Question

other properties of parallelograms:

• opposite sides are congruent.
• diagonals bisect each other.
• opposite angles are congruent.
• adacent angles are supplementary.

complete each proof.
4 given: ( j k l m ) is a parallelogram
prove: ( overline{j k} cong overline{l m} ) and ( overline{j m} cong overline{l k} )

Explanation:

Step1: Draw diagonal \(JL\)

In parallelogram \(JKLM\), draw diagonal \(JL\).

Step2: Use properties of parallelograms

Since \(JKLM\) is a parallelogram, \(JK\parallel LM\) and \(JM\parallel LK\). Then \(\angle KJL=\angle MLJ\) (alternate - interior angles) and \(\angle MJL=\angle KLJ\) (alternate - interior angles). Also, \(JL = JL\) (reflexive property).

Step3: Prove \(\triangle JKL\cong\triangle L MJ\)

By the ASA (Angle - Side - Angle) congruence criterion (\(\angle KJL=\angle MLJ\), \(JL = JL\), \(\angle MJL=\angle KLJ\)), \(\triangle JKL\cong\triangle L MJ\).

Step4: Use congruent triangles to find congruent sides

Since \(\triangle JKL\cong\triangle L MJ\), by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \(JK\cong LM\) and \(JM\cong LK\).

Answer:

The proof is completed by drawing a diagonal, using properties of parallelograms (parallel sides), ASA congruence for triangles, and then CPCTC to show \(JK\cong LM\) and \(JM\cong LK\).