1. **State the problem:** We are given a rhombus ABCD with $m\angle DAB = 110^\circ$. We need to find the measures of all angles in the rhombus and the angles $m\angle DEA$ and $m\angle DAE$ formed by the diagonals intersecting at point E. 2. **Recall properties of a rhombus:** - All sides are equal in length. - Opposite angles are equal. - Adjacent angles are supplementary (sum to $180^\circ$). - Diagonals bisect each other at right angles ($90^\circ$). 3. **Find all angles of the rhombus:** - Given $m\angle DAB = 110^\circ$. - Opposite angle $m\angle BCD = 110^\circ$. - Adjacent angles $m\angle ABC$ and $m\angle CDA$ are supplementary to $110^\circ$: $$m\angle ABC = m\angle CDA = 180^\circ - 110^\circ = 70^\circ$$ 4. **Analyze the diagonals:** - Diagonals intersect at E and bisect each other. - Diagonals are perpendicular, so $m\angle DEA = m\angle DAE = 90^\circ$ divided by 2 because E is the intersection point. 5. **Find $m\angle DEA$ and $m\angle DAE$:** - Since diagonals bisect the angles at vertices, and $m\angle DAB = 110^\circ$, the diagonal splits it into two equal angles: $$m\angle DAE = m\angle BAD / 2 = \frac{110^\circ}{2} = 55^\circ$$ - The diagonals intersect at right angles, so $m\angle DEA$ is the complement of $m\angle DAE$ in the right angle: $$m\angle DEA = 90^\circ - 55^\circ = 35^\circ$$ **Final answers:** - $m\angle DAB = m\angle BCD = 110^\circ$ - $m\angle ABC = m\angle CDA = 70^\circ$ - $m\angle DAE = 55^\circ$ - $m\angle DEA = 35^\circ$