Name ___________

Geometry 1Unit 6:

Quadrilaterals

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Geometry 1 Unit 6: Quadrilaterals6.1 PolygonsPolygon

3

Sides

Vertex

How to name a Polygon:

tri- 3 quadr- 4 penta- 5 hexa- 6hepta- 7 oct- 8 nona- 9 dec- 10

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polygons not polygons

A

D C

BE

Use the prefix chart above and the given definition towrite a definition of the boldfaced word.

1. triplet: One of three children born at one birth.Quadruplet

2. quadrennial: Happening once every four years.Octennial

3. decathalon: An athletic contest that consists of ten events foreach participant.Pentathalon

4. hexapod: Having six legs or feet.Tripod

5. octogenarian: A person between eighty and ninety years ofage.Nonagenarian

6. pentagram: A five-pointed star.Hexagram

7. octahedron: A solid geometric figure with eight plane faces.Decahedron

8. nonagon: A polygon with nine sides and nine angles.Heptagon

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9. heptameter: A line of verse consisting of seven metrical feet.Pentameter

Polygons

Number of Sides Name3456789101112n

Diagonals

Convex polygons

Concave polygons

Example 1

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Identify the polygon and state whether it is convex or concave.

Can a polygon be equiangular and not equilateral?Draw an example.

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Equilateral Polygon

Equiangular Polygon

Regular Polygon

Example 2Decide whether the polygon is regular.

Interior Angles of a Quadrilateral Theorem

__________________________________________

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1

4 3

2

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Example 3Find mÐF, mÐG, and mÐH.

Example 4Use the information in the diagram to solve for x

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x

x55°E

H

F

G

100°

2x + 30 3x – 5

120°

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Geometry 1 Unit 6: Quadrilaterals6.2 Properties of ParallelogramsParallelogram

Opposite Sides of a Parallelogram Theorem

Opposite Angles in a Parallelogram Theorem

Consecutive Angles in a Parallelogram Theorem

Diagonals in a Parallelogram Theorem

Example 1GHJK is a parallelogram. Find each unknown length

JH

LH

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P

Q

S

R

P

Q

S

R

Add to equal 180°

P

Q

S

RM

G

K

H

J

L68

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Example 2In parallelogram ABCD, mÐC = 105°. Find the measure of each angle.

mÐAmÐD

Example 3WXYZ is a parallelogram. Find the value of x.

Example 4Given:: ABCD is a

parallelogram.Prove: Ð2 @ Ð4

Statement ReasonsABCD is a parallelogram

 

AD || BC  

Ð2 @ Ð1  

AB || CD  

  Alternate interior angles theorem

Ð2 @ Ð4  

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3x + 18°

4x – 9°

A

D

B

C

1

4

2

3

Example 5

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Given: ACDF is a parallelogram.

ABDE is a parallelogram.Prove: ∆BCD @ ∆EFA

Statement ReasonACDF is a parallelogram.

 

ABDE is a parallelogram.

 

  Opposite sides of a parallelogram are congruent

AC = DF  AB = DE  AC = AB + BC  DF = DE + EF  AC = DE + DF  AB + BC = AB + EF

 

BC = EF    Def of Congruent∆BCD @ ∆EFA  

Example 6A four-sided concrete slab has consecutive angle measures of 85°, 94°, 85°, and 96°. Is the slab a parallelogram? Explain.

Investigating Properties of ParallelogramsCut 4 straws to form two congruent pairs.

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A B C

F DE

Partly unbend two paperclips, link their smaller ends, and insert the larger ends into two cut straws. Join the rest of the straws to form a quadrilateral with opposite sides congruent.Change the angles of your quadrilateral. Is your quadrilateral a parallelogram?

