Question

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

1. ( mangle2 = 30 ) \t1)
2. ( mangle1 = 2mangle2 ) \t2)
3. ( mangle1 = 2(30) ) \t3)
4. ( mangle1 = 60 ) \t4)
5. ( mangle3 = 30 ) \t5)

vertical angles theorem
addition property of equality
subtraction property of equality
definition of congruent angles

Explanation:

Step1: Identify the relationship between ∠2 and ∠3

∠2 and ∠3 are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent, which means their measures are equal.

Step2: Apply the Vertical Angles Theorem

Since ∠2 and ∠3 are vertical angles, \( m\angle 2 = m\angle 3 \). We know \( m\angle 2 = 30 \), so by the Vertical Angles Theorem, \( m\angle 3 = 30 \).

Answer:

Vertical Angles Theorem