Use the following diagram for questions 1-2.

Question

Use the Vertical Angles Theorem to prove the following:

Given: ∠1 ≅ ∠2

Prove: ∠1 ≅ ∠2 ≅ ∠3 ≅ ∠4

1. 2. How can you prove ∠1 ≅ ∠2 ≅ ∠3 ≅ ∠4 without using the vertical angles theorem? Explain.

Answer 1

∠1 and ∠4 are a linear pair therefore m∠1 + m∠4 = 180

∠2 and ∠3 are a linear pair therefore m∠2 + m∠3 = 180

Using substitution then m∠1 + m∠4 = m∠2 + m∠3

But since ∠1 ≅ ∠2 then m∠1 = m∠2

Using substitution then m∠1 + m∠4 = m∠1 + m∠3

Using the subtraction property of equality we can subtract m∠1 from both sides giving:

m∠4 = m∠3

If the measures are equal then ∠4 ≅ ∠3

Then you can repeat the same sort of process:

∠1 and ∠2 are a linear pair therefore m∠1 + m∠2 = 180

∠1 and ∠4 are a linear pair therefore m∠1 + m∠4 = 180

m∠1 + m∠2 = m∠1 + m∠4

m∠2 = m∠4

∠2 ≅ ∠4

So if ∠1 ≅ ∠2 and ∠2 ≅ ∠4 and ∠4 ≅ ∠3 then ∠1 ≅ ∠2 ≅ ∠3 ≅ ∠4