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!
