Question

2 - 56. for each triangle below, solve for the variable and answer the questions. homework help
a. solve for x. what kind of triangle is \\( \triangle a b c \\) ? be specific.
b. solve for y. what kind of triangle is shown in the figure? be specific.

Explanation:

Step1: Solve for \(x\) in \(\triangle ABC\)

In \(\triangle ABC\), since \(AB = AC\), the triangle is isosceles and \(\angle B=\angle C = x\). Using the angle - sum property of a triangle (\(\angle A+\angle B+\angle C = 180^{\circ}\)), we have \(2x+x + x=180^{\circ}\).

$$4x=180^{\circ}$$

$$x = 45^{\circ}$$

Since \(AB = AC\) and \(\angle A=2x = 90^{\circ}\), \(\triangle ABC\) is a right - isosceles triangle.

Step2: Solve for \(y\)

In the second triangle, since the two sides are equal, the base angles are equal. The non - adjacent interior angle to \(y\) is \(77^{\circ}\). Using the exterior angle property (exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles), and since the base angles are equal (\(77^{\circ}\) each). \(y=77^{\circ}+77^{\circ}\)

$$y = 154^{\circ}$$

The triangle is an isosceles triangle (two sides are equal).

Answer:

a. \(x = 45^{\circ}\), \(\triangle ABC\) is a right - isosceles triangle.
b. \(y = 154^{\circ}\), the triangle is an isosceles triangle.