In the figure below, what is the value of x if angles m and n are congruent?

We want to find the measure of x given that we know that m and n are congruent.

The measure of x is 116°.

Two angles are congruent if they have the same measure.

Then we have ∠n = ∠m

We also know that the sum of all interior angles of a triangle is equal to 180°, then if we look at the smaller triangle we have that:

114° + ∠n + ∠m = 180°

∠n + ∠m = 180° - 114° = 66°

Knowing that these measures are equal we can write:

2*∠m = 66°

∠m = 66°/2 = 33°

Now that we know the measure of m, we can do the same thing for the larger triangle:

∠x + ∠m + 31° = 180°

∠x + 33° + 31° = 180°

∠x = 180° - 64° = 116°

The measure of x is 116°.

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Applying the exterior angle theorem and the sum of triangles, the value of x in the figure is:

Recall:

Sum of triangles = 180 degrees

Exterior angle = sum of two interior opposite angles

Since we know that angles m and n are congruent, therefore:

Find x (exterior angle of triangle SUV):

(exterior angle of a triangle theorem)

Therefore, applying the exterior angle theorem and the sum of triangles, the value of x in the figure is:

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