Questions: The diagram below shows triangle ABD, with vector ABC , BE perpendicular to AD, and angle EBD congruent to angle CBD. If m angle ABE=52, what is m angle D?

The diagram below shows triangle ABD, with vector ABC , BE perpendicular to AD, and angle EBD congruent to angle CBD.

If m angle ABE=52, what is m angle D?

Transcript text: 6. 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 $m \angle A B E=52$, what is $m \angle D ?$

Solution

Solution Steps

Step 1: Identify Given Information

$ \triangle ABD $ is given.
$ \overline{BE} \perp \overline{AD} $, meaning $ \angle ABE $ is a right angle.
$ \angle EBD \cong \angle CBD $.
$ m\angle ABE = 52^\circ $.

Step 2: Determine $ m\angle EBD $

Since $ \angle ABE $ is a right angle, $ m\angle ABE = 90^\circ $.

Step 3: Calculate $ m\angle EBD $

Given $ m\angle ABE = 52^\circ $, we can find $ m\angle EBD $ by subtracting from 90°: \[ m\angle EBD = 90^\circ - 52^\circ = 38^\circ \]

Step 4: Use Congruent Angles

Since $ \angle EBD \cong \angle CBD $, $ m\angle CBD = 38^\circ $.

Step 5: Calculate $ m\angle D $

In $ \triangle ABD $, the sum of the angles is 180°: \[ m\angle D = 180^\circ - m\angle A - m\angle B \] Given $ m\angle A = 52^\circ $ and $ m\angle B = 38^\circ $: \[ m\angle D = 180^\circ - 52^\circ - 38^\circ = 90^\circ \]

Final Answer

\[ m\angle D = 90^\circ \]

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