Transcript Proving Properties of Parallelograms Adapted from Walch Education • A quadrilateral is a polygon with four sides. • A convex polygon is a polygon with no.

Proving Properties
of Parallelograms
Adapted from Walch Education
• 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.
Polygons
1.10.1: Proving Properties of Parallelograms
Convex polygon
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• 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.
Concave polygon
Polygons, continued
1.10.1: Proving Properties of Parallelograms
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• 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 .
Parallelogram
1.10.1: Proving Properties of Parallelograms
Parallelogram
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• If a polygon is a parallelogram, there are five theorems
associated with it.
• In a parallelogram, both pairs of opposite sides are
congruent.
• Parallelograms also have two pairs of opposite angles that
are congruent.
Parallelogram, continued
1.10.1: Proving Properties of Parallelograms
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Theorem
If a quadrilateral is a parallelogram, opposite sides are
congruent.
B
A
AB @ DC
AD @ BC
D
C
The converse is also true. If the opposite sides of a quadrilateral
are congruent, then the quadrilateral is a parallelogram.
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1.10.1: Proving Properties of Parallelograms
Theorem
If a quadrilateral is a parallelogram, opposite angles are
congruent.
B
A
ÐA @ ÐC
ÐB @ ÐD
D
C
The converse is also true. If the opposite angles of a quadrilateral
are congruent, then the quadrilateral is a parallelogram.
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1.10.1: Proving Properties of Parallelograms
• 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.
• The diagonals of a parallelogram bisect each other.
Parallelogram, continued
1.10.1: Proving Properties of Parallelograms
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Theorem
If a quadrilateral is a parallelogram, then consecutive angles are
supplementary.
B
A
mÐA + mÐB = 180
mÐB + mÐC = 180
mÐC + mÐD = 180
mÐD + mÐA = 180
D
C
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1.10.1: Proving Properties of Parallelograms
Theorem
The diagonals of a parallelogram bisect each other.
B
A
AP @ PC
P
BP @ PD
D
C
The converse is also true. If the diagonals of a quadrilateral
bisect each other, then the quadrilateral is a parallelogram.
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1.10.1: Proving Properties of Parallelograms
Theorem
The diagonal of a parallelogram forms two congruent triangles.
B
A
D
C
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1.10.1: Proving Properties of Parallelograms
• Use the parallelogram to verify
that the opposite angles in a
parallelogram are congruent
and consecutive angles are
supplementary given that
and
Practice
1.10.1: Proving Properties of Parallelograms
.
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Extend the lines in the parallelogram to show
two pairs of intersecting lines and label the
angles with numbers.
Step 1
1.10.1: Proving Properties of Parallelograms
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Prove
and
Given
Ð4 @ Ð13
Alternate Interior Angles Theorem
Ð13 @ Ð16
Vertical Angles Theorem
Ð16 @ Ð9
Alternate Interior Angles Theorem
Ð4 @ Ð9
Transitive Property
Step 2
1.10.1: Proving Properties of Parallelograms
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Prove
and
Given
Ð7 @ Ð10
Alternate Interior Angles Theorem
Ð10 @ Ð11
Vertical Angles Theorem
Ð11@ Ð14
Alternate Interior Angles Theorem
Ð7 @ Ð14
Transitive Property
Step 3
1.10.1: Proving Properties of Parallelograms
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Prove that consecutive angles of a parallelogram are
supplementary.
and
Given
∠4 and ∠14 are
supplementary.
Same-Side Interior
Angles Theorem
∠14 and ∠9 are
supplementary.
Same-Side Interior
Angles Theorem
∠9 and ∠7 are
supplementary.
Same-Side Interior
Angles Theorem
∠7 and ∠4 are
supplementary.
Same-Side Interior
Angles Theorem
Step 4
1.10.1: Proving Properties of Parallelograms
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We 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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1.10.1: Proving Properties of Parallelograms
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