Question

1. new theorem from a recent lesson... given: conclusion:

\\( \angle k \cong \angle l \\) state the reason for the conclusion. type a response

Explanation:

Step1: Recall triangle - congruence theorems

If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. This is based on the fact that the sum of the interior angles of a triangle is \(180^{\circ}\).
Let \(\triangle IJK\) and \(\triangle MNL\). We know that \(\angle I\cong\angle N\) and \(\angle J\cong\angle M\).
In \(\triangle IJK\), \(\angle I+\angle J+\angle K = 180^{\circ}\), so \(\angle K=180^{\circ}-\angle I - \angle J\).
In \(\triangle MNL\), \(\angle N+\angle M+\angle L = 180^{\circ}\), so \(\angle L=180^{\circ}-\angle N-\angle M\).

Step2: Use the angle - angle (AA) similarity - related angle congruence

Since \(\angle I\cong\angle N\) (given) and \(\angle J\cong\angle M\) (given), by substitution \(180^{\circ}-\angle I - \angle J=180^{\circ}-\angle N-\angle M\).

Answer:

If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.