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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

52°
52 degrees

38°

1 answer

To find angle \( b \), we can use the property of exterior angles in a triangle. The exterior angle is equal to the sum of the two opposite interior angles.

Given:

  • The exterior angle is \( 142° \)
  • The two interior angles opposite to the exterior angle are \( a \) and \( b \)

Using the relationship: \[ \text{Exterior Angle} = \text{Interior Angle } a + \text{Interior Angle } b \] This can be written as: \[ 142° = a + b \]

Since the triangle is a right triangle, we also know that: \[ a + b + 90° = 180° \] This implies: \[ a + b = 90° \]

Now we have two equations:

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

Since angle \( a + b = 90° \), we can use that to find angle \( b \): Substituting \( 90° \) from the right triangle property into the first equation. \[ 142° = 90° + b \] Rearranging gives: \[ b = 142° - 90° \] \[ b = 52° \]

Thus, the value of angle \( b \) is \( 52° \).

The correct response is: 52°