   Chapter 4.2, Problem 40E ### Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698

#### Solutions

Chapter
Section ### Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698
Textbook Problem
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# 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.

PQSR

QRPS

 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 ∆ABC,PQ¯∥AC¯ 3. The line segment joining the midpoints of two sides of a triangle is parallel to the third side 4. In ∆ADCSR¯∥AC¯ 4. The line segment joining the midpoints of two sides of a triangle is parallel to the third side 5. PQ¯∥SR¯ 5. PQ¯∥AC¯ and SR¯∥AC¯ 6

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