Geometry CP Section 6-1: Angles of Polygons Page 1 of 2

Main ideas: Find the sum of the measures of the interior angles of a polygon to classify figures and solve

problems.

Find the sum of the measures of the exterior angles of a polygon to classify figures and solve

problems.

Standard: 12.0

What is the sum of the angles in a triangle? _____________

A __________________ is a simple closed figure in a plane formed by three or more line segments.

A diagonal of a polygon is a segment that connects any two nonconsecutive vertices. For the polygons above,

draw the diagonals from one vertex of each figure, and complete the table below.

What pattern can you find in the table above?

__________________________________________________________

Example 1: A convex polygon has 13 sides. Find the sum of the measures of the interior angles.

Practice 1: Find the sum of the measures of the interior angles of each convex polygon.

1) 10-gon 2) 16-gon 3) 30-gon 4) 3x-gon

Example 2: The measure of an interior angle of a regular polygon is 120. Find the number of sides.

Practice 2:

The measure of an interior angle of a regular polygon is given. Find the number of sides in each polygon.

5) 150 6)160 7) 175

Example 3: Find x

Name of polygon Number of sides Number of triangles formed by diagonals Sum of the interior angles

Triangle 3 1 (1 • 180)= 180

Quadrilateral

Pentagon

Hexagon

Heptagon

Octagon

Nonagon

Decagon

Quadrilateral

Geometry CP Section 6-1: Angles of Polygons Page 2 of 2

Practice 3: Find x.

8) 9)

Measure the exterior angles in the polygons below, and complete the table that follows.

What pattern do you notice?

________________________________________________________________________

Example 4: Find the missing angle.

Example 5: Find the measures of an interior angle and an exterior angle for a regular 27-gon.

Practice 5: Find the measures of an interior angle and en exterior angle given the number of sides of each

regular polygon. Round to the nearest tenth if necessary.

10) 16 11) 23 12) 45

Polygon Number of sides Sum of exterior angles

Triangle

Quadrilateral

Pentagon

Geometry CP Lesson 6-2: Parallelograms Page 1 of 2

Main ideas: Recognize and apply properties of the sides and angles of parallelograms.

Recognize and apply properties of the diagonals of parallelograms.

Standard: 7.0

A parallelogram is a quadrilateral with both pairs of opposite sides ______________.

Here are four important properties of parallelograms:

Example 1: If ABCD is a parallelogram, find a and b.

Practice 1: Find x and y in each parallelogram.

Geometry CP Lesson 6-2: Parallelograms Page 2 of 2

Example 2: Complete each statement about parallelogram DEFG. Justify your answer.

Example 3: Find x and y in parallelogram ABCD.

Practice 3: Find x and y in each parallelogram.

Example 4:

Statements Reasons

1) 1) Given

2) 2) Opp. s of a quad. are

3) 3) Transitive Prop.

Geometry CP Lesson 6-3: Tests for Parallelograms Page 1 of 2

Main ideas: Recognize the conditions that ensure a quadrilateral is a parallelogram.

Prove that a set of points forms a parallelogram in a coordinate plane.

Standard: 7.0

By definition, the opposite sides of a parallelogram are ______________. So, if a quadrilateral has each pair of

opposite sides parallel, then it is a parallelogram. Other tests can be used to determine if a quadrilateral is a

parallelogram.

In other words:

Example 1: Determine whether the quadrilateral is a parallelogram. Justify your answer.

______________________________ _____________________

Practice 1: Determine whether each quadrilateral is a parallelogram. Justify your answer.

Example 2: Find x and y so that the quadrilateral is a parallelogram.

Geometry CP Lesson 6-3: Tests for Parallelograms Page 2 of 2

Practice 2:

Coordinate Geometry

Example 3: Determine whether a figure with the given vertices is a parallelogram. Use the method indicated.

Review: Write the following formulas:

Slope formula: ____________________________________________

Midpoint formula: _________________________________________

Distance formula: _________________________________________

Geometry CP Lesson 6-4: Rectangles Page 1 of 1

A rectangle is a quadrilateral with _________ ______________ angles.

Properties of a rectangle

Example 2:

Example 3:

Example 1: PRST is a rectangle.

Find each measure if m1=50o.

Geometry CP Lesson 6-5: Rhombi and Squares Page 1 of 2

Main ideas: Recognize and apply the properties of rhombi.

Recognize and apply the properties of squares.

Standard: 7.0

A rhombus is a quadrilateral with all four _____________ sides. Opposite sides are congruent, so a rhombus is

also a parallelogram and has all the properties of a parallelogram. Rhombi also have the following properties:

Example 1: In rhombus ABCD, mABC=32. Find the measure of each numbered angle.

