Question

1. the measures of the exterior angles of a triangle are $2x + 5$, $3x + 1$, and $5x + 4$. what is the value of $x$?

a. 35
b. 30
c. 17
d. 15

Explanation:

Step1: Recall the exterior angle sum of a triangle

The sum of the exterior angles of any triangle (and any polygon) is \(360^\circ\). So we set up the equation: \((2x + 5)+(3x + 1)+(5x + 4)=360\).

Step2: Combine like terms

Combine the \(x\) terms and the constant terms: \(2x+3x + 5x+5 + 1+4 = 360\), which simplifies to \(10x+10 = 360\).

Step3: Solve for \(x\)

Subtract 10 from both sides: \(10x=360 - 10=350\). Then divide both sides by 10: \(x=\frac{350}{10}=35\). Wait, but let's check again. Wait, no, the sum of exterior angles of a triangle is \(360^\circ\), but wait, actually, for any convex polygon, the sum of exterior angles is \(360^\circ\). But let's re - check the calculation.

Wait, \((2x + 5)+(3x + 1)+(5x + 4)=2x+3x+5x + 5 + 1+4=10x + 10\). Set equal to 360: \(10x+10 = 360\). Subtract 10: \(10x=350\), \(x = 35\). But wait, let's check the answer options. Option A is 35. But wait, maybe I made a mistake? Wait, no, the sum of exterior angles of a triangle is \(360^\circ\). Let's verify with \(x = 35\):

First angle: \(2\times35 + 5=70 + 5 = 75\)

Second angle: \(3\times35+1 = 105 + 1=106\)

Third angle: \(5\times35 + 4=175+4 = 179\)

Sum: \(75 + 106+179=360\). Yes, that works. Wait, but let's check the options again. Option A is 35. But wait, maybe the problem is about the exterior angles of a triangle, but sometimes, people confuse with interior angles. Wait, no, the sum of exterior angles of any polygon is \(360^\circ\) (for convex polygons). So the calculation seems correct.

Wait, but let's re - do the steps:

1. Sum of exterior angles of a triangle: \(360^\circ\). So the equation is \((2x + 5)+(3x + 1)+(5x + 4)=360\).

1. Combine like terms: \(2x+3x+5x=10x\), \(5 + 1+4 = 10\). So \(10x+10 = 360\).

1. Subtract 10 from both sides: \(10x=360 - 10=350\).

1. Divide by 10: \(x = 35\). So the answer should be A. 35.

Wait, but the initial calculation seems correct. So the value of \(x\) is 35.

Answer:

A. 35