There are several ways we can get from the given measure of ∠ 1 to the measure of ∠ 14, which is what we are interested in. We will follow the following route.
m∠ 1
Step 1 ⟶ m∠ 7
Step 2 ⟶ m∠ 6
Step 3 ⟶ m∠ 14
Let's look at the steps.
According to the Vertical Angles Theorem, ∠ 1 and ∠ 7 are congruent.
∠ 1≅ ∠ 7
Congruent angles have the same measure.
m ∠ 1=m ∠ 7
It is given that m ∠ 1=123, so ∠ 7 has the same measure.
m ∠ 7=123
Step 2
Angles ∠ 7 and ∠ 6 are consecutive interior angles.
According to the Consecutive Interior Angles Theorem, ∠ 7 and ∠ 6 are supplementary.
The measure of supplementary angles add to 180.
m ∠ 7+m ∠ 6=180
We know from Step 1 that m ∠ 7=123. Subtracting this from 180 gives the measure of ∠ 6.
m ∠ 6=180-123=57
According to the Corresponding Angles Postulate, ∠ 6 and ∠ 14 are congruent.
∠ 6≅ ∠ 14
Congruent angles have the same measure.
m ∠ 6=m ∠ 14
We know from Step 2 that m ∠ 6=57, so ∠ 14 has the same measure.
m ∠ 14=57
If in the figure m∠ 1=123, then the measure of ∠ 14 is 57.
Summary of Used Theorems
The following table is a summary of the claims and theorems used in finding the measure of ∠ 14.
