Explanation Given: A quadrilateral A B C D A parallelogram is a quadrilateral with two pairs of opposite congruent and parallel sides. The five ways to prove that a quadrilateral is a parallelogram are: Prove both pairs of opposite sides are parallel. If both he pairs of opposite sides are parallel is proven then it is a parallelogram. Prove that both pairs of opposite sides are congruent. Congruent means that their measures are same. If both pairs of opposite sides are proven congruent then automatically both the pairs will be parallel. Hence it is a quadrilateral. Prove that one pair of opposite sides is both congruent and parallel. If at least one pair of opposite sides is proven congruent and parallel then automatically the other pair will also be congruent and parallel. Hence proving the quadrilateral to be a parallelogram. Prove that the diagonals of the quadrilateral bisect each other. If the diagonals of the quadrilateral are proven to bisect each other then it is a parallelogram. Prove that both pairs of opposite angles are congruent. If both the pairs of opposite angles are proven to be congruent then the quadrilateral can be proven as a parallelogram. Conclusion: Hence, there are five known ways to prove a quadrilateral A B C D a parallelogram.

McDougal Littell Jurgensen Geometry: Student Edition Geometry

5th Edition

ISBN: 9780395977279

Author: Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

Publisher: Houghton Mifflin Company College Division

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Question

Chapter 5.3, Problem 5ST1

To determine

To determine the five ways to prove that quadrilateral ABCD is a parallelogram.

Expert Solution & Answer

Explanation of Solution

Given:

A quadrilateral ABCD

A parallelogram is a quadrilateral with two pairs of opposite congruent and parallel sides.

The five ways to prove that a quadrilateral is a parallelogram are:

Prove both pairs of opposite sides are parallel.
If both he pairs of opposite sides are parallel is proven then it is a parallelogram.
Prove that both pairs of opposite sides are congruent.
Congruent means that their measures are same. If both pairs of opposite sides are proven congruent then automatically both the pairs will be parallel. Hence it is a quadrilateral.
Prove that one pair of opposite sides is both congruent and parallel.
If at least one pair of opposite sides is proven congruent and parallel then automatically the other pair will also be congruent and parallel. Hence proving the quadrilateral to be a parallelogram.
Prove that the diagonals of the quadrilateral bisect each other.
If the diagonals of the quadrilateral are proven to bisect each other then it is a parallelogram.
Prove that both pairs of opposite angles are congruent.
If both the pairs of opposite angles are proven to be congruent then the quadrilateral can be proven as a parallelogram.

Conclusion:

Hence, there are five known ways to prove a quadrilateral ABCD a parallelogram.

Chapter 5.3, Problem 4ST1

Chapter 5.3, Problem 6ST1

Chapter 5 Solutions

McDougal Littell Jurgensen Geometry: Student Edition Geometry

Ch. 5.1 - Prob. 1CE Ch. 5.1 - Prob. 2CE Ch. 5.1 - Prob. 3CE Ch. 5.1 - Prob. 4CE Ch. 5.1 - Prob. 5CE Ch. 5.1 - Prob. 6CE Ch. 5.1 - Prob. 7CE Ch. 5.1 - Prob. 8CE Ch. 5.1 - Prob. 9CE Ch. 5.1 - Prob. 10CE

