Question

part 6 of 7
write a two-column proof for the following.
given: ( mangle 2 = 30 )
( mangle 1 = 2mangle 2 )
prove: ( mangle 3 + mangle 4 = 90 )

(do not include the degree symbol in your answers.)
statements reasons

1. ( mangle 2 = 30 ) | 1) given
2. ( mangle 1 = 2mangle 2 ) | 2) given
3. ( mangle 1 = 2( 30 ) ) | 3) substitution
4. ( mangle 1 = 60 ) | 4) simplification
5. ( mangle 3 = 30 ) | 5) vertical angles theorem
6. ( mangle 3 + mangle 4 = 30 + mangle 4 ) | 6) addition property of equality
7. ( mangle 4 = 60 ) | 7) vertical angles theorem
8. ( mangle 3 + mangle 4 = 30 + square ) | 8) substitution

Explanation:

Step1: Identify the value of \( m\angle4 \)

From step 7, we know that \( m\angle4 = 60 \).

Step2: Substitute \( m\angle4 \) into the equation

In step 8, we have \( m\angle3 + m\angle4 = 30 + \square \). Using the substitution property (since \( m\angle4 = 60 \)), we substitute \( 60 \) for \( m\angle4 \) in the equation. So the square should be filled with \( 60 \).

Answer:

60