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

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=√(45.25)\ units`, or
`PS=6.7\ units`

Part 6)
`TR=√(45.25)\ units`, or
`TR=6.7\ units`

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)=(SD)/(PD)`

`P=tan^(-1)((SD)/(PD))`

substitute the given values

`m\angle P=tan^(-1)((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=√(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=√(45.25)\ units`

therefore

`TR=√(45.25)\ units`

`TR=6.7\ units`