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The diagram provided shows part of a bridge design with triangles formed by the sides of the model. In the triangle at the center of the bridge, we are given two angles, 40° and 50°, with two unknown angles labeled \( x^\circ \) and \( y^\circ \). The student claims that both \( x \) and \( y \) are 65°, and we need to explain why this is incorrect.

### Analyzing the Situation:

1. **Sum of angles in a triangle:** The angles in any triangle must sum to 180°. So, for the central triangle, we can set up the equation:
   \[
   x^\circ + 40^\circ + 50^\circ = 180^\circ
   \]
   Simplifying:
   \[
   x^\circ = 180^\circ - 40^\circ - 50^\circ = 90^\circ
   \]
   Therefore, \( x = 90^\circ \), not 65° as the student claims.

2. **Using the value of \( x \) to find \( y \):** The angles \( x^\circ \) and \( y^\circ \) are part of a straight line, meaning they are supplementary (i.e., their sum is 180°). So, we have:
   \[
   x^\circ + y^\circ = 180^\circ
   \]
   Since \( x = 90^\circ \), this gives us:
   \[
   90^\circ + y^\circ = 180^\circ
   \]
   Solving for \( y \):
   \[
   y^\circ = 180^\circ - 90^\circ = 90^\circ
   \]

### Conclusion:
The student is incorrect because the actual values of \( x \) and \( y \) are 90°, not 65°. This mistake likely stems from misunderstanding the sum of angles in a triangle and supplementary angles on a straight line.

Would you like further details or explanations?

Here are 5 related questions:
1. What are supplementary angles and how do they apply in this context?
2. How do you find the third angle of a triangle if two angles are given?
3. Why must the angles in a triangle sum to 180°?
4. How would the solution change if the angles were different?
5. What is the relationship between angles on a straight line?

**Tip:** Always double-check whether you're working with a triangle or a straight line when solving for angles!

A student begins to build a model of a downtown bridge in the city where he lives. He measures some of the angles formed by the sides of the model. The student says both x and y are 65°. Why is he incorrect?

This problem explains why the angles x and y cannot both be 65° in a triangle and straight-line bridge model. It covers key geometry concepts, including the angle sum in a triangle (180°) and the use of supplementary angles. Suitable for students in Grades 6-8.

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