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

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

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calculista calculista · Answer 1 · 2025-03-24T01:04:47-04:00

Answer:

Part 1) [tex]m\angle P=48.01^o[/tex]

Part 2) [tex]m\angle PST=131.99^o[/tex]

Part 3) [tex]m\angle T=131.99^o[/tex]

Part 4) [tex]m\angle R=48.01^o[/tex]

Part 5) [tex]PS=\sqrt{45.25}\ units[/tex], or [tex]PS=6.7\ units[/tex]

Part 6) [tex]TR=\sqrt{45.25}\ units[/tex], or [tex]TR=6.7\ units[/tex]

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

[tex]PR=PD+ER+ST[/tex] ---> by addition segment postulate

[tex]PD=ER[/tex] ----> by isosceles trapezoid

so

[tex]PR=2PD+ST\\PD=(PR-ST)/2[/tex]

substitute the given values

[tex]PD=(24-15)/2[/tex]

[tex]PD=4.5\ units[/tex]

In the right triangle PSD

[tex]tan(P)=\frac{SD}{PD}[/tex]

[tex]P=tan^{-1}(\frac{SD}{PD})[/tex]

substitute the given values

[tex]m\angle P=tan^{-1}(\frac{5}{4.5})[/tex]

[tex]m\angle P=48.01^o[/tex]

Part 2) Find the measure of angle PST

we know that

In a trapezoid ST ║ RP

so

[tex]m\angle P+m\angle PST=180^o[/tex] ---> by consecutive interior angles

we have

[tex]m\angle P=48.01^o[/tex]

substitute

[tex]48.01^o+m\angle PST=180^o[/tex]

[tex]m\angle PST=180^o-48.01^o[/tex]

[tex]m\angle PST=131.99^o[/tex]

Part 3) Find the measure of angle T

we know that

In this problem we have an isosceles trapezoid

so

[tex]m\angle T=m\angle S[/tex]

we have

[tex]m\angle PST=m\angle S=131.99^o[/tex]

therefore

[tex]m\angle T=131.99^o[/tex]

Part 4) Find the measure of angle R

we know that

In this problem we have an isosceles trapezoid

so

[tex]m\angle R=m\angle P[/tex]

we have

[tex]m\angle P=48.01^o[/tex]

therefore

[tex]m\angle R=48.01^o[/tex]

Part 5) Find the measure of segment PS

we know that

In the right triangle PSD

Applying the Pythagorean Theorem

[tex]PS^2=SD^2+PD^2[/tex]

substitute the given values

[tex]PS^2=5^2+4.5^2[/tex]

[tex]PS^2=45.25[/tex]

[tex]PS=\sqrt{45.25}\ units[/tex]

[tex]PS=6.7\ units[/tex]

Part 6) Find the measure of segment TR

we know that

TR=PS ---> by isosceles trapezoid

we have

[tex]PS=\sqrt{45.25}\ units[/tex]

therefore

[tex]TR=\sqrt{45.25}\ units[/tex]

[tex]TR=6.7\ units[/tex]