Name Period Quadrilaterals Chapter 6 - GEOMETRYafhslewis.weebly.com/uploads/1/2/5/4/12543053/ch_6_notes.pdfName _____ Period _____ Quadrilaterals – Chapter 6 - GEOMETRY Section 6.1

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Name ________________________________________ Period _______

Quadrilaterals – Chapter 6 - GEOMETRY Section 6.1 Polygons

GOAL 1: Describing Polygons A polygon is a plane figure that meets the following conditions. ________________________________

1. It is formed by three or more segments called sides, such that no two sides

with a common endpoint are collinear. ___________________________________________

2. Each side intersects exactly two other sides, one at each endpoint. _____________________

Ex. 1 Decide whether the figure is a polygon. If not, explain why.

1. 2. 3. 4. 5.

A vertex is _______________________________________________. The plural of vertex is ________.

You can name a polygon by listing its vertices consecutively.

PQRST is one way to name this polygon. What is another way? ________________

Polygons are also named by the number of sides they have.

A polygon is convex if ___________________________________________________

______________________________________________________________________

A polygon is concave if __________________________________________________

Ex. 2 Use the number of sides to tell what kind of polygon the shape is. Then state whether the polygon is

convex orconcave.

6. 7. 8.

A diagonal of a polygon is a _____________________________________________________________.

# of sides Type of polygon # of sides Type of polygon # of sides Type of polygon

3 Triangle 7 Heptagon 12 Dodecagon

4 Quadrilateral 8 Octagon n n-gon

5 Pentagon 9 Nonagon

6 Hexagon 10 Decagon

2

Ex. 3 Use the diagram at the right to answer the following.

9. Name the polygon by the number of sides it has.

10. Polygon MNOPQR is one name. State two other names.

11. Name all of the diagonals that have vertex M as an endpoint.

12. Name the consecutive angles to .N

A polygon is equilateral if _______________________________________________________________.

A polygon is equiangular if ______________________________________________________________.

A polygon is regular if ____________________________________________________________.

Ex. 4 State whether the polygon is best described as equilateral, equiangular, regular, or none of these.

13. 14. 15. 16.

GOAL 2: Interior Angles of Quadrilaterals

Ex. 5 Use the information in the diagram to solve for x.

17. 18. 19.

Section 6.2 Properties of Parallelograms

GOAL 1: Properties of Parallelograms A parallelogram is a ___________________________________________________________________.

Theorem 6.1 Interior Angles of a Quadrilateral

The sum of the measures of the interior angles

of a quadrilateral is 360°.

.3604321 mmmm

3

Ex. 1 Mark the congruences for the theorems below.

Ex. 2 Lets prove Theorem 6.3 in a paragraph proof.

Given: ABCD is a parallogram.

Prove: DBCA and

Opposite sides of a parallelogram are congruent, so __________________ and __________________.

By the Reflexive Property of Congruence, __________________. CDBABD because of the

________ Congruence Postulate. Because _________________ parts of congruent triangles are congruent,

.CA Now draw diagonal AC. By use of the same reasoning, .DB

Ex. 3 Decide whether the figure is a parallelogram. If it is not, explain why not.

1. 2. 3.

Theorem 6.2

If a quadrilateral is a parallelogram, then its

opposite sides are congruent.

QRSPRSPQ and

Theorem 6.3

If a quadrilateral is a parallelogram, then its

opposite angles are congruent. SQRP and

Theorem 6.4

If a quadrilateral is a parallelogram, then its

consecutive angles are supplementary.

180 ,180

,180 ,180

PmSmSmRm

RmQmQmPm

Theorem 6.5

If a quadrilateral is a parallelogram, then its

diagonals bisect each other.

RMPMSMQM and

4

Ex. 4 Use the diagram of parallelogram MNOP at the right. Complete the statement and give a reason.

4. MN 5. MN P

6. ON 7. MPO

8. PQ 9. QM

10. MQN 11. NPO

Ex. 5 Find the measure in the parallelogram HIJK. Explain your reasoning.

12. HI 13. KH

14. GH 14. HJ

16. mKIH 17. mJIH

18. mKJI 19. mHKI

Ex. 6 Find the value of each variable in the parallelogram.

20. 21.

Section 6.3 Proving Quadrilaterals are Parallelograms

GOAL 1: Proving Quadrilaterals are Parallelograms Theorem 6.6 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Theorem 6.7 If both pairs of opposite angles of a quadrilateral are congruent,

then the quadrilateral is a parallelogram.

