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

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!

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

polygon.

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.

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

a heptagon?

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

measures of a 17-gon?

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!

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!!!

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

regular hexagon?

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

regular nonagon?

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?

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

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

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

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

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:

Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

Students will be able to:•Determine whether 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.

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

parallelogram?

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

parallelogram?

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

parallelogram?

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

parallelogram?

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

parallelogram?

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

on the given information? Explain!

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

on the given information? Explain!

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 6.4 – 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

A rhombus is a parallelogram with four congruent sides.

A rectangle is a parallelogram with four right angles.

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

Problem 1:Is Parallelogram ABCD a rhombus, rectangle or

square? Explain!

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

square? Explain!

Problem 2: What are the measures of the numbered angles

in rhombus ABCD?

in rhombus PQRS?

in the rhombus?

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

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

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

length of each diagonal

LN = 5x – 8 and MP = 2x + 1

Problem 5: Determine the most precise name for each

quadrilateral.

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

Chose among parallelogram, rhombus, rectangle, or square.

Opposite angles are congruent.

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

Diagonals are congruent

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

Each diagonal bisects opposite angles

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

Opposite sides are parallel

Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

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

or rectangle.

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

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

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

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

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

rhombus?

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

rectangle?

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

rhombus?

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

rectangle?

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

rectangle?

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?

Problem 4:Determine whether the quadrilateral can be a

parallelogram. Explain!

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

Problem 4b:Determine whether the quadrilateral can be a

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

Problem 4c:Determine whether the quadrilateral can be a

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

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.

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

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

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

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

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?

Problem 3:Find the measures of the numbered angles in

each isosceles trapezoid.

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

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

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.

Problem 4:Segment QR is the midsegment of trapezoid

LMNP. What is x?

Problem 4b:Find EF is the trapezoid.

Problem 4c:Find EF is the trapezoid.

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

expressions.

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

sides congruent.

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:

Section 6.6 – Trapezoids and KitesProblem 5d:

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.

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.

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

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

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 6.1 The Polygon Angle-Sum Theorem

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