ABCD is a parallelogram in which ∠BAD = 60° and ∠CBD = 70°. Find the measure of ∠BDC:

Question

Accepted Answer

To determine the measure of ∠BDC in the parallelogram ABCD, we need to use the properties of parallelograms and the given angles.

1. **Properties of a Parallelogram:**
   - Opposite angles are equal.
   - Adjacent angles are supplementary (sum up to 180°).

2. **Given Information:**
   - ∠BAD = 60°
   - ∠CBD = 70°

3. **Explanation:**
   - Since ABCD is a parallelogram, ∠BAD and ∠BCD are supplementary because they are consecutive angles. Therefore, ∠BCD = 180° - ∠BAD = 180° - 60° = 120°.
   - In triangle BCD, we know two angles: ∠CBD = 70° and ∠BCD = 120°.
   - The sum of angles in a triangle is always 180°. Therefore, we can find ∠BDC by subtracting the sum of the known angles from 180°:
     $$
     ∠BDC = 180° - (∠CBD + ∠BCD) = 180° - (70° + 120°) = 180° - 190° = 50°.
     $$

4. **Conclusion:**
   - Therefore, the measure of ∠BDC is 50°, which corresponds to Option 4.

5. **Why Other Options are Incorrect:**
   - **Option 1 (120°):** This is the measure of ∠BCD, not ∠BDC.
   - **Option 2 (70°):** This is the measure of ∠CBD, not ∠BDC.
   - **Option 3 (60°):** This is the measure of ∠BAD, not ∠BDC.

By following the properties of a parallelogram and the rules of angles in a triangle, we have shown that the correct measure of ∠BDC is indeed 50°, supporting Option 4 as the correct answer.