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 $m \angle D ?$
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Solution

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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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