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Use the angle relationships in the diagram below to determine the value of each variable. Name which geometric relationships you used.
A triangle has angles in degrees, B, 52 + C, & 37. One of two parallel lines cuts the triangle into 2 internal triangles. The top triangle has angles, B, top, 52, right, & unknown, left. Angle, d is the exterior angle of unknown. The other triangle has angles, c, 37, & unknown.  But angle d is opposite the unknown angle. The second parallel line is at the vertex of the 37 angle along with angle, a, between the parallel line & the triangle side.  Angle 63 is the angle opposite the sum of 37 & a.

Since the highlighted angles are vertical angles, and vertical angles are equal,

is an alternate interior angle to . Recall that alternate interior angles are equal.

Use the Triangle Angle Sum Theorem to calculate .

To find , notice that it is the exterior angle in the small triangle that contains and . What relationship do those angles have?

The diagram shows that 63 degrees is the same as 37 degrees + a.