Examining the diagram, we see that the sum of ∠a and 37^(∘) makes a vertical angle to the 63^(∘) angle. By the Vertical Angles Theorem, we know that these angles are congruent.
Since the two angles are congruent, we can equate their measures and solve the resulting equation for m∠a.
37^(∘)+m∠a=63^(∘) ⇔ m∠a=26^(∘)Next, we will identify the measure of ∠c. Notice that ∠a and ∠c are alternate interior angles and since the two lines cut by the transversal are parallel, we know by the Alternate Interior Angles Theorem that they must be congruent.
Now that we know the measure of ∠c, we can find the measure of ∠b. For this purpose, we will isolate one of the triangles we can see in the diagram.
Since the angles of a triangle sum to 180^(∘), we can write and solve an equation that involves m∠b.
When we know the measure of ∠b, we have enough information to find ∠d. For this purpose, we will isolate a second triangle we can see in the diagram.
Notice that ∠d is exterior angle to the triangle we see in the diagram. By the Exterior Angles Theorem, we know that ∠d equals the sum of the triangles non-adjacent angles.
m∠d=65^(∘)+52^(∘) ⇔ m∠d=117^(∘)
