PROOF Complete the paragraph proof to prove Theorem 7.10.
Given:ZK ZM, ZNZL
Prove: KLMN is a parallelogram.
N
M
Proof: LN divides quadrilateral KLMN into two triangles. The sum of the angle measures in each triangle is
measures for both triangles is
So, m2K+mZL+m2M+mZN=
so the sum of the angle
Because ZKM and
ZN2L, m2K = mM and m/NmL by the definition of congruence. By the Substitution Property of Equality, mK+mL+m2K+ml=
so
gives m/K+m&L=
Theorem. Likewise.
(m/K)+
(m2L) =
Dividing each side by
. The consecutive angles are supplementary, so KN || LM by the Converse of the Consecutive Interior Angles
(m2K)+
(MZN)
or m2K+m2N =
So these
consecutive angles are supplementary and KL || NM by the Converse of the Consecutive Interior Angle's Theorem. Opposite sides are parallel, so quadrilateral
KLMN is a parallelogram.