Let ABCD be a convex quadrilateral in which angle BAC = 50 deg angle CAD = 60 deg , angle CBD = 30 deg & angle BDC = 2 * S deg . If E is a point of intersection of AC and BD. find angle AEB

Let's solve the problem step by step.

Given:
- ABCD is a convex quadrilateral.
- \(\angle BAC = 50^\circ\)
- \(\angle CAD = 60^\circ\)
- \(\angle CBD = 30^\circ\)
- \(\angle BDC = 2S\)

We are asked to find the angle \(\angle AEB\).

### Step 1: Determine \(\angle ABC\) and \(\angle BCD\)
Using the sum of the angles in triangle \(ABC\), we have:
\[
\angle ABC = 180^\circ - \angle BAC - \angle CBD
\]
Substitute the given values:
\[
\angle ABC = 180^\circ - 50^\circ - 30^\circ = 100^\circ
\]

Similarly, for triangle \(ACD\), the sum of angles gives:
\[
\angle ACD = 180^\circ - \angle CAD - \angle BDC
\]
Substitute the values, and let \(\angle BDC = 2S\):
\[
\angle ACD = 180^\circ - 60^\circ - 2S = 120^\circ - 2S
\]

### Step 2: Use the given information about angles
Now, let's look at the point \(E\), where the diagonals \(AC\) and \(BD\) intersect. By the properties of angles in a convex quadrilateral and intersecting diagonals, we have:
\[
\angle AEB = 180^\circ - (\angle BAC + \angle CBD)
\]
Substituting the given angles:
\[
\angle AEB = 180^\circ - (50^\circ + 30^\circ) = 100^\circ
\]

So, the measure of \(\angle AEB\) is \(100^\circ\).

### Conclusion:
The angle \(\angle AEB\) is \(100^\circ\).

Would you like further details or have any questions about the solution?

Here are 5 related questions you can explore:
1. How does the intersection of diagonals affect the angles in a quadrilateral?
2. What is the significance of convex quadrilaterals in geometry?
3. Can you derive a formula for \(\angle AEB\) if the angles of the quadrilateral are different?
4. How do internal and external angles of quadrilaterals relate to each other?
5. What are the special properties of cyclic quadrilaterals?

**Tip:** Always verify angle calculations using multiple methods, such as the angle sum property and triangle rules, to avoid errors in geometric problems.

Learn how to find angle AEB in a convex quadrilateral ABCD with given angle measurements. This problem involves applying geometric principles to solve for the intersection point of diagonals AC and BD. Suitable for high school geometry students.

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