1

Name the vertices:

Name the sides:

Name the diagonals containing G:

Name 2 consecutive ’s:

Name 2 nonconsecutive sides:

6.1 What is a Polygon?

Regular Polygon-

Polygon Formulas: (n = # of sides)

Sum of the interior angles of a polygon = 180°(n - 2)

Sum of the exterior angles of a polygon = 360°

Polygon Names

1. Find the sum of the measures of the angles of a convex polygon with 14 sides.

2. For the given regular polygon, find the measure of each of its interior angles:

a) dodecagon b) 16 – gon

3. Find the degree measure of each exterior angle of a regular polygon with 20 sides.

4. For the following measures of an angle of a regular polygon, find the number of sides.

a) 160 b) 140

Sides Name

N

3 ____________________________

4 ____________________________

5 ____________________________

6 ____________________________

7 ____________________________

8 ____________________________

9 ____________________________

10 ____________________________

12 ____________________________

C O

U

G A

R

One interior angle: 180(𝑛−2)

𝑛

One exterior angle: 360

𝑛

2

5. The sum of the interior angles of a convex polygon is 2520. Find the number of sides.

6. Find the number of sides of a regular polygon if the measure of one of its interior angles

Is three times the measure of its adjacent exterior angle.

Find the sum of the measures of the angles of a convex polygon with the given # of sides.

1. 17 2. 20 3. 12

For each of the following, the measure of one angle of a regular convex polygon is given. Find the # of sides.

4. 150 5. 120 6. 156

For each of the following, the number of sides of a regular polygon is given. Find the measure of each angle.

7. 4 8. 8 9. 10

Find the degree measure of one exterior angle for a regular polygon with the given # of sides

10. 8 11. 5 12. 13

15. The sum of the measure of the interior angles of a convex polygon is 1260. Classify the polygon.

16. The measure of one exterior angle of a regular polygon is 45. Classify the polygon.

17. Find the number of sides of a regular polygon, if the measure of one of its interior angles equals the

measure of its adjacent exterior angle.

18. Find the number of sides of a regular polygon, if the measure of one of its interior angles equals twice the

measure of the adjacent exterior angle.

19. Classify the regular polygon, if the measure of one of its interior angles equals one-half the measure of the

adjacent exterior angle.

20. If the sum of the measures of six interior angles of a heptagon is 755, what is the measure of the

remaining angle?

Tell whether each figure is a polygon. If it is a polygon, name it by the number

of its sides.

1.

2.

3.

4. For a polygon to be regular, it must be both equiangular and equilateral.

Name the only type of polygon that must be regular if it is equiangular._____________________

Tell whether each polygon is regular or irregular. Then tell whether it is concave or

convex.

5.

6.

7.

8. Find the sum of the interior angle measures of a 14-gon. ____________________

9. Find the measure of each interior angle of hexagon ABCDEF.

_________________________________________________________

_________________________________________________________

10. Find the value of n in pentagon PQRST.

________________________________________

3

6.2-6.3

A Quadrilateral is any 4-sided polygon. The sum of interior angles for every quadrilateral is 360.

Kites

2 pairs of congruent

consecutive sides (unequal)

The diagonals are

perpendicular

Squares

All properties of the

parallelogram, the rectangle

and the rhombus

Trapezoids

Has only 1 pair of parallel sides

(base 1 & base 2)

Non-parallel sides are called legs

Rectangles

All properties of

parallelogram

Four right angles

Diagonals are congruent &

bisect each other

Consecutive sides are

perpendicular

Parallelograms

Opposite sides are congruent

Opposite sides are parallel

Opposite angles are congruent

Consecutive angles are

supplementary

The diagonals bisect each

other

Rhombus

All the properties of parallelogram

Four congruent sides

Consecutive sides are congruent

Diagonals bisect opposite angles

Diagonals perpendicular & bisect

each other

Example 1. m<DCA = 27˚

m<CAD = 38˚

m<ABD = 63˚

Find each

measure.

Solve for x and y.

4

E

D

C B

A

7

6

12 True/False 1. Every parallelogram is a quadrilateral.

2. Every quadrilateral is a parallelogram.

3. All angles of a parallelogram are

congruent.

4. Opposite sides of a parallelogram are

always congruent.

5. In APEX , PXAP // .

6. In CARY, AYCR .

7. In TOAD, TA and OD bisect each

other.

˚ ˚

4

Proving a Quadrilateral is a parallelogram

A quadrilateral is a parallelogram if:

1. both pairs of opposite sides are parallel (by definition)

2.

3.

4.

5.

5

6.4-6.5 Rectangles Definition: A rectangle is a quadrilateral with _______________________.

Definition: A rectangle is a parallelogram with _______________________.

