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!