Question

1. r is in the interior of ∠bin. a is in the interior of ∠rin. ∠bir ≅ ∠ain. if m∠air = (m∠ain − 12)° and m∠bin = 165°, find m∠air. a. 59° b. 51° c. 47° d. 38.3° e. none of these

Explanation:

Step1: Define Variables and Relationships

Let \( m\angle AIN = x \). Since \( \angle BIR \cong \angle AIN \), then \( m\angle BIR = x \). Also, \( m\angle AIR=(x - 12)^\circ \).

Step2: Analyze Angle Sum

The measure of \( \angle BIN \) is the sum of \( \angle BIR \), \( \angle AIR \), and \( \angle AIN \)? Wait, no. Wait, R is in the interior of \( \angle BIN \), A is in the interior of \( \angle RIN \). So \( \angle BIN=\angle BIR+\angle RIN \), and \( \angle RIN=\angle AIR+\angle AIN \). So substituting, \( \angle BIN=\angle BIR+\angle AIR+\angle AIN \).

Since \( \angle BIR = \angle AIN = x \) and \( \angle AIR=(x - 12) \), then:
\( m\angle BIN=x+(x - 12)+x \)
We know \( m\angle BIN = 165^\circ \), so:
\( x+(x - 12)+x=165 \)

Step3: Solve for x

Combine like terms:
\( 3x-12 = 165 \)
Add 12 to both sides:
\( 3x=165 + 12=177 \)
Divide by 3:
\( x=\frac{177}{3}=59 \)

Step4: Find \( m\angle AIR \)

Since \( m\angle AIR=(x - 12)^\circ \), substitute \( x = 59 \):
\( m\angle AIR=59 - 12=47^\circ \)

Answer:

c. \( 47^\circ \)