Section 6.1 The Polygon Angle-Sum Theorem

Students will be able to:•Find the sum of the measures of the interior

angles of a polygon.•Find the sum of the measures of the exterior

angles of a polygon.Lesson Vocabulary•Equilateral polygon

•Equiangular polygon•Regular polygon

Section 6.1 The Polygon Angle-Sum Theorem

List the names of all of the polygons with 3 sides to 13 sides:

3 sided: ____________ 8 sided: ____________4 sided: ____________ 9 sided: ___________5 sided: ____________ 10 sided: ___________6 sided: ____________ 11 sided: ___________7 sided: ____________ 12 sided: ___________

13 sided: ___________

Section 6.1 The Polygon Angle-Sum TheoremA diagonal is a segment that connects two

nonconsecutive vertices in a polygon!

Section 6.1 The Polygon Angle-Sum Theorem

The Solve It is related to a formula for the sum of the interior angle measures of a CONVEX

polygon.

Section 6.1 The Polygon Angle-Sum Theorem

Essential Understanding:The sum of the interior angle measures of a polygon depends on the number of sides the

polygon has.

By dividing a polygon with n sides into (n – 2) triangles, you can show that the sum of the interior angle measures of any polygon is a

multiple of 180.

Section 6.1 The Polygon Angle-Sum Theorem

Problem 1: Finding a Polygon Angle SumWhat is the sum of the interior angle measures of

a heptagon?

Section 6.1 The Polygon Angle-Sum Theorem

Problem 1b: Finding a Polygon Angle SumWhat is the sum of the interior angle

measures of a 17-gon?

Section 6.1 The Polygon Angle-Sum Theorem

Problem 1c: The sum of the interior angle measures of a

polygon is 1980. How can you find the number of sides in the polygon? Classify it!

Section 6.1 The Polygon Angle-Sum Theorem

Problem 1d: The sum of the interior angle measures of a

polygon is 2880. How can you find the number of sides in the polygon? Classify it!!!

Section 6.1 The Polygon Angle-Sum Theorem

Section 6.1 The Polygon Angle-Sum Theorem

Section 6.1 The Polygon Angle-Sum Theorem

Problem 2: What is the measure of each interior angle in a

regular hexagon?

Section 6.1 The Polygon Angle-Sum Theorem

Problem 2b: What is the measure of each interior angle in a

regular nonagon?

Section 6.1 The Polygon Angle-Sum Theorem

Problem 2c: What is the measure of each interior angle in a

regular 100-gon?

Explain what happens to the interior angles of a regular figure the more sides the figure has?

What is the value approaching but will never get to?

Section 6.1 The Polygon Angle-Sum Theorem

Problem 3: What is m<Y in pentagon TODAY?

Section 6.1 The Polygon Angle-Sum Theorem

Problem 3b: What is m<G in quadrilateral EFGH?

Section 6.1 The Polygon Angle-Sum Theorem

You can draw exterior angles at any vertex of a polygon. The figures below show that the sum of

the measures of exterior angles, one at each vertex, is 360.

Problem 4: What is m<1 in the regular octagon below?

Problem 4b: What is the measure of an exterior angle of a

regular pentagon?

Problem 5: What do you notice about the sum of the interior

angle and exterior angle of a regular figure?

Problem 6: If the measure of an exterior angle of a regular polygon is 18. Find the measure of the interior

angle. Then find the number of sides the polygon has.

Problem 6b: If the measure of an exterior angle of a regular polygon is 72. Find the measure of the interior

angle. Then find the number of sides the polygon has.

Problem 6c: If the measure of an exterior angle of a regular polygon is x. Find the measure of the interior

angle. Then find the number of sides the polygon has.

Section 6.2 – Properties of Parallelograms

Students will be able to:•Use relationships among sides and angles of

parallelograms•Use relationships among diagonals of

parallogramsLesson Vocabulary:

•Parallelogram•Opposite Angles•Opposite Sides

•Consecutive Angles

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

Essential Understanding:Parallelograms have special properties regarding

their sides, angles, and diagonals.

In a quadrilateral, opposite sides do not share a vertex and opposite angles do not share a side.

Angles of a polygon that share a side are consecutive angles. In the diagram, <A and <B

are consecutive angles because the share side AB.

Problem 1:What is <P in Parallelogram PQRS?

Problem 1b:Find the value of x in each parallelogram.

Problem 2:Solve a system of linear equations to find the

values of x and y in Parallelogram KLMN. What are KM and LN?

Problem 2b:Solve a system of linear equations to find the

values of x and y in Parallelogram PQRS. What are PR and SQ?

Problem 3:

Extra Problems:

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

Extra Problems:

Find the measures of the numbered angles for each parallelogram.

