Question

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

Explanation:

Part a:

Step1: Identify equal sides & angles

In $\triangle ABC$, $AB=AC$, so $\angle C = \angle B = x$.

Step2: Apply angle sum property

Sum of angles in triangle is $180^\circ$.
$$2x + x + x = 180^\circ$$

Step3: Simplify and solve for $x$

$$4x = 180^\circ$$
$$x = \frac{180^\circ}{4} = 45^\circ$$

Step4: Classify the triangle

Two sides are equal, and all angles are less than $90^\circ$.

Part b:

Step1: Identify equal sides & angles

The triangle has two equal sides, so it is isosceles. The base angles are equal: the unmarked top angle is $71^\circ$.

Step2: Find the interior angle adjacent to $y$

Sum of angles in triangle is $180^\circ$. Let the adjacent interior angle be $\angle I$:
$$\angle I = 180^\circ - 71^\circ - 71^\circ = 38^\circ$$

Step3: Solve for $y$ using linear pair

$y$ and $\angle I$ are supplementary.
$$y + 38^\circ = 180^\circ$$
$$y = 180^\circ - 38^\circ = 142^\circ$$

Step4: Classify the triangle

Two sides are equal, and all angles are less than $90^\circ$.

Answer:

a. $x = 45^\circ$; $\triangle ABC$ is an isosceles acute triangle.
b. $y = 142^\circ$; the figure is an isosceles acute triangle.