Question

2-56. for each triangle below, solve for the variable and answer the questions. a. solve for x. what kind of triangle is △abc? be specific. hint (a): b. solve for y. what kind of triangle is shown in the figure? be specific.

Explanation:

Step1: Use angle - sum property of a triangle for part a

The sum of angles in a triangle is $180^{\circ}$. In $\triangle ABC$, we have $2x + x+x=180^{\circ}$.
Combining like - terms, we get $4x = 180^{\circ}$.

Step2: Solve for $x$

Dividing both sides of the equation $4x = 180^{\circ}$ by 4, we have $x=\frac{180^{\circ}}{4}=45^{\circ}$.
Since two angles ($x$ and $x$) are equal, $\triangle ABC$ is an isosceles right - triangle (because one angle is $2x = 90^{\circ}$ and two angles are equal).

Step3: Use exterior - angle property for part b

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. In the second triangle, the two non - adjacent interior angles to the exterior angle $y$ are equal (since the triangle is isosceles as indicated by the equal side lengths). Let the equal interior angles be $a$. So $y=a + a$. And since one of the non - exterior interior angles is $71^{\circ}$, then $y = 2\times71^{\circ}=142^{\circ}$. The triangle is an isosceles triangle.

Answer:

a. $x = 45^{\circ}$, $\triangle ABC$ is an isosceles right - triangle.
b. $y = 142^{\circ}$, The triangle is an isosceles triangle.