In the given diagram, we are tasked with finding the measure of angle $\angle CAB$.

Known information:

• $AD \perp DC$, meaning that $\angle ADC = 90^\circ$.
• $m\angle C = 57^\circ$ (given).
• $m\angle B = 43^\circ$ (given).

Approach:

1. Understanding the structure:

   • $\triangle ABC$ is the triangle of interest.
   • $AD$ is perpendicular to $DC$, meaning $\triangle ADC$ is a right triangle.

2. Using angle sum property of triangles: In any triangle, the sum of the interior angles is always $180^\circ$. So, in $\triangle ABC$: $m\angle CAB + m\angle B + m\angle C = 180^\circ$ Substituting the given values: $m\angle CAB + 43^\circ + 57^\circ = 180^\circ$ Simplifying: $m\angle CAB + 100^\circ = 180^\circ$ $m\angle CAB = 180^\circ - 100^\circ = 80^\circ$

Thus, the measure of $\angle CAB$ is 80°.

Would you like a detailed breakdown of any of these steps?

Relative questions:

1. How can the angle sum property be used in different types of triangles?
2. Why is $AD \perp DC$ relevant in this problem?
3. How do perpendicular lines affect the angles in a triangle?
4. Can you explain how we find unknown angles in any triangle using the given angles?
5. What other properties of triangles could be useful in solving problems like this?

Tip: Always check if the sum of angles in any triangle adds up to 180° to verify the correctness of your answer.