The diagonals AC and BD of a parallelogram ABCD intersect each other at the…

Concept

In a parallelogram, each pair of opposite sides is parallel. A transversal creates equal alternate interior angles; a linear pair sums to 180°, and the three interior angles of a triangle also sum to 180°.

Application

1. Because AD is parallel to BC and AC is a transversal, ∠BCA = ∠DAC = 32°. Since C, O and A are collinear, ∠BCO is also 32°.

2. Since OA and OC are opposite rays, ∠AOB and ∠BOC form a linear pair. Thus ∠BOC = 180° − 70° = 110°.

3. In triangle BOC, ∠OBC = 180° − (110° + 32°) = 38°. Since B, O and D are collinear, ∠OBC = ∠DBC.

Cross-check

The three angles of triangle BOC are 110°, 32° and 38°; their sum is 180°, so all the angle relations are consistent.

Result: ∠DBC = 38°.