Ch. 5.1 - Prob. 11CE Ch. 5.1 - Prob. 12CE Ch. 5.1 - Prob. 13CE Ch. 5.1 - Prob. 14CE Ch. 5.1 - Prob. 15CE Ch. 5.1 - Prob. 16CE Ch. 5.1 - Prob. 17CE Ch. 5.1 - Prob. 18CE Ch. 5.1 - Prob. 1WE Ch. 5.1 - Prob. 2WE Ch. 5.1 - Prob. 3WE Ch. 5.1 - Prob. 4WE Ch. 5.1 - Prob. 5WE Ch. 5.1 - Prob. 6WE Ch. 5.1 - Prob. 7WE Ch. 5.1 - Prob. 8WE Ch. 5.1 - Prob. 9WE Ch. 5.1 - Prob. 10WE Ch. 5.1 - Prob. 11WE Ch. 5.1 - Prob. 12WE Ch. 5.1 - Prob. 13WE Ch. 5.1 - Prob. 14WE Ch. 5.1 - Prob. 15WE Ch. 5.1 - Prob. 16WE Ch. 5.1 - Prob. 17WE Ch. 5.1 - Prob. 18WE Ch. 5.1 - Prob. 19WE Ch. 5.1 - Prob. 20WE Ch. 5.1 - Prob. 21WE Ch. 5.1 - Prob. 22WE Ch. 5.1 - Prob. 23WE Ch. 5.1 - Prob. 24WE Ch. 5.1 - Prob. 25WE Ch. 5.1 - Prob. 26WE Ch. 5.1 - Prob. 27WE Ch. 5.1 - Prob. 28WE Ch. 5.1 - Prob. 29WE Ch. 5.1 - Prob. 30WE Ch. 5.1 - Prob. 31WE Ch. 5.1 - Prob. 32WE Ch. 5.1 - Prob. 33WE Ch. 5.1 - Prob. 34WE Ch. 5.1 - Prob. 35WE Ch. 5.1 - Prob. 36WE Ch. 5.1 - Prob. 37WE Ch. 5.1 - Prob. 38WE Ch. 5.1 - Prob. 39WE Ch. 5.2 - Prob. 1CE Ch. 5.2 - Prob. 2CE Ch. 5.2 - Prob. 3CE Ch. 5.2 - Prob. 4CE Ch. 5.2 - Prob. 5CE Ch. 5.2 - Prob. 6CE Ch. 5.2 - Prob. 7CE Ch. 5.2 - Prob. 8CE Ch. 5.2 - Prob. 9CE Ch. 5.2 - Prob. 10CE Ch. 5.2 - Prob. 11CE Ch. 5.2 - Prob. 12CE Ch. 5.2 - Prob. 13CE Ch. 5.2 - Prob. 1WE Ch. 5.2 - Prob. 2WE Ch. 5.2 - Prob. 3WE Ch. 5.2 - Prob. 4WE Ch. 5.2 - Prob. 5WE Ch. 5.2 - Prob. 6WE Ch. 5.2 - Prob. 7WE Ch. 5.2 - Prob. 8WE Ch. 5.2 - Prob. 9WE Ch. 5.2 - Prob. 10WE Ch. 5.2 - Prob. 11WE Ch. 5.2 - Prob. 12WE Ch. 5.2 - Prob. 13WE Ch. 5.2 - Prob. 14WE Ch. 5.2 - Prob. 15WE Ch. 5.2 - Prob. 16WE Ch. 5.2 - Prob. 17WE Ch. 5.2 - Prob. 18WE Ch. 5.2 - Prob. 19WE Ch. 5.2 - Prob. 20WE Ch. 5.2 - Prob. 21WE Ch. 5.2 - Prob. 22WE Ch. 5.2 - Prob. 23WE Ch. 5.2 - Prob. 24WE Ch. 5.2 - Prob. 25WE Ch. 5.3 - Prob. 1CE Ch. 5.3 - Prob. 2CE Ch. 5.3 - Prob. 3CE Ch. 5.3 - Prob. 4CE Ch. 5.3 - Prob. 5CE Ch. 5.3 - Prob. 6CE Ch. 5.3 - Prob. 7CE Ch. 5.3 - Prob. 8CE Ch. 5.3 - Prob. 9CE Ch. 5.3 - Prob. 1WE Ch. 5.3 - Prob. 2WE Ch. 5.3 - Prob. 3WE Ch. 5.3 - Prob. 4WE Ch. 5.3 - Prob. 5WE Ch. 5.3 - Prob. 6WE Ch. 5.3 - Prob. 7WE Ch. 5.3 - Prob. 8WE Ch. 5.3 - Prob. 9WE Ch. 5.3 - Prob. 10WE Ch. 5.3 - Prob. 11WE Ch. 5.3 - Prob. 12WE Ch. 5.3 - Prob. 