To find the measure of angle \( b \) in the given right triangle using the properties of exterior angles, we can use the following steps:
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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 \( b \).
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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: \[ a + b = 90° \]
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Set up the Equation: From the properties of exterior angles: \[ \text{Exterior angle} = a + b \] Given that the exterior angle is 142°: \[ 142° = a + b \]
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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)
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