To prove that a quadrilateral is a rectangle, prove that:

1) It is a quadrilateral with _______________________.

2) It is a parallelogram with ______________________.

3) It is a parallelogram with ______________________.

Which of the following quadrilaterals are rectangles? Justify your answer.

1. 2. 3.

For 4 – 10, ABCD is a parallelogram. From the information given, tell whether ABCD is a rectangle.

4. Given: ABAD

5. Given: DBAC

6. Given: BCD is a right angle.

7. Given: BDAC

8. Given: BDAC ; ADC is a right angle

9. Given: ADC BCD

10. Given: DAC BAC

11. Find x and y Given: Diagonals RP and SQ of rectangle PQRS meet at M.

If PM = x + 3y, SM = 4y – 2x and RM = 20.

Rhombus Definition: A quadrilateral is a rhombus iff _______________________.

Definition: A parallelogram is a rhombus iff _______________________.

Mark the rhombus. How many ’s?

What must be true about HBO?

Therefore, diagonals must be _____.

Theorem: A parallelogram is a rhombus iff _________________________.

A B

D C

R

H

O

M

B

6

To Prove that a quadrilateral is a rhombus, prove that:

1) It is a quadrilateral with _________________________________________.

2) It is a parallelogram with ________________________________________.

3) It is a parallelogram with ________________________________________.

4) It is a parallelogram with ________________________________________.

Find all interior angles of the following rhombus.

Which of the following are rhombuses? Justify each answer.

1. 2. 3.

For 4 – 10, ABCD is a parallelogram. From the info. Given tell whether ABCD is a rhombus.

__________4. Given: ADAB

__________5. Given: DBAC

__________6. Given: BCD is a right angle

__________7. Given: BDAC

__________8. Given: BDAC ; ADC is a right angle

__________9. Given: ADC BCD

__________10. Given: DAC BAC

11. In rhombus ABCD, mABD = 3x – 5 and mBAC = 11x – 3. Find the measures of all the angles of the

rhombus.

12. In parallelogram ABCD, AB = 17x – 3, BC = 13x + 5, and CD = 4x + 23. Find the lengths of the sides of

parallelogram ABCD. What type of parallelogram is ABCD?

23

A

B

C

D

7

A parallelogram is a square iff it has one right angle and 2 adjacent sides.

A square is both a ____________________ and a ____________________.

A square has all of the properties of a _____________, ______________, and ________________.

To prove a quadrilateral is a square, prove that:

1) It is a rectangle with ________________________________________.

2) It is a rectangle with ________________________________________.

3) It is a rectangle with _________________________________________.

4) It is a rhombus with _________________________________________.

5) It is a rhombus with _________________________________________.

6) It is a parallelogram with _____________________________________.

Complete the following.

1. Every rectangle is also a ______________. 2. Every rhombus is also a ___________.

3. Every square is also a ___________, and a _________________.

4. A with diagonals is a __________ or a ___________.

5. A with diagonals is a ___________ or a ____________.

6. A whose diagonals are the bisectors of each other is a _________ or a ________.

True or False.

___________7. All rhombi are parallelograms.

___________8. Some rectangles are squares.

___________9. All parallelograms are rectangles.

___________10. Some rhombi are rectangles. _________11. All rectangles are squares.

___________12. All squares are rectangles. _________13. Some squares are rectangles.

Use square ABCD and the given information to find each value.

14. If mAEB = 3x, find x.

15. If mBAC = 9x, find x.

16. If AB = 2x + 1 and CD = 3x – 5, find BC

17. If mDAC = y and mBAC = 3x, find x and y.

18. If AB = x2 -15 and BC = 2x, find x.

A

D

B

C

E

A

D

B

C

E

8

6.6 Kites and Trapezoids

A kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other. Properties:

1. 2. 3.

Find the value(s) of the variable(s) in each kite.

4. 5. 6.

Can two angles of a kite be as follows? 7. opposite and acute 8. consecutive and obtuse

9. opposite and supplementary 10. consecutive and supplementary

11. opposite and complementary 12. consecutive and complementary

13. The perimeter of a kite is 66 cm. The length of one of its sides is 3 cm less

that twice the length of another. Find the length of each side of the kite.

Trapezoids

A trapezoid is a quadrilateral with exactly two parallel sides.

Parts of a trapezoid:

Isosceles Trapezoid: A trapezoid with congruent legs.

15y

(2x-4)

2x

(x+6)

(4x- 30)

(2y-20)

(3x+5)

y

3x

2

6xy

45 3

2 1

7

6

35

5 3

4

2

1

Base

Base

Leg Leg

Base Base

Leg

Leg

9054

9

find LK

c. RS = x + 5

HJ + LK = 4x + 6

find RS

Theorem: The base angles of an isosceles trapezoid are congruent.