Example 2:

Example 3:

Geometry CP Lesson 6-5: Rhombi and Squares Page 2 of 2

Example 4:

Example 5:

Statements Reasons

1) KGH, HJK, GHJ, and JKG

are isosceles

1)

2) 2) Def. of isosceles

3) 3) Transitive Prop.

4) 4)

5)GHJK is a rhombus. 5)

Hint: Apply the slope

formula and the

distance formula to

both diagonals. If the

diagonals are , the

quadrilateral is a

rhombus. If the

diagonals are , the

quad. is a rectangle. If

the diagonals are both

and , the quad. is a

square.

Geometry CP Lesson 6-6: Kites and Trapezoids Page 1 of 2

Main ideas: Recognize and apply the properties of trapezoids.

Solve problems involving the medians of trapezoids.

Standard: 7.0

In this lesson, we will study two special quadrilaterals that are not parallelograms.

Kites

Definition: A quadrilateral with exactly two pairs of adjacent congruent sides.

Properties of kites:

Diagonals are perpendicular KT IE

Only one pair of opposite angles congruent. I E

Ex 1: If IX = 5 and XC = 12, find IC and EC. Ex 2:

Trapezoids

Definition: A quadrilateral with exactly one pair of parallel sides.

A trapezoid has two bases and two legs.

Corresponding angles are supplementary. R and T, A and P are supplementary

If its legs are congruent, then it’s an isosceles trapezoid.

In an isosceles trapezoid, both pairs of base angles are congruent.

In an isosceles trapezoid, the diagonals are congruent.

To prove that a quadrilateral is a trapezoid:

1. Find slopes of both bases and see if they are the same.

2. Find slopes of both legs and make sure they are different.

To prove that a trapezoid is isosceles:

1. Use the distance formula to check of the legs have the same length

Definition: The median (or midsegment) of a trapezoid is the segment that joins the midpoint of the legs.

The median is parallel to both bases. BE AD CF

The length of the median is equal to ½ the sum of the lengths of the bases. 1

2BE AD CF

K

I

T

E

KI KE and TI TE

N

I

C

E

X N

I

C

E

120

30

R A

P T

RA TP

A D

F C

E B

Given 4 points, you would use

the distance formula to prove

that the quad. is a kite.

mE= __________ mN=_________

Geometry CP Lesson 6-6: Kites and Trapezoids Page 2 of 2

Geometry CP Lesson 6-7: Coordinate Proofs with Quadrilaterals Page 1 of 2

Main ideas: Position and label quadrilaterals for use in coordinate proofs.

Prove theorems using coordinate proofs.

Standards: 7.0 and 17.0

Coordinate proofs use properties of lines and segments to prove geometric properties. The first step in writing a

coordinate proof is to place the figure on the coordinate plane in a convenient way. Use the following guidelines

for placing a figure on the coordinate plane.

Some examples of quadrilaterals placed on the coordinate plane are given below. Notice how the figures have

been placed so the coordinates of the vertices are as simple as possible.

Example 1: Position and label each quadrilateral on the coordinate plane.

Once a figure has been placed on the coordinate plane, we can prove theorems using the Slope, Midpoint, and

Distance formulas.

Let’s review again:

Slope formula: ____________________________ Midpoint formula: _____________________________

Distance formula: ________________________________________________________________________

isosceles triangle with base b units and

height a units

Geometry CP Lesson 6-7: Coordinate Proofs with Quadrilaterals Page 2 of 2

Example 2:

Example 3:

Position and label the figure on the coordinate plane. Then write a coordinate proof for the following.

7. The diagonals of a parallelogram bisect each other.

Practice

Write a coordinate proof to show that the diagonals of a square are perpendicular.

Geometry CP Hierarchy of Polygons Page 1 of 1

A hierarchy is a ranking of classes or sets of things. Examples of classes of polygons are rectangles, rhombi,

trapezoids, parallelograms, squares and quadrilaterals. Arrange the categories in the word bank inside the

hierarchy below.

Using the hierarchy above, answer the following questions:

1) Are rectangles squares? ______________

2) Are squares rhombi? ________________

3) To which category do isosceles trapezoids belong? _____________________

4) To which category do kites belong? ______________

5) Are squares rectangles? _____________

6) Are kites trapezoids? _________________

7) Are quadrilaterals parallelograms? ________

8) Are rectangles parallelograms? ___________

9) Are rhombi parallelograms?_____________________

10) What could a square be considered? _____________________

isosceles trapezoids polygons rhombi

kites quadrilaterals squares

parallelograms rectangles trapezoids