Question

write a two - column proof for the following
given: ( mangle2 = 30 )
( mangle1 = 2mangle2 )
prove: ( mangle3 + mangle4 = 90 )
(do not include the degree symbol in your answers.)

statements\t\t\t\treasons

1. ( mangle2 = 30 )\t\t\t1) given
2. ( mangle1 = 2mangle2 )\t\t2) given
3. ( mangle1 = 2(30) )\t\t3) substitution
4. ( mangle1 = 60 )\t\t\t4) simplification
5. ( mangle3 = 30 )\t\t\t5) vertical angles theorem
6. ( mangle3 + mangle4 = 30 + mangle4 )\t6) addition property of equality
7. ( mangle4 = 60 )\t\t\t7) vertical angles theorem
8. ( mangle3 + mangle4 = 30 + square )\t8) substitution

Explanation:

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

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

Step2: Substitute into the equation

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

Answer:

60