Introduction What does it mean to be opposite? What does it mean to be consecutive? Think about a rectangular room. If you put your back against one corner.

IntroductionWhat does it mean to be opposite? What does it mean to be consecutive? Think about a rectangular room. If you put your back against one corner of that room and looked directly across the room, you would be looking at the opposite corner. If you looked to your right, that corner would be a consecutive corner. If you looked to your left, that corner would also be a consecutive corner.

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1.10.1: Proving Properties of Parallelograms

Introduction, continuedThe walls of the room could also be described similarly. If you were to stand with your back at the center of one wall, the wall straight across from you would be the opposite wall. The walls next to you would be consecutive walls. There are two pairs of opposite walls in a rectangular room, and there are two pairs of opposite angles. Before looking at the properties of parallelograms, it is important to understand what the terms opposite and consecutive mean.

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Key Concepts• A quadrilateral is a polygon

with four sides. • A convex polygon is a

polygon with no interior angle greater than 180º and all diagonals lie inside the polygon.

• A diagonal of a polygon is a line that connects nonconsecutive vertices.

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Convex polygon

Key Concepts, continued• Convex polygons are contrasted

with concave polygons.

• A concave polygon is a polygon with at least one interior angle greater than 180º and at least one diagonal that does not lie entirely inside the polygon.

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Concave polygon

Key Concepts, continued• A parallelogram is a special

type of quadrilateral with two pairs of opposite sides that are parallel.

• By definition, if a quadrilateral has two pairs of opposite sides that are parallel, then the quadrilateral is a parallelogram.

• Parallelograms are denoted by the symbol .

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Parallelogram

Key Concepts, continued• If a polygon is a parallelogram, there are five

theorems associated with it.

• In a parallelogram, both pairs of opposite sides are congruent.

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Key Concepts, continued

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Theorem

If a quadrilateral is a parallelogram, opposite sides are congruent.

The converse is also true. If the opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

A

D C

B

Key Concepts, continued• Parallelograms also have two pairs of opposite

angles that are congruent.

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Theorem

If a quadrilateral is a parallelogram, opposite angles are congruent.

The converse is also true. If the opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

A

D C

B

Key Concepts, continued• Consecutive angles are angles that lie on the same

side of a figure.

• In a parallelogram, consecutive angles are supplementary; that is, they sum to 180º.

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Theorem

If a quadrilateral is a parallelogram, then consecutive angles are supplementary.

A

D C

B

Key Concepts, continued• The diagonals of a parallelogram have a relationship.

They bisect each other.

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13


Theorem

The diagonals of a parallelogram bisect each other.

The converse is also true. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

A

D C

B

P

Key Concepts, continued• Notice that each diagonal divides the parallelogram

into two triangles. Those two triangles are congruent.

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Theorem

The diagonal of a parallelogram forms two congruent triangles.

A

D C

B

Common Errors/Misconceptions• thinking that all angles in a parallelogram are

congruent even if the parallelogram isn’t a rectangle or square

• misidentifying opposite pairs of sides • misidentifying opposite pairs of angles and

consecutive angles

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Guided Practice

Example 1Quadrilateral ABCD has the following vertices: A (–4, 4), B (2, 8), C (3, 4), and D (–3, 0). Determine whether the quadrilateral is a parallelogram. Verify your answer using slope and distance to prove or disprove that opposite sides are parallel and opposite sides are congruent.

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Guided Practice: Example 1, continued

1. Graph the figure.

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2. Determine whether opposite pairs of lines are parallel.Calculate the slope of each line segment.

is opposite ; is opposite .

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Guided Practice: Example 1, continuedCalculating the slopes, we can see that the opposite sides are parallel because the slopes of the opposite sides are equal. By the definition of a parallelogram, quadrilateral ABCD is a parallelogram.

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3. Verify that the opposite sides are congruent. Calculate the distance of each segment using the distance formula.

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From the distance formula, we can see that opposite

sides are congruent. Because of the definition of

congruence and since AB = DC and BC = AD, then

and .

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Guided Practice

Example 2Use the parallelogram from Example 1 to verify that the opposite angles in a parallelogram are congruent and consecutive angles are supplementary given that

and .

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1. Extend the lines in the parallelogram to show two pairs of intersecting lines and label the angles with numbers.

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2. Prove .

We have proven that one pair of opposite angles in a parallelogram is congruent.

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and Given

Alternate Interior Angles Theorem

Vertical Angles Theorem


Transitive Property


3. Prove .

We have proven that both pairs of opposite angles in a parallelogram are congruent.

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and Given


Vertical Angles Theorem


Transitive Property


4. Prove that consecutive angles of a parallelogram are supplementary.

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and Given

∠4 and 14 are ∠supplementary.

Same-Side Interior Angles Theorem







Guided Practice: Example 2, continuedWe have proven consecutive angles in a parallelogram are supplementary using the Same-Side Interior Angles Theorem of a set of parallel lines intersected by a transversal.

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Introduction What does it mean to be opposite? What does it mean to be consecutive? Think about a rectangular room. If you put your back against one corner.

Documents

continued parallelograms

opposite corner

key concepts

pairs of opposite angles

pairs of opposite sides

pairs of opposite walls

consecutive angles

consecutive corner