To solve for the measure of $\angle BAD$ in the triangle shown, we can follow these steps:

Step 1: Understand the problem

• We are tasked to find the measure of $\angle BAD$, which is part of $\triangle ABC$.
• Given:
  • $\angle A = 90^\circ$ (right angle at A).
  • $\angle DCB = 65^\circ$.
  • $\angle BDC = 50^\circ$.

Step 2: Find $\angle BCD$ in $\triangle BCD$

In $\triangle BCD$, the sum of interior angles is $180^\circ$: $\angle BCD = 180^\circ - \angle DCB - \angle BDC$ $\angle BCD = 180^\circ - 65^\circ - 50^\circ = 65^\circ$

Step 3: Use angles in $\triangle ABC$

In $\triangle ABC$:

• $\angle BAD + \angle BCD + \angle A = 180^\circ$.

Substitute known values: $\angle BAD + 65^\circ + 90^\circ = 180^\circ$ $\angle BAD = 180^\circ - 155^\circ = 25^\circ$

Final Answer:

$\angle BAD = 25^\circ$

Let me know if you'd like a detailed explanation of any step!

Related Questions:

1. How do you find missing angles in a triangle with different given angles?
2. What is the triangle sum property and how is it used in geometry?
3. Why is a right angle always $90^\circ$?
4. How does knowing one triangle's angle help in solving for another?
5. What are supplementary angles, and how are they applied here?

Tip: Always verify that the sum of angles in a triangle equals $180^\circ$ to confirm your results!