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:
4 sided:
5 sided:
6 sided:
7 sided:
____________ 8 sided: ____________
____________ 9 sided: ___________
____________ 10 sided: ___________
____________ 11 sided: ___________
____________ 12 sided: ___________
13 sided: ___________
Section 6.1
The Polygon Angle-Sum Theorem
A 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 Sum
What 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 Sum
What 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
parallograms
Lesson 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
Rectangles
Lesson 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 Kites
Students 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 Kites
A 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 Kites
Problem 5:
Quadrilateral DEFG is a kite. What are m<1,
m<2, m<3?
Section 6.6 – Trapezoids and Kites
Problem 5b:
Find the measures of the numbered angles in
each kite.
Section 6.6 – Trapezoids and Kites
Problem 5c:
Find the measures of the numbered angles in
each kite.
Section 6.6 – Trapezoids and Kites
Problem 5d:
Find the measures of the numbered angles in
each kite.
Section 6.6 – Trapezoids and Kites
Problem 5e:
Find the value(s) of the variable(s) in each kite.
Section 6.6 – Trapezoids and Kites
Problem 5f:
Find the value(s) of the variable(s) in each kite.
Section 6.6 – Trapezoids and Kites
Section 6.6 – Trapezoids and Kites
Problem 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 Kites
Problem 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 Kites
Problem 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
Problem 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