Question

Which of the following statements must be true about this diagram? Check all that apply.

Answer 1

see the attached figure to better understand the problem

we know that

the sum of the internal angles of a triangle is equal to $180\ degrees$

so

$m1+m2+m3=180$ -----> equation $1$

$m3+m4=180$ ------> by suplementary angles

solve for m3

$m3=180-m4$ -----> equation $2$

substitute equation $2$ in equation $1$

$m1+m2+(180-m4)=180$

$m1+m2+(180-m4)=180$

$m1+m2=m4$ ----> equation $3$

let's analyze each case

$case\ a)\ m1 + m2 = m4$

see the equation $3$

$m1+m2=m4$

$This\ case\ is\ true$

$case\ b)\ m1 + m2 = m3$

see the equation $1$

$m1+m2+m3=180\\ m1+m2=180-m3$

$This\ case\ is\ not\ true$

$because\ (m1+m2)\ is\ equal\ to\ the\ suplementary\ angle\ of\ m3$

$case\ c)\ m4\ is\ greater\ than\ m2$

see the equation $3$

$m1+m2=m4$

$m4 > m2$

so

$This\ case\ is\ true$

$case\ d)\ m4\ is\ greater\ than\ m1$

see the equation $3$

$m1+m2=m4$

$m4 > m1$

so

$This\ case\ is\ true$

$case\ e)\ m3\ is\ greater\ than\ m2$

see the equation $1$

$m1+m2+m3=180$

so

$Not\ necessarily\ m3\ is\ greater\ than\ m2$

$case\ e)\ m2+m3=m4$

see the equation $3$

$m1+m2=m4$

when m1=m3--------> m3+m2=m4

so

$Not\ necessarily$

therefore

the answer is

$A.\ m1 + m2 = m4$

$C.\ m4\ is\ greater\ than\ m2$

$D.\ m4\ is\ greater\ than\ m1$

Answer 2

By looking at the diagram:

m3 = 180-m4

sum of the angles = m1 + m2 + m3 = 180 degrees

substituting the value of m3:
m1 + m2 + ( 180 - m4 ) = 180

m4= 180 - m3

m1 + m2 + (180 - m4) - (180-m4) = 180 - (180-m4)

Hence:
m1 + m2 = m4
so the right answers after concluding from the above calculations is:

A. m1 + m2 = m4
C.m4 is greater than m2
D. m4 is greater than m1