We are given the following diagram.

We are asked to find the value of $x .$ Notice that the $x^{\circ}$ angle is one of the exterior angles of the polygon in the diagram. To find the measure of the $x^{\circ}$ angle, we will first recall an important piece of information.

Exterior Angles of a Polygon
In a polygon, the sum of the measures of the exterior angles — one at each vertex — is $36 0^{\circ} .$

This means that the sum of the exterior angles of the given triangle is also $36 0^{\circ} .$ Let's mark an exterior angle at each vertex of this triangle.

The exterior angles of the triangle have measures of

x^{\circ},

15 0^{\circ},

and

y^{\circ} .

These measures add up to

36 0^{\circ} .

x^{\circ} + 15 0^{\circ} + y^{\circ} = 36 0^{\circ}

We can use this equation to find the value of

x,

but we need to find the value of

y

first. Notice that the

y^{\circ}

angle and the

10 0^{\circ}

angle form a straight line.

Therefore, these angles are supplementary and the sum of their measures is

18 0^{\circ} .

y^{\circ} + 10 0^{\circ} = 18 0^{\circ}

Now we can solve this equation for

y .

For simplicity, we will not write the degree symbol.

y + 100 = 180

SubEqn

$LHS - 100 = RHS - 100$

y + 100 - 100 = 180 - 100

SubTerms

Subtract terms

y = 80

Finally, we will substitue

80

for

y

into the first equation and solve it for

x .

x + 150 + y = 360

Substitute

$y = 80$

x + 150 + 80 = 360

▼

Solve for

x

AddTerms

Add terms

x + 230 = 360

SubEqn

$LHS - 230 = RHS - 230$

x + 230 - 230 = 360 - 230

SubTerms

Subtract terms

x = 130

We got that

x = 130 .

Exercise

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