Question

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

A. m1 + m2 = m4
B. m1 + m2 = m3
C. m4 is greater than m2
D. m4 is greater than m1
E. m3 is greater than m2
F. m2 + m3 = m4

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