Question

Determine the values of x and y in the diagram and use them to determine the measures of the eight angles

Answer 1

Answer:

Part 1) $x=8$

Part 2) $y=12$

Part 3) $m\angle a=105^o$

Part 4) $m\angle b=75^o$

Part 5) $m\angle c=105^o$

Part 6) $m\angle d=75^o$

Part 7) $m\angle e=105^o$

Part 8) $m\angle f=75^o$

Part 9) $m\angle g=105^o$

Part 10) $m\angle h=75^o$

Step-by-step explanation:

see the attached figure with letters to better understand the problem

Part 1) Determine the value of x

we know that

$m\angle a=m\angle g$ ----> by alternate exterior angles

substitute the given values

$(17x-31)^o=(12x+9)^o$

solve for x

$17x-12x=9+31\\5x=40\\x=8$

Part 2) Determine the value of y

we know that

$m\angle c+m\angle d=180^o$ ----> by supplementary angles (consecutive interior angles)

substitute the given values

$(7y+21)^o+(5y+15)^o=180^o$

solve for y

$(12y+36)^o=180^o$

$12y=180-36$

$12y=144$

$y=12$

Part 3) Find the measure of angle a

we have

$m\angle a=(17x-31)^o$

substitute the value of x

$m\angle a=(17(8)-31)=105^o$

Part 4) Find the measure of angle b

we know that

$m\angle a+m\angle b=180^o$ ----> by supplementary angles (consecutive interior angles)

we have

$m\angle a=105^o$

substitute

$105^o+m\angle b=180^o$

$m\angle b=180^o-105^o$

$m\angle b=75^o$

Part 5) Find the measure of angle c

we know that

$m\angle c=m\angle a$ ----> by vertical angles

we have

$m\angle a=105^o$

therefore

$m\angle c=105^o$

Part 6) Find the measure of angle d

we know that

$m\angle d=m\angle b$ ----> by vertical angles

we have

$m\angle b=75^o$

therefore

$m\angle d=75^o$

Part 7) Find the measure of angle e

we know that

$m\angle e=m\angle a$ ----> by corresponding angles

we have

$m\angle a=105^o$

therefore

$m\angle e=105^o$

Part 8) Find the measure of angle f

we know that

$m\angle f=m\angle b$ ----> by corresponding angles

we have

$m\angle b=75^o$

therefore

$m\angle f=75^o$

Part 9) Find the measure of angle g

we know that

$m\angle g=m\angle c$ ----> by corresponding angles

we have

$m\angle c=105^o$

therefore

$m\angle g=105^o$

Part 10) Find the measure of angle h

we know that

$m\angle h=m\angle d$ ----> by corresponding angles

we have

$m\angle d=75^o$

therefore

$m\angle h=75^o$