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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

232°
232 degrees

92°
92 degrees

38°
38 degrees

52°
52 degrees
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1 answer

To find angle \( b \) using the properties of exterior angles of triangles, we use the fact that the exterior angle is equal to the sum of the two opposite interior angles.

In this case, the exterior angle is \( 142^\circ \), and it is equal to the sum of angles \( a \) and \( b \). We can express this mathematically as:

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

Since the triangle is a right triangle, we also know that the sum of the angles in the triangle is \( 180^\circ \). Therefore, we have:

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

This simplifies to:

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

Now, we have two equations:

  1. \( a + b = 90^\circ \) (from the right triangle)
  2. \( a + b = 142^\circ \) (from the exterior angle)

We can set these equal to each other and solve for \( a \):

\[ 142^\circ = 90^\circ + a \]

Rearranging gives:

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

Now that we know \( a \), we can find \( b \) using the first equation:

\[ a + b = 90^\circ \] \[ 52^\circ + b = 90^\circ \] \[ b = 90^\circ - 52^\circ = 38^\circ \]

Thus, the value of angle \( b \) is \( 38^\circ \). So, the correct response is:

38°