9. The diagram below shows \( \triangle A B D \), with \( \overrightarrow{A B C}, \overline{B E} \perp \overline{A D} \), and \( \angle E B D \cong \angle C B D \). If \( \mathrm{m} \angle A B E=52 \), what is m \( \angle D \) ?

To solve for \( m \angle D \) in triangle \( \triangle ABD \), we start by analyzing the given information. 1. We know that \( \overline{B E} \perp \overline{A D} \). This means that \( \angle A B E \) is a right angle, which implies: \[ m \angle A B E = 90^\circ \] 2. We are given that \( m \angle A B E = 52^\circ \). This seems contradictory because \( \angle A B E \) should be \( 90^\circ \). However, we can interpret this as \( \angle A B E \) being the angle between line segments \( \overline{A B} \) and \( \overline{B E} \). 3. Since \( \angle E B D \cong \angle C B D \), we can denote both angles as \( x \). 4. In triangle \( \triangle A B D \), we can apply the angle sum property of triangles, which states that the sum of the angles in a triangle is \( 180^\circ \): \[ m \angle A B D + m \angle A B E + m \angle D = 180^\circ \] 5. We know: - \( m \angle A B E = 52^\circ \) - \( m \angle A B D = x + 90^\circ \) (since \( \overline{B E} \perp \overline{A D} \)) 6. Therefore, we can substitute into the angle sum equation: \[ (x + 90^\circ) + 52^\circ + m \angle D = 180^\circ \] 7. Simplifying this gives: \[ x + 142^\circ + m \angle D = 180^\circ \] 8. Rearranging to solve for \( m \angle D \): \[ m \angle D = 180^\circ - (x + 142^\circ) \] \[ m \angle D = 38^\circ - x \] 9. Since \( \angle E B D \cong \angle C B D \), we can conclude that \( x \) is equal to \( 52^\circ \) (the angle \( A B E \)). Thus: \[ m \angle D = 38^\circ - 52^\circ \] \[ m \angle D = 38^\circ \] Therefore, the measure of angle \( D \) is: \[ \boxed{38^\circ} \]