Let's mark the congruent angles in the given diagram.
We have that ∠XCA and ∠ACB are supplementary, and also that ∠YBA and ∠ABC. By the definition of supplementary angles, we get two important equations.
m∠XCA + m ∠ACB &= 180^(∘)
m∠YBA + m ∠ABC &= 180^(∘)
Since ∠ACB ≅ ∠ABC, we have that m ∠ACB = m ∠ABC.
m∠XCA + m ∠ABC &= 180^(∘)
m∠YBA + m ∠ABC &= 180^(∘)
Next, we subtract the second equation from the first one.
m∠XCA + m ∠ABC &= 180^(∘)
^- m∠YBA + m ∠ABC &= 180^(∘)
m∠XCA - m∠YBA &= 0
The latter equation tells us that m∠XCA = m∠YBA and so, ∠XCA ≅ ∠YBA.
We will summarize the proof in the following two-column table.
Statements
Reasons
1.
∠ACB ≅ ∠ABC
1.
Given
2.
m∠ACB = m∠ABC
2.
Definition of Congruent Polygons
3.
∠ACB and ∠XCA, and ∠ABC and ∠YBA, are supplementary angles
