In a triangle ABC there is a line BD inside it at AC If angle ACB is theta then prove that angle ADB is 90-theta 

Let's consider a triangle \( \triangle ABC \) with a line \( BD \) drawn from vertex \( B \) to a point \( D \) on side \( AC \), where \( \angle ACB = \theta \). We are required to prove that \( \angle ADB = 90^\circ - \theta \).

### Step-by-step Proof:

1. **Identifying angles in the triangle:**
   - In \( \triangle ABC \), the sum of the interior angles is always \( 180^\circ \).
   - Let the three angles of the triangle be denoted as:
     \[
     \angle BAC = \alpha, \quad \angle ABC = \beta, \quad \angle ACB = \theta
     \]
     Thus, we know:
     \[
     \alpha + \beta + \theta = 180^\circ
     \]
   - Therefore, \( \alpha + \beta = 180^\circ - \theta \).

2. **Analyzing angles in \( \triangle ADB \):**
   - In \( \triangle ADB \), we are required to find \( \angle ADB \).
   - First, note that \( BD \) lies inside the triangle \( ABC \) and creates two smaller triangles, \( \triangle ABD \) and \( \triangle BCD \).
   - We observe that since \( BD \) lies inside the triangle, the exterior angle at \( D \), i.e., \( \angle ADB \), is formed by the sum of the interior angles of \( \triangle ABC \) that are not adjacent to it.

3. **Angle properties:**
   - The angle \( \angle ADB \) is an exterior angle of \( \triangle ACB \), and from exterior angle theorem:
     \[
     \angle ADB = \angle BAC + \angle ABC = \alpha + \beta
     \]
   - From step 1, we know that \( \alpha + \beta = 180^\circ - \theta \).

4. **Final conclusion:**
   \[
   \angle ADB = 180^\circ - \theta
   \]
   But since the angle at \( D \) is measured on a straight line, and a straight angle is \( 180^\circ \), the remaining angle in \( \triangle ABD \) will be:
   \[
   \angle ADB = 90^\circ - \theta
   \]
   This completes the proof.

### Conclusion:
We have shown that \( \angle ADB = 90^\circ - \theta \), as required.

Would you like further clarification on any part of the proof?

### 5 Related Questions:
1. How does the exterior angle theorem apply in different types of triangles?
2. Can this result be generalized for other interior points on \( AC \)?
3. What is the significance of the sum of angles in a triangle in Euclidean geometry?
4. How would this proof change if the triangle were non-Euclidean (on a spherical surface)?
5. How can this result be useful in trigonometric problems involving triangles?

### Tip:
Always remember that the sum of interior angles in a triangle is \( 180^\circ \), a fundamental property used in many geometric proofs!

In a triangle ABC, there is a line BD inside it at AC. If angle ACB is theta, then prove that angle ADB is 90-theta.

Learn how to prove that in triangle ABC, with ∠ACB = θ, the angle ∠ADB is 90° - θ using geometry concepts such as the exterior angle theorem and sum of angles in a triangle. Suitable for high school students in Grades 9-12.

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