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In this problem, we are given a diagram with angles, and the task is to find the measure of ∠1. We are provided with the following information:

- \( m \angle 7 = 95.3^\circ \)
- \( m \angle 5 = 32.4^\circ \)

Let's analyze the diagram. The structure consists of two intersecting triangles, forming a shape similar to an "X." Here are a few geometric properties we can use to solve the problem:

1. **Vertical angles**: Opposite angles formed by intersecting lines are congruent. For example:
   - \( \angle 1 = \angle 6 \)
   - \( \angle 3 = \angle 7 \)
   - \( \angle 2 = \angle 5 \)

2. **Triangle angle sum**: The sum of the angles in any triangle is always 180°.

From the diagram:
- In triangle 527 (upper triangle), the sum of the interior angles is \( m \angle 2 + m \angle 5 + m \angle 7 = 180^\circ \).
  
We know:
- \( m \angle 5 = 32.4^\circ \)
- \( m \angle 7 = 95.3^\circ \)

So, \( m \angle 2 = 180^\circ - (95.3^\circ + 32.4^\circ) \).

Let me calculate that for you.We find that \( m \angle 2 = 52.3^\circ \).

Since \( \angle 1 \) and \( \angle 2 \) are vertical angles, they are congruent. Therefore, \( m \angle 1 = 52.3^\circ \).

Do you want further details or have any questions? Here are five related questions for further exploration:

1. How do vertical angles help in solving angle problems?
2. What is the significance of the triangle angle sum property?
3. Can you find the measure of other angles in the diagram using the same approach?
4. How would this change if the figure had parallel lines?
5. What other theorems could be applied to this type of diagram?

**Tip**: Always remember that vertical angles are congruent when two lines intersect. This can simplify many geometry problems.

In the picture below, m∠7 = 95.3°, m∠5 = 32.4°, find the m∠1. (Do not put degrees for your answer because that will be assumed. Picture is not drawn to scale!)

Solve a geometry problem where m∠7 = 95.3° and m∠5 = 32.4° to find m∠1 using vertical angles and triangle angle sum theorems. Suitable for students in Grades 8-10.

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