Question

1. in the diagram below, ray $overrightarrow{ab}$ bisects $angle mat$. if the measure of $angle mat$ equals $64^{circ}$ and the measure of $angle bat$ is given by the expression $2x - 10$, then which of the following is the value of $x$? (1) 14 (2) 15 (3) 21 (4) 37 4. in the diagram shown, ray $overrightarrow{cd}$ is perpendicular to ray $overrightarrow{ce}$. if the measure of $angle dcf$ is $115^{circ}$ then what is the measure of $angle ecf$?

Explanation:

Step1: Recall angle - bisector property

Since ray $\overline{AB}$ bisects $\angle MAT$, then $\angle MAT = 2\angle BAT$. Given $\angle MAT=64^{\circ}$, so $\angle BAT=\frac{1}{2}\angle MAT = 32^{\circ}$.

Step2: Set up an equation

We know that $\angle BAT = 2x - 10$. Since $\angle BAT = 32^{\circ}$, we set up the equation $2x-10 = 32$.

Step3: Solve the equation for $x$

Add 10 to both sides of the equation: $2x-10 + 10=32 + 10$, which simplifies to $2x=42$. Then divide both sides by 2: $x=\frac{42}{2}=21$.

Step4: Solve for the second - part (angle measure)

Since $\overrightarrow{CD}\perp\overrightarrow{CE}$, then $\angle DCE = 90^{\circ}$. We know that $\angle DCF=115^{\circ}$ and $\angle DCF=\angle DCE+\angle ECF$.
So $\angle ECF=\angle DCF-\angle DCE$. Substitute the values: $\angle ECF = 115^{\circ}-90^{\circ}=25^{\circ}$.

Answer:

1. (3) 21
2. $25^{\circ}$