ are opposite rays

Since $\overrightarrow{YX}$ and $\overrightarrow{YZ}$ are opposite rays, $\angle XYZ = 180^{\circ}$. And $\angle XYZ=m\angle AYX + m\angle AYB+m\angle CYB+m\angle CYZ$. So $180^{\circ}=x + x + y + y=2x + 2y$.

Step4: Solve for $x + y$

Dividing both sides of the equation $180^{\circ}=2x + 2y$ by 2, we get $x + y = 90^{\circ}$.

Step5: Find $m\angle AYC$

Since $m\angle AYC=m\angle AYB+m\angle CYB=x + y$, then $m\angle AYC = 90^{\circ}$.

Answer:

$m\angle AYC = 90^{\circ}$