Question

se the image to answer the question.



An illustration shows a right triangle. The right angle is at the left; the top angle is labeled a; and the angle on the right is labeled b. A line extending outward from angle a forms an exterior angle of 142 degrees.
Use the properties of exterior angles of triangles to find angle b
.

(1 point)
Responses

52°
52 degrees

38°
38 degrees

92°
92 degrees

232°

Answer 1

To find the measure of angle \( b \) in the given right triangle using the properties of exterior angles, we can use the following steps:

1. **Understand Exterior Angles**: The measure of an exterior angle is equal to the sum of the measures of the two opposite interior angles. In this case, the exterior angle (142°) is equal to the sum of angle \( a \) and angle \( b \).

2. **Identify Angle Relationships**: Since it’s a right triangle, one of the angles is 90°. Let’s denote the angles as follows:
- The right angle: \( 90° \)
- Angle \( a \)
- Angle \( b \)

We know that:
\[
a + b + 90° = 180°
\]
Therefore:
\[
a + b = 90°
\]

3. **Set up the Equation**: From the properties of exterior angles:
\[
\text{Exterior angle} = a + b
\]
Given that the exterior angle is 142°:
\[
142° = a + b
\]

4. **Solve the System of Equations**:
We have two equations:
- \( a + b = 90° \)
- \( a + b = 142° \)

But since both equal \( a + b \) we have a contradiction unless we rearrange and think carefully:
If we isolate \( b \):
\[
b = 142° - a
\]

But we know \( a + b = 90° \):
Rearranging this gives us \( b = 90° - a \).
Now replace \( b \) with \( 90° - a \) in the exterior angle equation:
\[
142° = a + (90° - a)
\]
This means:
\[
142° = 90°
\]
Which leads us to realize that if we solely think through our angles.
To find angle \( b \) from angle \( a \):
Assuming (and that \( 90 - b = \text{some exterior angle}\)):
We isolate:
\[
b = 142° - 90°
\]
Thus
\[
b = 52°
\]

So, angle \( b \) is **52 degrees**.

**Final Answer:** 52° (52 degrees)