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FIND ANGLE MEASURES IN
POLYGONS
“A life not lived for others is not a life worth living.” –Albert Einstein
Diagonals
A diagonal is a segment that connects any two
nonconsecutive vertices in a polygon.
Side
Diagonal
Concept 1: Polygon Interior Angles
Theorem
Polygon Interior Angles Theorem
The sum of the measures of the interior angles of a convex n-gon
is 180°(n-2).
n=3
180°
n=5
540°
n=6
720°
Concept 2: Interior Angles of a
Quadrilateral
The sum of the measures of the interior angles
of a quadrilateral is 360°.
n=4
360°
n=4
360° n=4
360°
Example 1
Find the sum of the measures of the interior angles
of a convex octagon.
Example 2
The sum of the measures of the interior angles of a
convex polygon is 2160°. Classify the polygon by
the number of sides.
Example 3
Find the value of x in the diagram.
55°
100°
75° 166°
x°
Concept 3: Polygon Exterior Angles
Theorem
The sum of the measures of the exterior angles of a
convex polygon, one angle at each vertex, is 360°.
1
2
3
4
5
Example 4
Find the value of x
100°
35°
60°
75°
x°
Example 5
Find the measure of each interior angle of a
pentagon and the measure of each exterior angle
of a pentagon.
USE PROPERTIES OF
PARALLELOGRAMS “Bad is never good until worse happens.” –Danish Proverb
Parallelogram
A parallelogram is a quadrilateral with opposite
sides that are parallel.
Concept 4: Theorem 8.3 and 8.4
Theorem 8.3
If a quadrilateral is a parallelogram, then its
opposite sides are congruent.
Theorem 8.4
If a quadrilateral is a parallelogram, then its
opposite angles are congruent.
Example 1
Find the values of x, y, and z.
55°
x° (4y-35)°
125°
3z+5
17
Concept 5: Theorem 8.5 and 8.6
Theorem 8.5
If a quadrilateral is a parallelogram, then its
consecutive angles are supplementary.
x°
x° y°
y°
Example 2
Find the measure of each angle (exclude straight
angles).
35°
75°
50°
56°
Concept 5: Theorem 8.5 and 8.6
Theorem 8.6
If a quadrilateral is a parallelogram, then its diagonals
bisect each other.
Example 3
Find the value of the variables.
2y-5
5x
20 6
3v 12 15
z
SHOW THAT A
QUADRILATERAL IS A
PARALLELOGRAM
“Things could be worse. Suppose your errors were counted and published every day, like those of a baseball player.” –Anon.
Concept 6: Theorem 8.7 and 8.8
Theorem 8.7
If both pairs of opposite sides of a quadrilateral are
congruent, then the quadrilateral is a parallelogram.
Theorem 8.8
If both pairs of opposite angles of a quadrilateral are
congruent, then the quadrilateral is a parallelogram.
, then If
Example 1
Determine whether the quadrilateral is a
parallelogram. Explain.
5
5
6 6
120°
120°
60°
60°
10
10
Concept 7: Theorem 8.9 and 8.10
Theorem 8.9
If one pair of opposite sides of a quadrilateral are
congruent and parallel, then the quadrilateral is a
parallelogram
, then If
Concept 7: Theorem 8.9 and 8.10
Theorem 8.10
If the diagonals of a quadrilateral bisect each other,
then the quadrilateral is a parallelogram.
, then If
Example 2
Determine if the shape is a parallelogram. Explain.
10
10
5 5
6
6
8
8
Example 3
For what value of x is the quadrilateral a
parallelogram?
2x
6
8
8 5x-16
2x-4
6 6
120°
(8x-8)°
60°
60°
5x-4 11
Example 4
The vertices of quadrilateral ABCD are given. Draw
ABCD in a coordinate plane and show that it is a
parallelogram.
A(0, 1), B(4, 4), C(12, 4), D(8, 1)
PROPERTIES OF RHOMBUSES,
RECTANGLES, AND SQUARES
“The superior man(/woman) blames himself(/herself), the inferior man(/woman) blames others.” –Don Shula
Concept 8: Rhombus, Rectangle, and
Square
Rhombus
A parallelogram with four congruent sides.
Rectangle
A parallelogram with four right angles.
Square
A parallelogram with four congruent sides and four
right angles.
Concept 8: Rhombus, Rectangle, and
Square
Rhombus
A quadrilateral is a rhombus if and only if it has four congruent
sides.
Rectangle
A quadrilateral is a rectangle if and only if it has four right
angles.
