We want to find the angle measures of the numbered angles in the diagram. To do this we will first find the angle measure of one of the angles that forms a linear pair with the angle that measures 63^(∘). Let's call the angle A.
Measure of ∠A
A linear pair is formed by ∠A and the angle that measures 63. Angles that form a linear pair add up to a measure of 180.
63+ m ∠A =180 ⇔ m ∠A = 117
Measure of ∠2
Next, notice that ∠A, ∠2, and the angle that measures 37^(∘) are three interior angles of a triangle. Therefore, by the Triangle Angle-Sum Theorem their measures add up to 180.
m ∠A + m ∠2 + 37 =180
We know that m ∠A=117, so we can find m ∠2 by substituting 117 for m ∠A and solving the equation.
We see that ∠2 and ∠3 are alternate interior angles. Opposite sides of a parallelogram are parallel, so by the Alternate Interior Angles Theorem we can conclude that they are congruent.
∠2 ≅ ∠3
The definition of congruent angles tells us that their measures are equal, so m ∠2 = m ∠3. We already know that m ∠2 = 26, so we can conclude that m ∠3 = 26.
Measure of ∠1
Also ∠1 and the angle that measures 37^(∘) are alternate interior angles in the parallelogram. Therefore, the angles must be congruent.
∠1 ≅ 37 ^(∘)
Their measures are equal, so m ∠1 = 37^(∘). Let's gather the results we found!