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Geometry 1 Unit 6: Quadrilaterals6.3 Proving Quadrilaterals are Parallelograms

Converse of the Opposite Sides of a Parallelogram Theorem

Converse of the Opposite Angles in a Parallelogram Theorem

Converse of the Consecutive Angles in a Parallelogram Theorem

Converse of the Diagonals in a Parallelogram Theorem

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D

A

C

B

D

A

C

B

D

A

C

B(180 – x)°

x°

x°

D

A

C

BM

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Example 1Given: ∆PQT @ ∆RSTProve: PQRS is a parallelogram. Statements Reasons

∆PQT @ ∆RST    CPCTCPT = RT  ST = QT    Def. of bisect

PQRS is a parallelogram  

Example 2A gate is braced as shown. How do you know that opposite sides of the gate are congruent?

Congruent and Parallel Sides Theorem

To determine if a quadrilateral is a parallelogram, you need to know one of the following:

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P Q

RS

T

A

B

D

C

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Example 3Show that A(-1,2), B(3,2), C(1,-2), and D(-3,-2) are the vertices of a parallelogram.

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Answer Sometimes, Always or Never

1. A square is a rectangle.

2. A rectangle is a square.

3. A rhombus is a rectangle.

4. A square is a rhombus.

5. A rhombus is a rectangle.

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Geometry 1 Unit 6: Quadrilaterals6.4 Rhombuses, Rectangles, and Squares

Rectangle

Rhombus

Square

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Example 1Decide if each statement is always, sometimes or never true.

A rhombus is a rectangle

A parallelogram is a rectangle

A rectangle is a square

A square is a rhombus

Example 2Given FROG is a rectangle, what else do you know about FROG?

Example 3EFGH is a rectangle. K is the midpoint of FH. EG = 8z – 16,

What is the measure of segment EK?What is the measure of segment GK?

Rhombus Corollary

Rectangle Corollary

SquareCorollary

Perpendicular Diagonals of a Rhombus Theorem

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F R

G O

B

A

C

D

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Diagonals Bisecting Opposite Angles Theorem.

Diagonals in a Rectangle Theorem

Example 4You cut out a parallelogram shaped quilt piece and measure the diagonals to be congruent. What is the shape?

An angle formed by the diagonals of the quilt piece measures 90°. Is the shape a square?

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A

B

D

C

A B

D C

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Geometry 1 Unit 6: Quadrilaterals6.5 Trapezoids and Kites

Trapezoid

Bases

Pairs of Base Angles

Legs

Isosceles Trapezoid

Base Angles of an Isosceles Trapezoid TheoremCongruent Base Angles in a Trapezoid Theorem.

Diagonals in an Isosceles Trapezoid Theorem

Midsegment of a trapezoid

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D

A BC

D

A B

C

D

A B

C

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Midsegment Theorem for Trapezoids

Kite

Diagonals of a Kite Theorem

Opposite Angles in a Kite Theorem

Example 4GHJK is a kite. Find HP.

Example 5RSTU is a kite. Find mÐR, mÐS, and mÐT.

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B CD

NMA

5√29

G

H

P J

K

x + 30°

125°

x°R

S

T

U

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Geometry 1 Unit 6: Quadrilaterals6.6 Special Quadrilaterals

Property Parallelogram Rectangle Rhombus Square Trapezoid Kite

1.Both pairs of opposite sides are congruent

           

2. Diagonals are congruent

           

3. Diagonals are perpendicular

           

4. Diagonals bisect each other

           

5. Consecutive angles are supplementary

           

6. Both pairs of opposite angles are congruent

           

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Geometry 1 Unit 6: Quadrilaterals6.7 Areas of Triangles and Quadrilaterals

Example 1

Example 2What is the base of a triangle that has an area of 48 and a height of 3?

Example 3A rectangle has an area of 100 square meters and a height of 25 meters. Are all the rectangles with these dimensions congruent?

Example 4

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Example 5What is the height of a parallelogram that has an area of 96 square feet and a base length of 8 feet?

Example 6Find the area of trapezoid EFGH.

E(-2, 3), F(2, 4), G(2, -2), H(-2, -1)

Example 7

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Example 8

Example 9

Example 10Find the area of rhombus EFGH if EG = 10 and FH = 15.

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