Extra Problems:

Extra Problems:

Extra Problems:

Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

Students will be able to:•Determine whether a quadrilateral is a

parallelogram

Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

Essential Understanding:You can decide whether a quadrilateral is a

parallelogram if its sides, angles, and diagonals have certain properties.

In Lesson 6-2, you learned theorems about the properties of parallelograms. In this lesson, you will learn the converses of those theorems. That is, if a quadrilateral has certain properties, then it

must be a parallelogram.

Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

Problem 1: For what value of y must PQRS be a

parallelogram?

Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

Problem 1b: For what value of x and y must ABCD be a

parallelogram?

Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

Problem 1c: For what value of x and y must ABCD be a

parallelogram?

Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

Problem 1d: For what value of x and y must ABCD be a

parallelogram?

Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

Problem 1e: For what value of x and y must ABCD be a

parallelogram?

Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

Problem 2: Can you prove that the quadrilateral is a parallelogram based

on the given information? Explain!

Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

Problem 2: Can you prove that the quadrilateral is a parallelogram based

on the given information? Explain!

Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

Problem 3: A truck sits on the platform of a vehicle lift. Two moving

arms raise the platform until a mechanic can fit underneath. Why will the truck always remain parallel to the ground as it

is lifted? Explain!

Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

Students will be able to:•Define and Classify special types of

parallelograms•Use the Properties of Rhombuses and

RectanglesLesson Vocabulary

•Rhombus•Rectangle

•Square

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

A rhombus is a parallelogram with four congruent sides.

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

A rectangle is a parallelogram with four right angles.

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

A square is a parallelogram with four congruent sides and four right angles.

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

Problem 1:Is Parallelogram ABCD a rhombus, rectangle or

square? Explain!

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

Problem 1b:Is Parallelogram EFGH a rhombus, rectangle or

square? Explain!

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

Problem 2: What are the measures of the numbered angles

in rhombus ABCD?

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

Problem 2: What are the measures of the numbered angles

in rhombus PQRS?

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

Problem 2: What are the measures of the numbered angles

in the rhombus?

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

Problem 2: What are the measures of the numbered angles

in the rhombus?

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

Problem 3: In rectangle RSBF, SF = 2x + 15 and

RB = 5x – 12. What is the length of a diagonal?

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

Problem 4: LMNP is a rectangle. Find the value of x and the

length of each diagonal

LN = 5x – 8 and MP = 2x + 1

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

Problem 5: Determine the most precise name for each

quadrilateral.

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

Problem 6: List all quadrilaterals that have the given property.

Chose among parallelogram, rhombus, rectangle, or square.

Opposite angles are congruent.

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

Problem 6b: List all quadrilaterals that have the given property.

Chose among parallelogram, rhombus, rectangle, or square.

Diagonals are congruent

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

Problem 6c: List all quadrilaterals that have the given property.

Chose among parallelogram, rhombus, rectangle, or square.

Each diagonal bisects opposite angles

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

Problem 6d: List all quadrilaterals that have the given property.

Chose among parallelogram, rhombus, rectangle, or square.

Opposite sides are parallel

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

Students will be able to:Determine whether a parallelogram is a rhombus

or rectangle.

Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

Problem 1:Can you conclude that the parallelogram is a rhombus, a rectangle, or a square? Explain!

Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

Problem 1b:Can you conclude that the parallelogram is a rhombus, a rectangle, or a square? Explain!

Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

Problem 1c:Can you conclude that the parallelogram is a rhombus, a rectangle, or a square? Explain!

Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

Problem 1d:Can you conclude that the parallelogram is a rhombus, a rectangle, or a square? Explain!

Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

Problem 2:For what value of x is parallelogram ABCD a

rhombus?

Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

Problem 2b:For what value of x is the parallelogram a

rectangle?

Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

Problem 2c:For what value of x is the parallelogram a

rhombus?

Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

Problem 2d:For what value of x is the parallelogram a

rectangle?

Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

Problem 2e:For what value of x is the parallelogram a

rectangle?

Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

Problem 3:Builders use properties of diagonals to “square up” rectangular

shapes like building frames and playing-field boundaries. Suppose you are on the volunteer building team at the right. You are helping to lay out a rectangular patio for a youth center. How

can you use the properties of diagonals to locate the four corners?

Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

Problem 4:Determine whether the quadrilateral can be a

parallelogram. Explain!

The diagonals are congruent, but the quadrilateral has no right angles.

Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

Problem 4b:Determine whether the quadrilateral can be a

parallelogram. Explain!

Each diagonal is 3 cm long and two opposite sides are 2 cm long.

Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

Problem 4c:Determine whether the quadrilateral can be a

parallelogram. Explain!

