Question

in the figure, $overrightarrow{tp}$ and $overrightarrow{ts}$ are opposite rays. $overrightarrow{tq}$ bisects $angle rtp$. if $mangle ptq = 12x + 4$ and $mangle rtq = 15x - 5$, find $mangle rtp$.

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{TQ}$ bisects $\angle RTP$, then $m\angle PTQ=m\angle RTQ$. So we set up the equation $12x + 4=15x-5$.

Step2: Solve the equation for $x$

Subtract $12x$ from both sides: $4 = 3x-5$. Then add 5 to both sides: $3x=9$, and $x = 3$.

Step3: Find $m\angle RTP$

Since $m\angle RTP=m\angle PTQ + m\angle RTQ$ and $m\angle PTQ=m\angle RTQ$, we can use either expression for $m\angle PTQ$ or $m\angle RTQ$ to find the measure of one of the sub - angles and then double it. Using $m\angle PTQ=12x + 4$, substitute $x = 3$: $m\angle PTQ=12\times3+4=36 + 4=40$. Then $m\angle RTP=2\times m\angle PTQ=80$.

Answer:

$80$