Question

50 POINTS [GEOMETRY] If m∠DEG= 5x - 4 m∠GEF=7x - 8 m∠DEH= 9y + 5 find the values of x and y

Please answer this question and tell me how to do it please.

Answer 1

Answer: x=16 and y=11

Explanation:

In the given figure , we two lines GH and DF intersecting each other at E.

m∠DEG= 5x - 4 , m∠GEF=7x - 8, m∠DEH= 9y + 5 (1)

Since ∠DEG and ∠GEF lies on line DF.

Then, m∠DEG + m∠GEF=180° [Linear pair]

`\Rightarrow\ 5x - 4+7x - 8=180` (from (1))

`\Rightarrow\ 12x-12=180`

`\Rightarrow\ 12x=180+12`

`\Rightarrow\ 12x=192`

`\Rightarrow\ x=(192)/(12)=16`

Then, m∠GEF=7x - 8= 7(16)-8=104° (2)

Also, ∠GEF and ∠DEH are vertical opposite angles.

And measure of vertical angles are equal.

∴ m∠GEF= m∠DEH

`104=9y+5\ \ \text{[Using (1) and (2) ]}\\\\\Rightarrow\ 9y=104-5\\\\\Rightarrow\ 9y=99\\\\\Rightarrow\ y=11`

Hence, the values of x and y are 16 and 11 respectively.

Answer 2

we are given

DEF is a line

so, sum of angles must be 180

angle(DEG)+angle(GEF)=180

now, we can plug values

`5x-4+7x-8=180`

`12x-12=180`

`12x=192`

`x=16`.........Answer

vertically opposite angle must be equal

angle(GEF)=angle(DEH)

now, we can plug values

`9y+5=7x-8`

now, we can plug x=16

`9y+5=7*16-8`

`9y+5=104`

`9y=99`

`y=11`...................Answer