Solve this question for me---**ID:**
6d99b141
**Geometric Figure Description:**
* **Type:** Geometric figure, a triangle with internal lines and points.
* **Main Elements:**
* **Points:** Labeled D, C, B, E, A.
* **Lines:** Line segment DA is a horizontal base. Point E is on DA, between D and A. Point C is above DA and to the left. Point B is above DA and to the right. Line segments are drawn connecting D to C, C to A, A to B, B to D, C to B, and B to E. Line segment CB intersects DA at point E.
* **Triangles:** Triangles such as DAB, DCB, ACB, ACD, BCE, ACE, ABE, DBE are formed.
* **Angles:** An angle labeled with "x°" is shown near point E, formed by line segments AE and BE. This angle is ∠AEB.
* **Relative Position:** Points D, E, and A are collinear on a horizontal line. C and B are located above the line DA. Line segment CB passes through point E.
* **Labels and Annotations:** Points are labeled D, C, B, E, A. An angle is labeled x°.
**Note:**
Figure not drawn to scale.
**Question Stem:**
In the figure, AC = CD. The measure of angle EBC is 45°, and the measure of angle ACD is 104°. What is the value of x?
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Let's analyze this geometry problem step by step. We have a figure with points D, C, B, E, and A, where AC equals CD, angle EBC is 45 degrees, and angle ACD is 104 degrees. We need to find the value of x, which is angle AEB.
Since AC equals CD, triangle ACD is isosceles. In an isosceles triangle, the base angles are equal, so angle CAD equals angle CDA. Let's call each of these angles alpha. Using the angle sum property of triangles, we have 2 alpha plus 104 degrees equals 180 degrees. Solving this equation, we get alpha equals 38 degrees.
Now let's analyze triangle BCE. We know angle EBC is 45 degrees from the given information. Angle BCE is the same as angle ACD, which is 104 degrees. Using the angle sum property in triangle BCE, we have angle BEC plus 45 degrees plus 104 degrees equals 180 degrees. Therefore, angle BEC equals 31 degrees.
The key insight is that angles AEB and BEC form a linear pair on line DA. By the linear pair property, these two angles must sum to 180 degrees. Since we found that angle BEC equals 31 degrees, we can write x plus 31 degrees equals 180 degrees. Solving this equation, we get x equals 149 degrees.
Let's summarize our solution. First, we used the fact that AC equals CD to determine that triangle ACD is isosceles, giving us angles CAD and CDA equal to 38 degrees each. Then we found angle BEC equals 31 degrees using the angle sum in triangle BCE. Finally, we applied the linear pair property to find that x equals 149 degrees. Therefore, the value of x is 149 degrees.