We are asked to determine whether the given triangles are similar.
Let's consider the triangle on the left.
Note that we know the measures of two of the angles of the left triangle. To find the measure of the third angle, remember that the sum of the measures of the interior angles of a triangle is 180. This lets us write an equation for m∠3.
56+ 76+ m ∠3 = 180
Let's solve this equation for m∠3.
As we can see, two angles of the triangle on the left are congruent to two corresponding angles of the triangle on the right. Specifically, the two angles of measure 48^(∘) and the two angles of measure 76^(∘) are congruent to each other. Therefore, the triangles are similar by the Angle-Angle Criterion.
Finding the Value of x
Now we will find the value of x. Let's take a look at the given diagram.
Note that the two angles of measure 48^(∘) and the angle of measure (4x)^(∘) form a line, or a straight angle. Therefore, the sum of their measures is 180^(∘). This lets us write an equation for x.
48 + 4x + 48 = 180
Let's solve this equation by isolating x on the left-hand side and simplifying.
