paralleogram proofs

[ayuban2]

Complete the proof below. Be sure to number your responses 1, 2, 3, and 4 (one point each). Theorem 5-8 If one pair of opposite sides of a quadrilateral is both congruent and parallel, then it is a parallelogram. Parallelogram I J H K. Segment I J and segment H K are both parallel and congruent to each other. Diagonals I H and J K intersect at point L. Angles 1 and 4 are alternate interior angles. Angles 2 and 3 are alternate interior angles. Given: Segment J I is parallel to Segment H K and Segment J I is congruent to Segment H KProve: Quadrilateral H K I J is a parallelogram. Paragraph Proof: We are given that Segment J I is parallel to Segment H K and Segment J I is congruent to Segment H K. Diagonals Segment I H and Segment J K are drawn intersecting at point L. angle 1 is congruent to angle 4 and angle 2 is congruent to angle 3 because __(1)__ are congruent. This allows us to prove that triangle IJL is congruent to HKL by __(2)__. This allows us to state that Segment I L is congruent to Segment H L and because __(3)__. In other words, diagonals Segment I H and Segment J K __(4)__ each other by the definition of bisector. According to Theorem 5-7, quadrilateral HKIJ is a parallelogram because the diagonals Segment I H and Segment J K bisect each other.

 

[mmm4444bot]

I don't understand your request for help. What would you like us to do for you?

Please read the post titled "Read Before Posting" for guidelines on how to ask for help on these boards.

Cheers :cool:

 

[Deleted member 4993]

Complete the proof below. Be sure to number your responses 1, 2, 3, and 4 (one point each). Theorem 5-8 If one pair of opposite sides of a quadrilateral is both congruent and parallel, then it is a parallelogram. Parallelogram I J H K. Segment I J and segment H K are both parallel and congruent to each other. Diagonals I H and J K intersect at point L. Angles 1 and 4 are alternate interior angles. Angles 2 and 3 are alternate interior angles. Given: Segment J I is parallel to Segment H K and Segment J I is congruent to Segment H KProve: Quadrilateral H K I J is a parallelogram. Paragraph Proof: We are given that Segment J I is parallel to Segment H K and Segment J I is congruent to Segment H K. Diagonals Segment I H and Segment J K are drawn intersecting at point L. angle 1 is congruent to angle 4 and angle 2 is congruent to angle 3 because __(1)__ are congruent. This allows us to prove that triangle IJL is congruent to HKL by __(2)__. This allows us to state that Segment I L is congruent to Segment H L and because __(3)__. In other words, diagonals Segment I H and Segment J K __(4)__ each other by the definition of bisector. According to Theorem 5-7, quadrilateral HKIJ is a parallelogram because the diagonals Segment I H and Segment J K bisect each other.

One of the most un-readable statement of problem.

 

[soroban]

Hello, ayuban2!

Complete the proof below.

Theorem 5-8:
If one pair of sides of a quadrilateral is both congruent and parallel,
. . then the quadrilateral is a parallelogram.

\(\displaystyle \text{Given: quadrilateral }HKIJ.\)
. . . . . . .\(\displaystyle \text{Sides }IJ\text{ and }HK\text{ are parallel and equal.}\)
. . . . . . .\(\displaystyle \text{Diagonals }IH\text{ and }JK\text{ intersect at }L.\)
. . . . . . .\(\displaystyle \angle 1\text{ and }\angle 4\text{ are alternate-interior angles.}\)
. . . . . . .\(\displaystyle \angle 2\text{ and }\angle 3 \text{ are alternate-interior angles.}\)

\(\displaystyle \text{Prove: quadrilateral }HKIJ\text{ is a parallelogram.}\)

           I                   J
            o  *  *  *  *  *  o
           *  * 4       3 *  *
          *     *     *     *
         *        o        *
        *     *   L *     *
       *  * 2       1 *  *
      o  *  *  *  *  *  o
    K                    H

Paragraph Proof:

We are given that \(\displaystyle IJ \parallel HK\) and \(\displaystyle IJ = HK.\)
Diagonals \(\displaystyle IH\) and \(\displaystyle JK\) are drawn, intersecting at \(\displaystyle L.\)

\(\displaystyle \angle 1 = \angle 4\,\text{ and }\,\angle 2 = \angle 3\) because: (1) alternate-interior angles are congruent.

This allows us to prove that \(\displaystyle \Delta IJL\,\cong\,\Delta HKL\) by (2) a.s.a.

This allows us to state that \(\displaystyle IL = HL\) because (3) "congruent parts".

In other words, diagonals \(\displaystyle IH\text{ and }JK\) (4) bisect each other by the definition of bisector.

According to Theorem 5-7, quadrilateral \(\displaystyle HKIJ\) is a parallelogram
. . because the diagonals \(\displaystyle IH\) and \(\displaystyle JK\) bisect each other.