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Chapter 5
Quadrilaterals
Mid-Term Exam Review Project
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Chapter 5: Break Down
• 5-1: Properties of Parallelograms
• 5-2: Ways to Prove that Quadrilaterals are Parallelograms
• 5-3: Theorems Involving Parallelograms
• 5-4 : Special Quadrilaterals
• 5-5: Trapezoids
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Chapter 5: Section 1
Properties of Parallelograms
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Parallelograms
A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
But there are more characteristics of parallelograms than just that…
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These additional characteristics are applied in several theorems
throughout the section.
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Theorem 5-1
“Opposite sides of a parallelogram are congruent.”
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This theorem states that the sides in a parallelogram that are opposite of each other are congruent. This means that in ABCD, side AB is congruent to side CD and side AC to side BD.
A B
C D
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Theorem 5-2
“Opposite Angles of a parallelogram are congruent”
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This theorem states that in a parallelogram, the opposite angles of a
parallelogram are congruent.
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Theorem 5-3
“Diagonals of a parallelogram bisect each other”
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This theorem states that in any parallelogram the diagonals bisect
each other.
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Section 5-2
Ways to Prove that Quadrilaterals are Parallelograms
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5 Ways to Prove that a Quadrilateral is a Parallelogram
• Show that both pairs of opposite sides are parallel
• Show that both pairs of opposite sides are congruent
• Show that one pair of opposite sides are both congruent and parallel
• Show that both pairs of opposite angles are congruent
• Show that the diagonals bisect each other
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Theorem 5-4
“If both pairs of opposite sides of a quadrilateral are congruent, then the
quadrilateral is a parallelogram.”
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Theorem 5-4T S
Q R
This theorem states that if side TS is congruent to side QR and side QT is congruent to side SR, then the quadrilateral is a parallelogram.
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Theorem 5-5
“If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a
parallelogram.”
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Theorem 5-5
• Theorem 5-5 states that if side VL and side AD are both parallel and congruent, then the quadrilateral is a parallelogram.
V L
DA
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Theorem 5-6
“If both pairs of opposite angles of a quadrilateral are congruent, then the
quadrilateral is a parallelogram.”
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Theorem 5-6
• This theorem states that if angle M is congruent to angle E, and angle I is congruent to angle K, then quadrilateral MIKE is a parallelogram.
M
EK
I
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Theorem 5-7
If the diagonals of a quadrilateral bisect each other, then the
quadrilateral is a parallelogram.
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Theorem 5-7
• This theorem states that if segment HA is congruent to segment AT, and segment MA is congruent to segment AD, that quadrilateral HDMT is a parallelogram.
D
A
M T
H
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Chapter 5: Section 3
• Theorems Involving Parallel Lines
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Theorem 5-8
• “If two lines are parallel, then all points on one line are equidistant from the other line.”
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All of the points on a line are equidistant from points on
the perpendicular bisector of a parallel line.
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Theorem 5-9
• “If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.”
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Theorem 5-10
• “A line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side.”
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Theorem 5-11
• “The segment that joins the midpoints of two sides of a triangle.”
1) is parallel to the third side
2) is half the length of the third side
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Section 5-4
Special Parallelograms
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Special Parallelograms
• There are four distinct special parallelograms that have unique characteristics.
• These special parallelograms include:
• Rectangles
• Rhombi
• Squares
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Rectangles
• A rectangle is a quadrilateral with four right angles. Every rectangle, therefore, is a parallelogram.
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Why?
• Every rectangle is a parallelogram because all angles in a rectangle are right, and all right angles are congruent.
• If the both pairs of opposite angles in a quadrilateral are congruent, then the quadrilateral is a parallelogram.
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Additionally:
• If a rectangle is a parallelogram, then it retains all of the characteristics of a parallelogram. If this is true then name all congruencies in the following diagram.
S A
ED
R
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Solution:
S A
R
D E
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Rhombi
• A rhombus is a quadrilateral with four congruent sides. Therefore, every rhombus is a parallelogram.
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Why?
• Every rhombus is a parallelogram because all four sides are congruent, and therefore both pairs of opposite sides are congruent.
