Question

problem 14: (first taught in lesson 30) from this given statement, select the definition, property, postulate, or theorem that justifies the prove statement. given: ∠d ≅ ∠q, ∠c ≅ ∠p prove: ∠e ≅ ∠n

Explanation:

Step1: Recall angle - sum property of triangles

The sum of interior angles of a triangle is \(180^{\circ}\). So for \(\triangle CDE\), \(\angle C+\angle D+\angle E = 180^{\circ}\), and for \(\triangle PQN\), \(\angle P+\angle Q+\angle N=180^{\circ}\).

Step2: Use given congruent angles

We know that \(\angle D\cong\angle Q\) and \(\angle C\cong\angle P\), which means \(m\angle D = m\angle Q\) and \(m\angle C=m\angle P\).

Step3: Deduce the third - angle congruence

Substitute the equal angle measures into the angle - sum equations. If \(m\angle C+\angle D+\angle E = 180^{\circ}\) and \(m\angle P+\angle Q+\angle N = 180^{\circ}\), and \(m\angle C=m\angle P\), \(m\angle D = m\angle Q\), then \(m\angle E=m\angle N\), so \(\angle E\cong\angle N\) by the Third - Angle Theorem.

Answer:

The Angle - Angle (AA) Similarity Postulate (assuming the triangles are being considered for similarity and the third - angle theorem for congruence). If we are working in the context of triangle congruence, since the sum of the interior angles of a triangle is 180 degrees, in \(\triangle CDE\) and \(\triangle PQN\), if \(\angle D\cong\angle Q\) and \(\angle C\cong\angle P\), then \(\angle E\cong\angle N\) by the Third - Angle Theorem.