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.

Question

Use the properties of exterior angles of triangles to find angle b
.

(1 point)
Responses

38°
38 degrees

52°
52 degrees

232°
232 degrees

92°

Answer 1

In a triangle, the exterior angle is equal to the sum of the two remote interior angles. In this case, the exterior angle is \(142^\circ\), and the two remote interior angles are \(a\) and \(b\).

The relationship can be expressed with the formula:

\[ \text{Exterior angle} = \text{Angle } a + \text{Angle } b \] Given that:

\[ 142^\circ = a + b \]

We also know that in a right triangle, the sum of angles \(a\), \(b\), and the right angle (which is \(90^\circ\)) is:

\[ a + b + 90^\circ = 180^\circ \]

This simplifies to:

\[ a + b = 90^\circ \]

Now, we already have the equation from the exterior angle:

1. \(a + b = 142^\circ\)
2. \(a + b = 90^\circ\)

These two equations cannot hold true simultaneously since the value of \(a + b\) cannot be both \(142^\circ\) and \(90^\circ\). This means we need to calculate one of the angles based on the values we've gotten.

Let's use the fact that in a right triangle, angle \(a\) and angle \(b\) together must equal \(90^\circ\) (since it's a right triangle).

Thus from the two equations, since \(a + b = 90^\circ\): Using:

\[ 142 = 90 + b \]

Rearranging gives:

\[ b = 142 - 90 = 52^\circ \]

Thus, angle \(b\) is:

52°