Question

in $\triangle xyz$, $\angle x \cong \angle y$ and $\angle z$ is 4 times the measure of $\angle y$. what is the measure of $\angle z$?\
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a. $30^\circ$\
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b. $60^\circ$\
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c. $120^\circ$\
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d. $150^\circ$

Explanation:

Step1: Define Variables for Angles

Let \( m\angle Y = x \). Since \( \angle X \cong \angle Y \), \( m\angle X = x \). And \( \angle Z \) is 4 times \( \angle Y \), so \( m\angle Z = 4x \).

Step2: Use Triangle Angle Sum Theorem

The sum of angles in a triangle is \( 180^\circ \). So, \( m\angle X + m\angle Y + m\angle Z = 180^\circ \). Substitute the values: \( x + x + 4x = 180^\circ \).

Step3: Solve for \( x \)

Combine like terms: \( 6x = 180^\circ \). Divide both sides by 6: \( x = 30^\circ \).

Step4: Find \( m\angle Z \)

Since \( m\angle Z = 4x \), substitute \( x = 30^\circ \): \( m\angle Z = 4 \times 30^\circ = 120^\circ \).

Answer:

C. \( 120^\circ \)