Given: Parallelogram KLMN

Mathematics

2 answers:

Dvinal [7]3 years ago

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Answer:

It is followed by properties of parallelogram and the proof is explained in step-by-step solution.

Step-by-step explanation:

Given a parallelogram KLMN

Parallelogram is a quadrilateral with parallel opposite sides.

To Prove: ∠N≅∠L and ∠M≅∠K

In Parallelogram KLMN the opposite sides are parallel therefore, the opposite angles are vertical angles. And the vertical angles are always equal and congruent. Hence,

∠N≅∠L and ∠M≅∠K

For any quadrilateral sum of all angles equal to 360°

m∠K + m∠L + m∠M + m∠N = 360°

Now given quadrilateral is parallelogram therefore opposite angles are equal. So, we can write m∠L = m∠N, m∠M = m∠K

⇒ m∠K + m∠N + m∠K + m∠N = 360°

2 m∠K + 2 m∠N = 360°

m∠K + m∠N = 180°

For the same parallelogram, because opposite sides are equal therefore use m∠K = m∠L, m∠N = m∠M using above equation, we get

m∠L + m∠M = 180°

Because m∠M = m∠K

m∠K + m∠L = 180°

Same Side Interior Angles Theorem states that if the two parallel lines intersect or cut by a transversal then the interior angles on the same side are supplementary that means their sum is equal to 180°

Hence, by above theorem m∠K + m∠N = 180°

& m∠K + m∠L = 180°

By above two equations, we get

m∠K + m∠N = m∠K + m∠L

because m∠K = m∠M & ∠N = ∠L

⇒ m∠L + m∠M = m∠K + m∠L

Because opposite angles of a parallelogram are alternate angles therefore are equal. hence,

m∠N = m∠L

m∠M = m∠K

Angle congruency postulate

Opposite angles of parallelogram are vertical angles and the vertical angles are congruent as well as equal.

∠N≅∠L and ∠M≅∠K

Substitution Property of equality states that if one angle is equal to another angle the one can be substituted in the place of another that means if ∠M=∠K then we can substituted ∠K in place of ∠M

Transitive property of congruence states that if if the two angles or segments or triangles are congruent to third then they are congruent to each other.

Subtraction Property of Equality states that if if two sides of any equation are equal then subtraction of same value from both sides results in equality of equation.

Parallel lines are those lines which do not intersect or cut each other at the end.

Parallelogram is a quadrilateral have opposite sides equal and parallel.