To solve for $m \angle D$ in the diagram, let's follow these steps:

Given Information:

1. $BE \perp BD$, which means $\angle EBD = 90^\circ$.
2. $AC \cong BC$, implying that $\triangle ABC$ is isosceles.
3. $m \angle A = 67^\circ$.

Step 1: Calculate Base Angles of $\triangle ABC$

Since $AC \cong BC$ and $\triangle ABC$ is isosceles, the base angles $\angle B$ and $\angle C$ are equal. Using the angle sum property of a triangle: $\angle B = \angle C = \frac{180^\circ - 67^\circ}{2} = \frac{113^\circ}{2} = 56.5^\circ$

Step 2: Consider $\triangle EBD$

Given that $BE$ is perpendicular to $BD$, we know: $\angle EBD = 90^\circ$ Since $\angle B$ was calculated as $56.5^\circ$, and $D$ is an external point with respect to $\triangle ABC$, $\angle D$ forms an exterior angle with $\angle C$ and the line $BD$.

Step 3: Calculate $m \angle D$

$\angle D$ is formed by the extension of side $BC$ of $\triangle ABC$ and the angle adjacent to it. Using the external angle theorem: $\angle D = \angle A + \angle C = 67^\circ + 56.5^\circ = 123.5^\circ$

Final Answer:

$m \angle D = 123.5^\circ$

This is the measure of $m \angle D$ based on the provided diagram and the given relationships.

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Would you like more details on any specific part of the calculation or a breakdown of the geometric theorems used?

Related Questions:

1. What is the external angle theorem, and how does it apply to this problem?
2. How can the angle sum property of a triangle help in solving other angle-related problems?
3. What are the properties of an isosceles triangle, and how do they influence angle calculations?
4. How do perpendicular lines relate to angle calculations in a triangle?
5. Can this problem be solved using alternate interior angles? If yes, how?

Tip:

Always cross-check your angle calculations with the properties of the triangle and any known geometric theorems, especially when dealing with isosceles triangles and perpendicular lines.