Question

Given: ST ║ RP , SD ⊥ PR , SP = TR, SD = 5, ST = 15, PR = 24 Find: m∠P, m∠PST, m∠T, m∠R, PS, TR

Answer 1

Answer:

Part 1) $m\angle P=48.01^o$

Part 2) $m\angle PST=131.99^o$

Part 3) $m\angle T=131.99^o$

Part 4) $m\angle R=48.01^o$

Part 5) $PS=\sqrt{45.25}\ units$, or $PS=6.7\ units$

Part 6) $TR=\sqrt{45.25}\ units$, or $TR=6.7\ units$

Step-by-step explanation:

see the attached figure to better understand the problem

In this problem we have an isosceles trapezoid

Part 1) Find the measure of angle P

Find the value of side PD

we have that

$PR=PD+ER+ST$ ---> by addition segment postulate

$PD=ER$ ----> by isosceles trapezoid

so

$PR=2PD+ST\\PD=(PR-ST)/2$

substitute the given values

$PD=(24-15)/2$

$PD=4.5\ units$

In the right triangle PSD

$tan(P)=\frac{SD}{PD}$

$P=tan^{-1}(\frac{SD}{PD})$

substitute the given values

$m\angle P=tan^{-1}(\frac{5}{4.5})$

$m\angle P=48.01^o$

Part 2) Find the measure of angle PST

we know that

In a trapezoid ST ║ RP

so

$m\angle P+m\angle PST=180^o$ ---> by consecutive interior angles

we have

$m\angle P=48.01^o$

substitute

$48.01^o+m\angle PST=180^o$

$m\angle PST=180^o-48.01^o$

$m\angle PST=131.99^o$

Part 3) Find the measure of angle T

we know that

In this problem we have an isosceles trapezoid

so

$m\angle T=m\angle S$

we have

$m\angle PST=m\angle S=131.99^o$

therefore

$m\angle T=131.99^o$

Part 4) Find the measure of angle R

we know that

In this problem we have an isosceles trapezoid

so

$m\angle R=m\angle P$

we have

$m\angle P=48.01^o$

therefore

$m\angle R=48.01^o$

Part 5) Find the measure of segment PS

we know that

In the right triangle PSD

Applying the Pythagorean Theorem

$PS^2=SD^2+PD^2$

substitute the given values

$PS^2=5^2+4.5^2$

$PS^2=45.25$

$PS=\sqrt{45.25}\ units$

$PS=6.7\ units$

Part 6) Find the measure of segment TR

we know that

TR=PS ---> by isosceles trapezoid

we have

$PS=\sqrt{45.25}\ units$

therefore

$TR=\sqrt{45.25}\ units$

$TR=6.7\ units$