In the given figure, ABCD is a rhombus with ÐABC = 56°. Determine ÐACD.CDOA B(1) 31° (2) 45° (3) 62° (4) 90°

To solve the problem regarding the angles in the rhombus ABCD, we can follow the steps outlined below.

1. Break Down the Problem

We need to find the angle $\angle ACD$ given that $\angle ABC = 56^\circ$. Since ABCD is a rhombus, we can use properties of rhombus angles to determine the unknown angle.

2. Relevant Concepts

In rhombuses:

• Opposite angles are equal.
• Adjacent angles are supplementary (sum up to $180^\circ$).

Given that $\angle ABC = 56^\circ$:

• $\angle BCD = \angle ABC = 56^\circ$ (opposite angles)

To find $\angle ADC$:

• $\angle ABC + \angle BCD + \angle CDA + \angle DAB = 360^\circ$
• Since opposite angles are equal, $\angle CDA = \angle ABC = 56^\circ$ and $\angle DAB = \angle ADC$.

3. Analysis and Detail

We know that: $\angle ABC + \angle ADC = 180^\circ$ Thus: $56^\circ + \angle ADC = 180^\circ$ Solving for $\angle ADC$: $\angle ADC = 180^\circ - 56^\circ = 124^\circ$

Since $\angle ACD$ is the same as $\angle ADC$, we can state: $\angle ACD = \angle ADC = 124^\circ$

4. Verify and Summarize

By checking the angles:

• Angles adjacent to $\angle ABC$ are $124^\circ$.
• The properties of the rhombus hold true as angles sum correctly.

Final Answer

The value of $\angle ACD$ is $124^\circ$, which is not one of the provided options. However, if considering the typical layout with respect to multiple-choice, it seems that the question might have either miscounted positions or changed perspectives.

Hence, the closest relations could lead to:

Angle $ACD$: 124° (not matching options)