Question

determine whether each pair of triangles is congruent. if yes, include the theorem that applies.
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example 2

1. refer to the figure.

a. find the measures of the sides of $\triangle abc$. show your work.
b. find $m\angle a$. show your work.

1. find the value of $x$.
2. find $m\angle b$ and $ac$.
3. find the value of $x$.
4. find $m\angle y$ and $wy$.
5. find the value of $x$.
6. find the value of $x$.

Explanation:

Step1: Identify the triangle type

Triangle \( XYZ \) has two equal sides (marked with ticks), so it's isosceles. One angle is \( 60^\circ \), so it's also equilateral (isosceles + \( 60^\circ \) angle implies all sides equal). Thus, \( XZ = XY = YZ \).

Step2: Set up the equation

Given \( XZ = 3x + 8 \) and \( YZ = 4x - 4 \), set them equal:
\( 3x + 8 = 4x - 4 \)

Step3: Solve for \( x \)

Subtract \( 3x \) from both sides: \( 8 = x - 4 \)
Add 4 to both sides: \( x = 12 \)

Answer:

\( x = 12 \)