Question

in $\triangle efg$ and $\triangle yxz$, $m\angle f \cong m\angle x$ and $m\angle e \cong m\angle y$. if $m\angle e = 62^\circ$ and $m\angle x = 80^\circ$, what is the measure of $\angle z$?
options: $38^\circ$, $62^\circ$, $80^\circ$, $142^\circ$

Explanation:

Step1: Use the property of congruent triangles

Since \(\triangle EFG\cong\triangle YXZ\), then \(\angle E\cong\angle Y = 62^{\circ}\), \(\angle F\cong\angle X=80^{\circ}\), \(\angle G\cong\angle Z\).

Step2: Apply the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). For \(\triangle EFG\), we know that \(m\angle E + m\angle F+m\angle G=180^{\circ}\). Substitute \(m\angle E = 62^{\circ}\) and \(m\angle F = 80^{\circ}\) into the formula: \(62^{\circ}+80^{\circ}+m\angle G = 180^{\circ}\).

Step3: Solve for \(m\angle G\)

\(m\angle G=180^{\circ}-(62^{\circ} + 80^{\circ})=180^{\circ}-142^{\circ}=38^{\circ}\). Since \(\angle G\cong\angle Z\), then \(m\angle Z = 38^{\circ}\).

Answer:

\(38^{\circ}\)