Question

find m∠x: triangle wxy with angles at w: (4x + 11)°, at x: (2x + 22)°, and an exterior angle at y: (9x - 15)° (formed by extension to z).

Explanation:

Step1: Identify Triangle Type

The triangle \( \triangle WXY \) has two equal angles (marked by arcs), so it's isosceles with \( \angle W = \angle X \)? Wait, no—wait, the exterior angle at \( Y \) is \( (9x - 15)^\circ \), and in a triangle, an exterior angle equals the sum of the two non-adjacent interior angles. Wait, actually, the arcs on \( \angle W \) and \( \angle X \)? Wait, no, looking at the diagram: \( \angle W = (4x + 11)^\circ \), \( \angle X = (2x + 22)^\circ \), and the exterior angle at \( Y \) (extending to \( Z \)) is \( (9x - 15)^\circ \). In a triangle, the exterior angle is equal to the sum of the two remote interior angles. So \( \angle W + \angle X = \) exterior angle at \( Y \). Wait, no—wait, maybe the triangle is isosceles with \( \angle W = \angle X \)? Wait, \( 4x + 11 = 2x + 22 \)? Let's check: \( 4x + 11 = 2x + 22 \) → \( 2x = 11 \) → \( x = 5.5 \), but then exterior angle would be \( 9(5.5) - 15 = 49.5 - 15 = 34.5 \), and \( \angle W + \angle X = (22 + 11) + (11 + 22) = 33 + 33 = 66 \), which doesn't match. So maybe the exterior angle is equal to the sum of the two non-adjacent interior angles. So \( (4x + 11) + (2x + 22) = 9x - 15 \).

Step2: Set Up Equation

Sum of interior angles \( \angle W + \angle X = \) exterior angle \( \angle Y \) (exterior angle theorem). So:

$$ (4x + 11) + (2x + 22) = 9x - 15 $$

Simplify left side: \( 6x + 33 = 9x - 15 \)

Step3: Solve for \( x \)

Subtract \( 6x \) from both sides:

$$ 33 = 3x - 15 $$

Add 15 to both sides:

$$ 48 = 3x $$

Divide by 3:

$$ x = 16 $$

Step4: Find \( m\angle X \)

\( \angle X = 2x + 22 \). Substitute \( x = 16 \):

$$ 2(16) + 22 = 32 + 22 = 54 $$

Answer:

\( 54^\circ \)