Two opposite angles are right angles but the quadrilateral is not a rectangle.

Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

Section 6.6 – Trapezoids and KitesStudents will be able to:

Verify and use properties of trapezoids and kites.

Lesson Vocabulary•Trapezoid

•Base•Leg

•Base angle•Isosceles trapezoid

•Midsegment of a trapezoid•Kite

Section 6.6 – Trapezoids and KitesA trapezoid is a quadrilateral with exactly one pair of

parallel sides. The parallel sides are of trapezoid are called bases.

The nonparallel sides are called legs. The two angles that share a base of a trapezoid are

called base angles. A trapezoid has two pairs of base angles.

Section 6.6 – Trapezoids and Kites

An isosceles trapezoid is a trapezoid with legs that are congruent. ABCD below is an isosceles trapezoid. The angles of an isosceles trapezoid

have some unique properties.

Section 6.6 – Trapezoids and Kites

Section 6.6 – Trapezoids and Kites

Section 6.6 – Trapezoids and Kites

Problem 1:CDEF is an isosceles trapezoid and m<C = 65.

What are m<D, m<E, and m<F?

Section 6.6 – Trapezoids and Kites

Problem 1b:PQRS is an isosceles trapezoid and m<R = 106.

What are m<P, m<Q, and m<S?

Section 6.6 – Trapezoids and Kites

Problem 2:The second ring of the paper fan consists of 20 congruent

isosceles trapezoids that appear to form circles. What are the measures of the base angles of these trapezoids?

Section 6.6 – Trapezoids and Kites

Problem 3:Find the measures of the numbered angles in

each isosceles trapezoid.

Section 6.6 – Trapezoids and Kites

Problem 3b:Find the measures of the numbered angles in

each isosceles trapezoid.

Section 6.6 – Trapezoids and Kites

Problem 3c:Find the measures of the numbered angles in

each isosceles trapezoid.

Section 6.6 – Trapezoids and Kites

In lesson 5.1 you learned about the midsegments of triangles…What are they????

Trapezoids also have midsegments.

The midsegment of a trapezoid is the segment that joins the midpoints of its legs. The

midsegment has two unique properties.

Section 6.6 – Trapezoids and Kites

Section 6.6 – Trapezoids and Kites

Problem 4:Segment QR is the midsegment of trapezoid

LMNP. What is x?

Section 6.6 – Trapezoids and Kites

Problem 4b:Find EF is the trapezoid.

Section 6.6 – Trapezoids and Kites

Problem 4c:Find EF is the trapezoid.

Section 6.6 – Trapezoids and Kites

Problem 4e:Find the lengths of the segments with variable

expressions.

Section 6.6 – Trapezoids and Kites

A kite is a quadrilateral with two pairs of consecutive sides congruent and no opposite

sides congruent.

Section 6.6 – Trapezoids and Kites

Section 6.6 – Trapezoids and KitesProblem 5:

Quadrilateral DEFG is a kite. What are m<1, m<2, m<3?

Section 6.6 – Trapezoids and KitesProblem 5b:

Find the measures of the numbered angles in each kite.

Section 6.6 – Trapezoids and KitesProblem 5c:

Find the measures of the numbered angles in each kite.

Section 6.6 – Trapezoids and KitesProblem 5d:

Find the measures of the numbered angles in each kite.

Section 6.6 – Trapezoids and KitesProblem 5e:

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

Section 6.6 – Trapezoids and KitesProblem 5f:

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

Section 6.6 – Trapezoids and Kites

Section 6.6 – Trapezoids and KitesProblem 6:

Determine whether each statement is true or false. Be able to justify your answer.

•All squares are rectangles•A trapezoid is a parallelogram

•A rhombus can be a kite•Some parallelograms are squares

•Every quadrilateral is a parallelogram•All rhombuses are squares.

Section 6.6 – Trapezoids and KitesProblem 7:

Name each type of quadrilateral that can meet the given condition.

•Exactly one pair of congruent sides•Two pairs of parallel sides

•Four right angles•Adjacent sides that are congruent

•Perpendicular diagonals•Congruent diagonals

Section 6.6 – Trapezoids and KitesProblem 8:

Can two angles of a kite be as follows? Explain!•Opposite and acute

•Consecutive and obtuse•Opposite and supplementary

•Consecutive and supplementary•Opposite and complementary

•Consecutive and complementary

Section 6.6 – Trapezoids and KitesProblem 8:

Can two angles of a kite be as follows? Explain!•Opposite and acute

•Consecutive and obtuse•Opposite and supplementary

•Consecutive and supplementary•Opposite and complementary

•Consecutive and complementary

Section 6.6 – Trapezoids and Kites

Section 6.6 – Trapezoids and Kites