Question

given: parallelogram klmn prove: ∠n≅∠l and ∠m≅∠k statement parallelogram klmn reason given (overline{kl}paralleloverline{nm}) and (overline{kn}paralleloverline{lm}) definition of parallelogram (mangle k + mangle n=180^{circ}) (mangle l + mangle m = 180^{circ}) (mangle k + mangle l=180^{circ})

Explanation:

Step1: Recall parallel - side property

Since \(KLMN\) is a parallelogram, by the definition of a parallelogram, \(\overline{KL}\parallel\overline{NM}\) and \(\overline{KN}\parallel\overline{LM}\).

Step2: Use consecutive - angles supplementary

For parallel lines \(\overline{KL}\parallel\overline{NM}\), \(\angle K\) and \(\angle N\) are consecutive interior angles, so \(m\angle K + m\angle N=180^{\circ}\). For \(\overline{KN}\parallel\overline{LM}\), \(\angle L\) and \(\angle M\) are consecutive interior angles, so \(m\angle L + m\angle M = 180^{\circ}\). Also, for \(\overline{KL}\parallel\overline{NM}\) and \(\overline{KN}\parallel\overline{LM}\), \(\angle K\) and \(\angle L\) are consecutive interior angles, so \(m\angle K + m\angle L=180^{\circ}\).

Step3: Prove angle congruence

From \(m\angle K + m\angle N=180^{\circ}\) and \(m\angle K + m\angle L=180^{\circ}\), we can conclude that \(m\angle N=m\angle L\) (by the subtraction property of equality, subtracting \(m\angle K\) from both equations), so \(\angle N\cong\angle L\). Similarly, from \(m\angle L + m\angle M=180^{\circ}\) and \(m\angle K + m\angle L=180^{\circ}\), we can conclude that \(m\angle M=m\angle K\) (by subtracting \(m\angle L\) from both equations), so \(\angle M\cong\angle K\).

Answer:

We have proved that \(\angle N\cong\angle L\) and \(\angle M\cong\angle K\) using the properties of parallel - lines and the definition of a parallelogram.