Question

Question

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parallelogram K L M N where K N is parallel to L M and K L is parallel to M N

Given: Parallelogram KLMN

Prove: ∠N≅∠L and ∠M≅∠K

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Statement Reason
Parallelogram KLMN Given
KL¯¯¯¯¯∥NM¯¯¯¯¯¯¯ and KN¯¯¯¯¯¯∥LM¯¯¯¯¯¯ Definition of parallelogram
m∠K+m∠N=180°

m∠L+m∠M=180°

m∠K+m∠L=180°

Response area
m∠K+m∠N=m∠K+m∠L

m∠L+m∠M=m∠K+m∠L

Response area
m∠N=m∠L

m∠M=m∠K

Subtraction Property of Equality
∠N≅∠L and ∠M≅∠K Response area

Answer 1

In parallelogram KLMN, opposite angles are proved congruent by showing that consecutive angles are supplementary, and then using the Subtraction Property of Equality to deduce that ∠N ≅ ∠L and ∠M ≅ ∠K.

Step-by-step explanation:

To prove that in parallelogram KLMN, the angles ∠N ≅ ∠L and ∠M ≅ ∠K, we follow a series of logical statements based on the properties of parallelograms and congruence relationships.

1. Parallelogram KLMN is given.
2. The opposite sides of a parallelogram are parallel by definition, so KL ∥∥ NM and KN ∥∥ LM.
3. In a parallelogram, consecutive angles are supplementary, so m∠K + m∠N = 180° and m∠L + m∠M = 180°.
4. However, opposite angles of a parallelogram are congruent. This means that m∠K + m∠N = m∠K + m∠L and m∠L + m∠M = m∠K + m∠L.
5. Using the Subtraction Property of Equality, we deduce that m∠N = m∠L and m∠M = m∠K.
6. Therefore, ∠N ≅ ∠L and ∠M ≅ ∠K.