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\treasons

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

(diagram of intersecting lines with angles 1, 2, 3, 4 is present.)

Explanation:

Step1: Identify the relationship between ∠2 and ∠3

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

Step2: Determine the reason for m∠3 = 30

Since m∠2 = 30 and ∠2 ≅ ∠3 (Vertical Angles Theorem), we can conclude that m∠3 = m∠2 = 30. So the reason is "Vertical Angles are Congruent" (or "Vertical Angles Theorem").

Answer:

Vertical Angles are Congruent (or Vertical Angles Theorem)