Theorem 6.8 If an angle of a quadrilateral is supplementary to both of its consecutive

angles, then the quadrilateral is a parallelogram.

Theorem 6.9 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Theorem 6.10 If one pair of opposite sides of a quadrilateral are congruent and parallel,

then the quadrilateral is a parallelogram.

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We can also use the definition of a parallelogram to prove that a quadrilateral is a parallelogram.

If both pairs of opposite sides are parallel,

then the quadrilateral is a parallelogram.

Ex. 1 Name 6 ways to prove that a quadrilateral is a parallelogram.

Ex. 2 Are you given enough information to determine whether the quadrilateral is a parallelogram?

1. 2. 3.

4. 5. 6.

Ex. 3 What additional information is needed in order to prove that quadrilateral ABCD is a parallelogram?

7. ABPDC 8. AB DC

9. DCA BAC 10. DE EB

11. mCDAmDAB 180

Ex. 4 What value of x and y will make the polygon a parallelogram?

12. 13. 14.

6

GOAL 2: Using Coordinate Geometry When a figure is in the coordinate plane, you can use the Distance Formula to prove that sides are

congruent and you can use the slope formula to prove that sides are parallel.

Ex. 5 Prove that the points represent the vertices of a parallelogram. Use two different methods.

A( 2, -1), B( 1, 3), C( 6, 5), D( 7, 1)

Ex. 6 Draw the quadrilateral ABCD.

If the hat rack were expanded outward, would

ABCD still be a parallelogram? Explain.

Section 6.4 Rhombuses, Rectangles, and Squares

GOAL 1: Properties of Special Parallelograms In this lesson you will study three special types of parallelograms: rhombuses, rectangles, and squares.

A rhombus is a A rectangle is a A square is a

______________________ ______________________ ________________________

______________________ ______________________ ________________________

You can use the following corollaries to prove that a quadrilateral is a rhombus, rectangle, or square without

proving first that the quadrilateral is a parallelogram.

Rhombus Corollary

A quadrilateral is a rhombus if and only if it has four congruent sides.

Rectangle Corollary

A quadrilateral is a rectangle if and only if it has four right angles.

Square Corollary

A quadrilateral is a square if and only if it is a rhombus and a rectangle.

Ex. 1 Decide whether the statement is sometimes, always, or never true.

1. A square is a rectangle. 2. A parallelogram is a rhombus.

3. A rectangle is a square. 4. A rhombus is a rectangle.

5. A parallelogram is a rectangle. 6. A square is a parallelogram.

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GOAL 2: Using Diagonals of Special Parallelograms

The following theorems are about diagonals of rhombuses and rectangles.

Theorem 6.11

A parallelogram is a rhombus if and only if its diagonals are perpendicular. ABCD is a rhombus if and only if AC BD

Theorem 6.12

A parallelogram is a rhombus if and only if each diagonal bisects a pair

of opposite angles. ABCD is a rhombus if and only if

AC bisects DAB and BCD and

BD bisects ADC and CBA

Theorem 6.13 A parallelogram is a rectangle if and only if its diagonals are congruent. ABCD is a rectangle if and only if AC BD

Remember that is a square is both a rectangle and a rhombus.

Ex. 2 List everything you know about squares. (Hint: List everything about parallelograms, rectangles

and rhombuses.

Ex. 3 Match the properties of a quadrilateral with all of the types of quadrilateral which have that property.

7. The diagonals are congruent. A. Parallelogram

8. Both pairs of opposite sides are congruent. B. Rectangle

9. Both pairs of opposite sides are parallel. C. Rhombus

10. All angles are congruent. D. Square

11. All sides are congruent.

12. Diagonals bisect the angles.

Ex. 4 Decide whether the statement is sometimes, always, or never true.

13. A rhombus is equilateral.

14. The diagonals of a rectangle are perpendicular.

15. The opposite angles of a rhombus are supplementary.

16. The diagonals of a rectangle bisect each other.

17. The consecutive angles of a square are supplementary.

8

Ex. 5 Find the value of x.

