Chapter 4.2, Problem 44E

### Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Chapter
Section

### Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

# Prove that when the midpoints of consecutive sides of a quadrilateral are joined in order, the resulting quadrilateral is a parallelogram.

To determine

The quadrilateral that results by joining the midpoints of consecutive sides of a quadrilateral in order is a parallelogram.

Explanation

Formal proof:

Statement:

The quadrilateral that results by joining the midpoints of consecutive sides of a quadrilateral in order is a parallelogram.

Drawing:

Given:

ABCD is a quadrilateral and PQRS is the quadrilateral that formed by joining the midpoints of the sides of the quadrilateral ABCD taken in order.

Prove:

PQRS is a parallelogram.

PQâˆ¥SR

QRâˆ¥PS

 PROOF Statements Reasons 1. P, Q, R and S are the midpoints of the sidesAB, BC, CD and DA of the quadrilateralABCD respectively 1. Given 2. Draw the diagonal (AC), joining A and C. 2. Separate the quadrilateral into two triangles 3. In âˆ†ABC,PQÂ¯âˆ¥ACÂ¯ 3. The line segment joining the midpoints of two sides of a triangle is parallel to the third side 4. In âˆ†ADCSRÂ¯âˆ¥ACÂ¯ 4. The line segment joining the midpoints of two sides of a triangle is parallel to the third side 5. PQÂ¯âˆ¥SRÂ¯ 5. PQÂ¯âˆ¥ACÂ¯ and SRÂ¯âˆ¥ACÂ¯ 6

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