Question

find the value of x in this convex polygon.
(2x + 38)°
121°
133°

Explanation:

Step1: Recall the exterior angle sum of a polygon

The sum of the exterior angles of any convex polygon is \( 360^\circ \). So we can set up an equation with the given angles and the expression involving \( x \).
The angles are \( 121^\circ \), \( 133^\circ \), \( (2x + 38)^\circ \), and we need to find the fourth exterior angle? Wait, no, wait. Wait, a triangle? Wait, no, the figure is a triangle? Wait, no, the exterior angles of a triangle? Wait, no, the sum of exterior angles of any convex polygon is \( 360^\circ \), but if it's a triangle, the sum of exterior angles is also \( 360^\circ \). Wait, let's check the given angles. Wait, the three exterior angles? Wait, no, maybe it's a triangle, and we have three exterior angles? Wait, no, the figure shows three angles: \( 121^\circ \), \( 133^\circ \), and \( (2x + 38)^\circ \), and we need to find \( x \) such that the sum of exterior angles is \( 360^\circ \). Wait, but a triangle has three exterior angles, so sum should be \( 360^\circ \). So:
\( 121 + 133 + (2x + 38) + \) wait, no, wait, maybe I made a mistake. Wait, no, the figure is a triangle, so three exterior angles. Wait, but the sum of exterior angles of any convex polygon is \( 360^\circ \), so for a triangle, it's also \( 360^\circ \). So:
\( 121 + 133 + (2x + 38) = 360 \)? Wait, no, wait, maybe there's a typo, or maybe it's a triangle with three exterior angles. Wait, let's calculate:
First, sum the known angles and the expression:
\( 121 + 133 + 2x + 38 = 360 \)

Step2: Simplify the left - hand side

Combine like terms: \( 121+133 = 254 \), \( 254 + 38=292 \), so the equation becomes \( 2x+292 = 360 \)

Step3: Solve for \( x \)

Subtract 292 from both sides: \( 2x=360 - 292 \)
\( 360-292 = 68 \), so \( 2x = 68 \)
Divide both sides by 2: \( x=\frac{68}{2}=34 \)

Wait, but wait, maybe I misinterpreted the figure. Wait, maybe it's a triangle, and the exterior angles. Wait, but let's check again. The sum of exterior angles of any convex polygon is \( 360^\circ \), so if we have three exterior angles: \( 121^\circ \), \( 133^\circ \), and \( (2x + 38)^\circ \), then their sum is \( 360^\circ \). So the calculation above is correct.

Answer:

\( x = 34 \)