https://file.aiquickdraw.com/mathbot/file/1725682429217fdzjf9p3.jpg

The problem in the image asks to calculate the angle \( x \), given the following conditions:
- The triangle is divided into two smaller triangles by the height.
- \( \overline{AB} = \overline{AC} \), meaning that triangle \( ABC \) is isosceles.
- \( \overline{AM} = \overline{MC} \), meaning that \( M \) is the midpoint of \( \overline{BC} \).
- The angle \( \angle AMC = \overline{AM}C \), indicating that angle \( AMC \) and \( AMB \) are congruent.
- The angle \( \angle BMA = 20^\circ \).

Let me proceed with solving the problem.

### Step 1: Set up the known angles and relationships.
- Since \( \triangle ABC \) is isosceles, we know that \( \angle ABC = \angle ACB \).
- The problem states \( M \) is the midpoint, so triangles \( \triangle AMB \) and \( \triangle AMC \) are congruent.
- \( \angle BMA = 20^\circ \).

### Step 2: Calculate angle \( x \).
Since the sum of angles in a triangle is always \( 180^\circ \), we can calculate \( \angle A \) and use symmetry of the isosceles triangle to find angle \( x \).

I'll now calculate the angles.The angle \( x \), which represents \( \angle BAC \), is \( 90^\circ \).

If you need further details or have any questions, feel free to ask!

Here are 5 related questions for further exploration:
1. What properties of an isosceles triangle were used to solve for the angle \( x \)?
2. How does the midpoint \( M \) divide the triangle \( ABC \)?
3. What is the significance of the equal angles in triangle \( ABC \)?
4. Can you prove that the sum of angles in any triangle is always \( 180^\circ \)?
5. How would the solution change if the given angle \( \angle BMA \) were different?

**Tip:** Always check the properties of triangles (e.g., isosceles, equilateral) as they often simplify complex angle calculations.

In the shown diagram, if AB = AC and AM = MC, calculate angle x.

Solve the geometry problem involving an isosceles triangle where AB = AC and AM = MC. Learn how to calculate the unknown angle x using the isosceles triangle theorem and the angle sum rule. Suitable for students in Grades 8-10.

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