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Squares
• A square is a quadrilateral with four right angles and four congruent sides. Therefore, every square is a rectangle, a rhombus, and a parallelogram.
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Why?
• A square is a rectangle a rhombus, and a parallelogram because all four sides are congruent and all four angles are right angles.
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Theorems
• Theorem’s 5-12 through 5-13 are very simple and self explanatory. They read as follows:
• 5-12: The diagonals of a rectangle are congruent
• 5-13: The diagonals of a rhombus are perpendicular.
• 5-14: Each diagonal of a rhombus bisects two angles of the rhombus.
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Theorem 5-12
• According to theorem 5-12, segment SE is congruent to segment DA
S A
D E
R
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Theorem 5-13
• According to theorem 5-13, segment AD is congruent to segment CB.
D
B C
A
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Theorem 5-14
• According to theorem 5-14, segment AD bisects angle BAC and angle BCD, and segment BC bisects angle ABD and angle ACD.
D
B C
A
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Theorem 5-15
“The Midpoint of the hypotenuse of a right triangle is equidistant from the
three vertices.”
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Theorem 5-15
M
According to this theorem, M is equidistant from points A, B, and C.
A
BC
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Theorem 5-16
If an angle of a parallelogram is a right angle, then the parallelogram is
a rectangle.
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Theorem 5-16
• In MATH, we know that angle H is a right angle. This means that all of the angles are right angles, and that MATH is a rectangle. (According to Theorem 5-16)
TH
AM
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Theorem 5-17
If two consecutive sides of a parallelogram are congruent, then the
parallelogram is a rhombus.
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Theorem 5-17
• In MATH, if we know that segment MA is congruent to segment AT, then we know that MATH is a rhombus, because segment MA is congruent to segment TH and segment AT is congruent to segment MH. Therefore, all sides are congruent to each other.
M
H
T
A
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Chart of Special QuadrilateralsProperty Parallelogram Rectangle Rhombus Square
Opp. Sides Parallel ♫ ♫ ♫ ♫
Opp. Sides Congruent ♫ ♫ ♫ ♫
Opp. Angles Congruent ♫ ♫ ♫ ♫
A diag. forms two congruent ∆s ♫ ♫ ♫ ♫
Diags. Bisect each other ♫ ♫ ♫ ♫
Diags are congruent ♫ ♫
Diags. are perpendicular ♫ ♫
A diag. bisects two angles ♫ ♫
All angles are Right Angles ♫ ♫
All sides are congruent ♫ ♫
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Section 5-5
Trapezoids
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Trapezoid
• A trapezoid is defined as a quadrilateral with exactly one pair of parallel sides.
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• The parallel sides are called the bases.
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• The other sides are the legs
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Isosceles Trapezoid
• A trapezoid with congruent legs is called an Isosceles Trapezoid.
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Theorems
• The Following Theorems Concern Trapezoids and their dimensions.
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Theorem 5-18
Base angles of an isosceles trapezoid are congruent.
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• This theorem states that trapezoid HAIR is isosceles, then angle R is congruent to angle I, and angle H is congruent to angle A.
H
R I
A
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Theorem 5-19
The Median of a trapezoid:
(1) Is parallel to the bases;
(2) Has a length equal to the average of the bases.
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The Median of a Trapezoid
• The median of a trapezoid is the segment that joins the midpoints of the legs.
• In Trapezoid ANDR, EW is the median because it joins the midpoints, E and W, of the legs.
A
W E
N
R D
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Theorem 5-19
• According to this theorem, Segment EW (a) is parallel to AN and DR, and a length equal to the average of the lengths of AN and DR.
A
W E
N
R D
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Use the Theorems to Complete:
• Given: AN = 10; DR = 20; AR is congruent to ND
• WE =?• Angle R is congruent to ?• Angle A is congruent to ?
A
W E
N
R D
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Solution:
• WE = 15
• Angle R is congruent to Angle D
• Angle A is congruent to Angle N
A
W E
N
R D
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The End
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Brought to you by:•ReuvenSmells™ Productions and
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This presentation by:
• Chris O’Connell
• Brent Sneider
• Jason Fernandez
• Mike
• Russell Waldman
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FIN