18. MNOP is a square 19. DEFG is a rhombus. 20. WZYZ is a rectangle.

Section 6.5 Trapezoids and Kites

GOAL 1: Using Properties of Trapezoids

A trapezoid is a ____________________________________________________

____________________________________________. The parallel sides are the

___________. A trapezoid has two pairs of ____________________. The

nonparallel sides are the ________ of the trapezoid. If the legs of a trapezoid are

congruent, then the trapezoid is an ____________________________________.

Ex. 1 Match the pairs of segments or angles with the term, which describes them in trapezoid PQRS.

1. S and P A. bases

2. QS and PR B. legs

3. QR and PS C. diagonals

4. Q and S D. base angles

5. PQ and RS E. opposite angles

Theorem 6.14 If a trapezoid is isosceles, then each pair of base angles is congruent.

A B, C D

Theorem 6.15

If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid.

ABCD is an isosceles trapazoid.

Theorem 6.16

A trapezoid is isosceles if and only if its diagonals are congruent.

ABCD is isosceles if and only if AC BD

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Ex. 2 Complete the statement with always, sometimes or never.

6. A trapezoid is _________________________ a parallelogram.

7. The bases of a trapezoid are ____________________parallel.

8. The base angles of an isosceles trapezoid are ____________________ congruent.

9. The legs of a trapezoid are _____________________ congruent.

Ex. 3 Find the angle measures of ABCD.

10. 11.

The midsegment of a trapezoid is the ________________________________________________________.

Theorem 6.17 Midsegment Theorem for Trapezoids

The midsegment of a trapezoid is parallel to each base and its length is

one half the sum of the lengths of the bases.

MN PAD, MN PBC, MN 1

2(AD BC)

Ex. 4 Find the length of the midsegment .RT

12. 13. 14.

GOAL 2: Using Properties of Kites

A kite is a _________________________________________________________

___________________________________________.

Theorem 6.18 If a quadrilateral is a kite, then its diagonals are perpendicular.

Theorem 6.19 If a quadrilateral is a kite, then exactly one pair of opposite

angles are congruent.

10

Ex. 5 Find the length of the sides to the nearest hundredth or the measure of the angles in kite KITE.

15. 16. 17.

Section 6.6 Special Quadrilaterals

GOAL 1: Summarizing Properties of Quadrilaterals

Ex. 1 Summarize the seven special types of quadrilaterals in a diagram.

Ex. 2 Put an X in the box if the shape always has the given property.

Property gram Rectangle Rhombus Square Kite Trapezoid Isosceles

Trapezoid Both pairs of opp.

sides are

Exactly 1 pair of

opp. sides are

Diagonals are

Diagonals are

Diagonals bisect

each other

Both pairs of opp.

Sides are

Exactly 1 pair of

opp. Sides are

All sides are

Both pairs of opp.

's are

Exactly 1 pair of

opp. 's are

All 's are

Area

11

Ex. 3 Identify the special quadrilateral. Use the most specific name.

1. 2. 3.

Ex. 4 What quadrilateral meet the conditions shown? ABCD is not drawn to scale.

4. 5. 6.

Section 6.7 Areas of Triangles and Quadrilaterals

GOAL 1: Using Area Formulas

Area Postulates

Postulate 22 Area of a Square Postulate

The area of a square is the square of the length of its side, or A s2

Postulate 23 Area Congruence Postulate

If two polygons are congruent, then they have the same area.

Postulate 24 Area Addition Postulate

The area of a region is the sum of the areas if its nonoverlapping parts.

Area Theorems

Theorem 6.20 Area of a Rectangle

The area of a rectangle is the product of its base and height.

A bh

Theorem 6.21 Area of a Parallelogram

The area of a parallelogram is the product of a base and its corresponding height. A bh

Theorem 6.22 Area of a Triangle

The area of a triangle is one half the product of a base and its corresponding height.

A 1

2bh

12

Ex. 1 Find the area of the polygon.

1. 2. 3.

GOAL 2: Areas of Trapezoids, Kites, and Rhombuses

Theorem 6.23 Area of a Trapazoid

The area of a trapezoid is on half the product of the height and the sum of the basses.

A 1

2h(b1 b2 )

Theorem 6.24 Area of a Kite

The area of a kite is one half the product of the lengths of its diagonals.

A 1

2d1d2

Theorem 6.25 Area of a Rhombus

The area of a rhombus is equal to one half the product of the lengths of the diagonals.

A 1

2d1d2

Ex. 2 Find the area of the polygon.

4. 5. 6.

7. 8. 9.