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WCHS 2024 HS: Geometry [M] (Prescriptive) (GP)/ 5:Congruent Triangles: Part 2
4. In an A-frame house, the two congruent sides extend from the ground to form a 34° angle at the peak. What angle does each side form with the ground?
目
C
146°

Verified Answer :

GinnyAnswer

In this problem, we are dealing with an isosceles triangle formed by the two congruent sides of an A-frame house. Let's break down the solution step-by-step:

1. Understanding the Triangle:
- We have an isosceles triangle where the two congruent sides form an angle of 34° at the peak (the top vertex of the triangle).

2. Sum of Angles in a Triangle:
- The sum of all internal angles in any triangle is 180°.

3. Identifying the Known Angle:
- The angle at the peak of the triangle is given as 34°.

4. Calculating the Base Angles:
- Since the triangle is isosceles, the two angles at the base (i.e., where the sides meet the ground) are congruent.
- Let each of these base angles be represented as \(\( x \)\).

5. Forming an Equation:
- We use the fact that the sum of the angles in a triangle is 180°:
\(\[ x + x + 34° = 180° \]\)
- Simplifying this equation:
\(\[ 2x + 34° = 180° \]\)

6. Solving for \(\( x \)\):
- Subtract 34° from both sides of the equation:
\(\[ 2x = 180° - 34° \]\)
- This simplifies to:
\(\[ 2x = 146° \]\)

7. Divide Both Sides by 2:
- To find \(\( x \)\), divide 146° by 2:
\(\[ x = \frac{146°}{2} \]\)
- Thus, \(\( x = 73° \)\).

Therefore, each side of the A-frame house forms a 73° angle with the ground. The angles are \(\( 34° \)\) at the peak and \(\( 73° \)\) at each base with the ground.

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