Question

the angle bisector pf angle ACD in rhombus ABCD makes a 64 degrees angle with diagonal BD. Find the measure of angle BAD.

Answer 1

In a rhombus, the diagonals bisect each other at right angles and also bisect the angles of the rhombus. Let's denote the vertices of the rhombus as follows:

- A, B, C, D are the vertices of the rhombus.
- The diagonals intersect at point O.

Given that the angle bisector of angle ACD makes a 64-degree angle with diagonal BD, we need to find the measure of angle BAD.

1. **Analyze Angles in Rhombus**:
- Since ABCD is a rhombus, angle ACD is equal to angle ABC. We denote these angles as \( \angle ACD = \angle ABC = y \).
- Therefore, \( \angle CAB = \angle ADC = x \).

2. **Properties of a Rhombus**:
- The sum of the angles in quadrilateral ABCD is \( 360^\circ \):
\[
2x + 2y = 360^\circ \quad \Rightarrow \quad x + y = 180^\circ
\]

3. **Relations of the Angles**:
- Since the diagonals bisect the angles, angle ACD is bisected by the angle bisector resulting in two angles: \( \angle ACE \) and \( \angle DCE \), where \( E \) is the point where the angle bisector intersects diagonal AC.
- Specifically, each of these angles (from \( \angle ACD \)) measures \( \frac{y}{2} \).

4. **Finding angle BAD**:
- The angle bisector of \( \angle ACD \) creates an angle of \( 64^\circ \) with diagonal BD.
- Since diagonal BD bisects angle ACD also, we can express the relationship as:
\[
\angle ACO + 64^\circ = 90^\circ
\]
- Therefore, we can write:
\[
\angle ACO = 90^\circ - 64^\circ = 26^\circ
\]

5. **Extracting Values**:
- Now, since the whole angle ACD is equal to \( y \), and the diagonal bisects it, we have:
\[
\frac{y}{2} = 26^\circ \quad \Rightarrow \quad y = 52^\circ
\]
- Using the relationship \( x + y = 180^\circ \):
\[
x + 52^\circ = 180^\circ \quad \Rightarrow \quad x = 180^\circ - 52^\circ = 128^\circ
\]

6. **Conclusion**:
- The measure of angle BAD, which is \( x \), is therefore:
\[
\boxed{128^\circ}
\]