Question

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

Answer 1

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

Answer 2

Answer:  x=16   and y=11

Step-by-step 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))

$\Rightarrow\ 12x-12=180$

$\Rightarrow\ 12x=180+12$

$\Rightarrow\ 12x=192$

$\Rightarrow\ x=\dfrac{192}{12}=16$

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.