Theorem: The diagonals of an isosceles trapezoid are congruent.

Every trapezoid contains two pairs of consecutive angles that are supplementary.

Example 1: Given the trapezoid HLJK. If the 65Jm and the 95Km , the

measure of angles H and L .

Example 2: Use Isosceles Trapezoid ABCD with

length of AD = BC.

Definition: Am altitude is a line segment from one vertex of one base of the trapezoid and perpendicular to the

opposite base.

Theorem: The length of the median of a trapezoid equals one-half the sum of the bases.

212

1bbm

Example 3: Find the missing measures of the given trapezoid.

a. mIRD

b. YR

c. DR

d. AC

Example 4: HJKL is an isosceles trapezoid with bases HJ and LK , and median RS . Use the given

information to solve each problem.

a. LK = 30

HJ = 42

find RS

b. RS = 17

HJ = 14

A B

C D

//

H

R

L K

S

J

I B 7

A C

D X Y R 75

3

a. mDAB = 75. Find the mADC.

b. AC = 40. Find BD.

c. If 256 xAm and

158 xBm , find the measures

of angle C and D.

10

Example 5: Find the length of each side of the isosceles trapezoid below.

6x + 5 B C

D A

5x + 12 28 - 3x

14x

11

Algebraic Formulas Used to Determine the Type of Quadrilateral To Show that a quadrilateral is a Parallelogram

Method 1: Both pairs of opposite sides are congruent (find distance)

Method 2: Both pairs of opposite sides are parallel (find slope)

Method 3: One pair of opposite sides are both parallel and congruent (find distance and slope)

To show that a quadrilateral is a Rhombus ****FIRST show that it is a parallelogram****

Method 1: All 4 sides are congruent

Method 2: Diagonals are perpendicular (find slope of diagonals)

To show that a quadrilateral is a Rectangle ****FIRST show that it is a parallelogram****

Method 1: All angles are right angles

Method 2: Diagonals are congruent (find distance of diagonals)

To show that a quadrilateral is an Isosceles Trapezoid

Graph first o Legs are congruent (find distance)

o Bases are parallel (find slope)

Diagonals are congruent

To show that a quadrilateral is a Kite

Two pairs of consecutive congruent sides that are not congruent to each other (find the distance)

Practice Determining Quadrilaterals (Most Precise Name)

1) Given coordinates of quadrilateral EFGH are E (6, 5), F (6, 11), G (14, 18), and H (14, 12)

A) Determine if it is a parallelogram by checking to see if opposite sides are parallel. (slope)

B) Determine if it is a parallelogram by checking to see if opposite sides are . (distance)

C) Determine if it is a parallelogram by checking to see if diagonals bisect each other. (midpoint)

2) Given quadrilateral ABCD with coordinates A (-1, -2), B (4, 4), C (10, -1), and D (5, -7).

Is quad ABCD a rectangle? __________ (Note: Use the slope formula.)

Slope of = _____ Slope of = _____

Slope of = _____ Slope of = _____

3) The coordinates of quadrilateral QRST are: Q (-2, -1), R (-1, 2), S (2, 3), and T (1, 0).

a) Find the slopes of the diagonals of quad QRST. Are they perpendicular? ________

Slope of = _______ Slope of = _______

b) Find the midpoints of each of the diagonals.

Midpoint of = (_____, _____) Midpoint of = (_____, _____)

Do they bisect each other? _________

Why or why not? _______________________________________

c) What are all the possible classifications for quad QRST? _________________________

d) The most precise name?_____________________________

AB CD

BC AD

QS RT

QS RT

12

4) Given: A (-1,-6), B (1,-3), C (11, 1) and D (9,-2)

Show that Quad. ABCD is a parallelogram.

5) Given: EFGH is a parallelogram with E(-4,1), F(2,3), G(4,9) and H(-2,7)

Show that EFGH is a rhombus.

6) Given: RSTU is a parallelogram with R (-4, 5), S (-1, 9), T (7, 3) and U (4,-1)

Show that RSTU is a rectangle.

7) Given: Quad. ABCD with A (6,-4), B (6, 2), C (3, 2) and D (3,-4). Is this a parallelogram?

HW For #8–10, use slope, midpoint and/or the distance formulas to determine the most precise name for the

quadrilateral with the given vertices.

8) A (-4, 3), B (-4, 8), C (3, 10) D (3, 5)

9) A (-3, 7), B (1, 10), C (1, 5), D (-3, 2)

10) A (6, -5), B (3, 10), C (0, -5), D (3, -9)

11) A (-3, -3), B (3, 4), C (5, 0), D (-4, -1)

12) A (6, 0), B (0, 6), C (-6, 0), D (0, -6)