Question

triangle \\( \triangle a b c \\) is reflected across line \\( n \\) to create \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\).
what is the measure of \\( \angle c \\)?

Explanation:

Step1: Use reflection property

Since \(\triangle ABC\) is reflected across line \(n\) to create \(\triangle A'B'C'\), \(\triangle ABC\cong\triangle A'B'C'\). So \(AC = A'C'=61\), \(BC = B'C' = 57\), \(\angle B=\angle B' = 67^{\circ}\), \(\angle A=\angle A'=59^{\circ}\)

Step2: Apply triangle angle - sum formula

The sum of the interior angles of a triangle is \(180^{\circ}\). For \(\triangle ABC\), we know that \(\angle A+\angle B+\angle C=180^{\circ}\)

Step3: Solve for \(\angle C\)

Substitute \(\angle A = 59^{\circ}\) and \(\angle B=67^{\circ}\) into the formula \(\angle C=180^{\circ}-\angle A - \angle B\)

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Answer:

\(54^{\circ}\)