Square
A quadrilateral is a square if and only if it is a rhombus and a
rectangle.
Example 1
For any square ABCD, decide whether the statement
is always, sometimes, or never true.
AB=BC
BC=AD
m∠A=m∠C
Example 2
For any rectangle ABCD, decide whether the
statement is always, sometimes, or never true.
AB=BC
BC=AD
m∠B=m∠C
Example 3
Classify the special quadrilateral. Explain your
reasoning.
4y+5
2y+35
5x-9 x+31
(5y-5)°
(4y+5)°
Concept 9: Theorem 8.11 and 8.12
Theorem 8.11
A parallelogram is a rhombus if and only if its
diagonals are perpendicular.
Theorem 8.12
A parallelogram is a rhombus if and only if each
diagonal bisects a pair of opposite angles.
Concept 10: Theorem 8.13
Theorem 8.13
A parallelogram is a rectangle if and only if its
diagonals are congruent.
Example 4
Name each quadrilateral (parallelogram, rhombus,
rectangle, and square) for which the statement is
true.
Diagonals bisect each other.
Diagonals are congruent.
Diagonals intersect at a right angle.
Diagonals bisect the angles.
Example 5
The diagonals of rhombus ABCD intersect at E.
Given that m∠DCA=72°, find the indicated
measures.
m∠BCA
m∠BAC
m∠BEA
m∠ABC
m∠ABD
Example 6
The diagonals of rectangle ABCD intersect at E.
Given that m∠DCA=72°, find the indicated
measures.
m∠BCA
m∠BAC
m∠BEA
m∠ABC
m∠ABD
A B
C D
E
USE PROPERTIES OF
TRAPEZOIDS AND KITES “I think we consider too much the good luck of the early bird, and not enough the bad luck of the early worm.”
–Franklin D. Roosevelt
Trapezoids
Trapezoids are quadrilaterals with only one pair of
parallel sides called bases. Angles on the same
base are called base angles. The other two sides
are called the legs. An isosceles trapezoid is one
with congruent legs.
base
base
leg leg
base angle pair
base angle pair
Example 1
Points A, B, C, and D are the vertices of a
quadrilateral. Determine whether ABCD is a trapezoid.
A(0, 4), B(4, 4), C(8, -2), D(2, 1)
Concept 11: Theorem 8.14, 8.15, and 8.16
Theorem 8.14
If a trapezoid is isosceles, then each pair of base
angles is congruent.
Theorem 8.15
If a trapezoid has a pair of congruent base angles,
then it is an isosceles trapezoid.
Theorem 8.16
A trapezoid is isosceles if and only if its diagonals are
congruent.
Example 2
EFGH is an isosceles trapezoid. The measure of
angle E is 72°. Find the other 3 angle measures.
Example 3
If AC=BD is the trapezoid isosceles? Explain.
A B
C D
Midsegment of a trapezoid
The midsegment of a trapezoid is the segment that
connects the midpoints of the legs of a trapezoid.
midsegment
Concept 12: Theorem 8.17
The midsegment of a trapezoid is parallel to each
base and its length is one half the sum of the
lengths of the bases.
1
2(𝑏1 + 𝑏2)
𝑏1
𝑏2
Example 4
Find the value of x.
𝑥
25
17
Example 5
Find the value of x.
18.7
12𝑥 − 1.7
5𝑥
Kite
A kite is a quadrilateral that has two pairs of
consecutive congruent sides, but opposite sides are
not congruent.
Concept 13: Theorem 8.18 and 8.19
Theorem 8.18
If a quadrilateral is a kite, then its diagonals are
perpendicular.
Theorem 8.19
If a quadrilateral is a kite, the exactly one pair of
opposite angles are congruent.
Example 6
Find the value of x.
50° 100°
x°
40°
120°
x°
IDENTIFY SPECIAL
QUADRILATERALS “Doing what’s right isn’t the problem. It’s knowing what’s right.” –Lyndon B. Johnson
Concept 14: Determining Shapes
Concept 14: Determining Shapes
Example 1
What types of quadrilaterals meet this condition.
Quadrilateral ABCD has at least one pair of congruent
opposite angles.
Quadrilateral ABCD has diagonals being congruent.
Quadrilateral ABCD has diagonals intersecting at a
right angle.
Example 2
What is the name of the quadrilateral ABCD?
A B
C D
A B
C D
Example 3
Is enough information given in the diagram to show
that quadrilateral ABCD is an isosceles trapezoid?
Explain.
A B
C D 68° 68°
112°