13WE Ch. 5.3 - Prob. 14WE Ch. 5.3 - Prob. 15WE Ch. 5.3 - Prob. 16WE Ch. 5.3 - Prob. 17WE Ch. 5.3 - Prob. 18WE Ch. 5.3 - Prob. 19WE Ch. 5.3 - Prob. 20WE Ch. 5.3 - Prob. 21WE Ch. 5.3 - Prob. 22WE Ch. 5.3 - Prob. 23WE Ch. 5.3 - Prob. 24WE Ch. 5.3 - Prob. 25WE Ch. 5.3 - Prob. 1ST1 Ch. 5.3 - Prob. 2ST1 Ch. 5.3 - Prob. 3ST1 Ch. 5.3 - Prob. 4ST1 Ch. 5.3 - Prob. 5ST1 Ch. 5.3 - Prob. 6ST1 Ch. 5.3 - Prob. 7ST1 Ch. 5.3 - Prob. 8ST1 Ch. 5.3 - Prob. 1E Ch. 5.3 - Prob. 2E Ch. 5.3 - Prob. 3E Ch. 5.3 - Prob. 4E Ch. 5.3 - Prob. 5E Ch. 5.3 - Prob. 6E Ch. 5.4 - Prob. 1CE Ch. 5.4 - Prob. 2CE Ch. 5.4 - Prob. 3CE Ch. 5.4 - Prob. 4CE Ch. 5.4 - Prob. 5CE Ch. 5.4 - Prob. 6CE Ch. 5.4 - Prob. 7CE Ch. 5.4 - Prob. 8CE Ch. 5.4 - Prob. 9CE Ch. 5.4 - Prob. 10CE Ch. 5.4 - Prob. 11CE Ch. 5.4 - Prob. 1WE Ch. 5.4 - Prob. 2WE Ch. 5.4 - Prob. 3WE Ch. 5.4 - Prob. 4WE Ch. 5.4 - Prob. 5WE Ch. 5.4 - Prob. 6WE Ch. 5.4 - Prob. 7WE Ch. 5.4 - Prob. 8WE Ch. 5.4 - Prob. 9WE Ch. 5.4 - Prob. 10WE Ch. 5.4 - Prob. 11WE Ch. 5.4 - Prob. 12WE Ch. 5.4 - Prob. 13WE Ch. 5.4 - Prob. 14WE Ch. 5.4 - Prob. 15WE Ch. 5.4 - Prob. 16WE Ch. 5.4 - Prob. 17WE Ch. 5.4 - Prob. 18WE Ch. 5.4 - Prob. 19WE Ch. 5.4 - Prob. 20WE Ch. 5.4 - Prob. 21WE Ch. 5.4 - Prob. 22WE Ch. 5.4 - Prob. 23WE Ch. 5.4 - Prob. 24WE Ch. 5.4 - Prob. 25WE Ch. 5.4 - Prob. 26WE Ch. 5.4 - Prob. 27WE Ch. 5.4 - Prob. 28WE Ch. 5.4 - Prob. 29WE Ch. 5.4 - Prob. 30WE Ch. 5.4 - Prob. 31WE Ch. 5.4 - Prob. 32WE Ch. 5.4 - Prob. 33WE Ch. 5.4 - Prob. 34WE Ch. 5.4 - Prob. 35WE Ch. 5.4 - Prob. 36WE Ch. 5.4 - Prob. 37WE Ch. 5.4 - Prob. 38WE Ch. 5.4 - Prob. 39WE Ch. 5.4 - Prob. 40WE Ch. 5.4 - Prob. 41WE Ch. 5.4 - Prob. 42WE Ch. 5.4 - Prob. 1MRE Ch. 5.4 - Prob. 2MRE Ch. 5.4 - Prob. 3MRE Ch. 5.4 - Prob. 4MRE Ch. 5.4 - Prob. 5MRE Ch. 5.4 - Prob. 6MRE Ch. 5.4 - Prob. 7MRE Ch. 5.4 - Prob. 8MRE Ch. 5.4 - Prob. 9MRE Ch. 5.5 - Prob. 1CE Ch. 5.5 - Prob. 2CE Ch. 5.5 - Prob. 3CE Ch. 5.5 - Prob. 4CE Ch. 5.5 - Prob. 5CE Ch. 5.5 - Prob. 6CE Ch. 5.5 - Prob. 7CE Ch. 5.5 - Prob. 8CE Ch. 5.5 - Prob. 9CE Ch. 5.5 - Prob. 10CE Ch. 5.5 - Prob. 11CE Ch. 5.5 - Prob. 1WE Ch. 5.5 - Prob. 2WE Ch. 5.5 - Prob. 3WE Ch. 5.5 - Prob. 4WE Ch. 5.5 - Prob. 5WE Ch. 5.5 - Prob. 6WE Ch. 5.5 - Prob. 7WE Ch. 5.5 - Prob. 8WE Ch. 5.5 - Prob. 9WE Ch. 5.5 - Prob. 10WE Ch. 5.5 - Prob. 11WE Ch. 5.5 - Prob. 12WE Ch. 5.5 - Prob. 13WE Ch. 5.5 - Prob. 14WE Ch. 5.5 - Prob. 15WE Ch. 5.5 - Prob. 16WE Ch. 5.5 - Prob. 17WE Ch. 5.5 - Prob. 18WE Ch. 5.5 - Prob. 19WE Ch. 5.5 - Prob. 20WE Ch. 5.5 - Prob. 21WE Ch. 5.5 - Prob. 22WE Ch. 5.5 - Prob. 23WE Ch. 5.5 - Prob. 24WE Ch. 5.5 - Prob. 25WE Ch. 5.5 - Prob. 26WE Ch. 5.5 - Prob. 27WE Ch. 5.5 - Prob. 28WE Ch. 5.5 - Prob. 29WE Ch. 5.5 - Prob. 30WE Ch. 5.5 - Prob. 31WE Ch. 5.5 - Prob. 32WE Ch. 5.5 - Prob. 33WE Ch. 5.5 - Prob. 34WE Ch. 5.5 - Prob. 35WE Ch. 5.5 - Prob. 36WE Ch. 5.5 - Prob. 1ST2 Ch. 5.5 - Prob. 2ST2 Ch. 5.5 - Prob. 3ST2 Ch. 5.5 - Prob. 4ST2 Ch. 5.5 - Prob. 5ST2 Ch. 5.5 - Prob. 6ST2 Ch. 5.5 - Prob. 7ST2 Ch. 5.5 - Prob. 8ST2 Ch. 5 - Prob. 1CR Ch. 5 - Prob. 2CR Ch. 5 - Prob. 3CR Ch. 5 - Prob. 4CR Ch. 5 - Prob. 5CR Ch. 5 - Prob. 6CR Ch. 5 - Prob. 7CR Ch. 5 - Prob. 8CR Ch. 5 - Prob. 9CR Ch. 5 - Prob. 10CR Ch. 5 - Prob. 11CR Ch. 5 - Prob. 12CR Ch. 5 - Prob. 13CR Ch. 5 - Prob. 14CR Ch. 5 - Prob. 15CR Ch. 5 - Prob. 16CR Ch. 5 - Prob. 17CR Ch. 5 - Prob. 18CR Ch. 5 - Prob. 19CR Ch. 5 - Prob. 20CR Ch. 5 - Prob. 21CR Ch. 5 - Prob. 22CR Ch. 5 - Prob. 1CT Ch. 5 - Prob. 2CT Ch. 5 - Prob. 3CT Ch. 5 - Prob. 4CT Ch. 5 - Prob. 5CT Ch. 5 - Prob. 6CT Ch. 5 - Prob. 7CT Ch. 5 - Prob. 8CT Ch. 5 - Prob. 9CT Ch. 5 - Prob. 10CT Ch. 5 - Prob. 11CT Ch. 5 - Prob. 12CT Ch. 5 - Prob. 13CT Ch. 5 - Prob. 14CT Ch. 5 - Prob. 15CT Ch. 5 - Prob. 16CT Ch. 5 - Prob. 17CT Ch. 5 - Prob. 18CT Ch. 5 - Prob. 19CT Ch. 5 - Prob. 1CUR Ch. 5 - Prob. 2CUR Ch. 5 - Prob. 3CUR Ch. 5 - Prob. 4CUR Ch. 5 - Prob. 5CUR Ch. 5 - Prob. 6CUR Ch. 5 - Prob. 7CUR Ch. 5 - Prob. 8CUR Ch. 5 - Prob. 9CUR Ch. 5 - Prob. 10CUR Ch. 5 - Prob. 11CUR Ch. 5 - Prob. 12CUR Ch. 5 - Prob. 13CUR Ch. 5 - Prob. 14CUR Ch. 5 - Prob. 15CUR Ch. 5 - Prob. 16CUR Ch. 5 - Prob. 17CUR Ch. 5 - Prob. 18CUR Ch. 5 - Prob. 19CUR Ch. 5 - Prob. 20CUR Ch. 5 - Prob. 21CUR Ch. 5 - Prob. 22CUR Ch. 5 - Prob. 23CUR Ch. 5 - Prob. 24CUR Ch. 5 - Prob. 25CUR Ch. 5 - Prob. 26CUR Ch. 5 - Prob. 27CUR Ch. 5 - Prob. 28CUR Ch. 5 - Prob. 29CUR Ch. 5 - Prob. 30CUR Ch. 5 - Prob. 31CUR

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