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Chapter 9
Parallel Lines
Chin-Sung Lin
Angles Formed by
Intersecting Lines
Mr. Chin-Sung Lin
Transversal
A transversal is a line that intersects
two coplanar lines at two distinct
points
k
m
Line k is a transversal
n
Mr. Chin-Sung Lin
Angles of Transversal and Lines
Eight angles are formed by the
transversal and the two lines
k
1
3
5
7
8
2
4
m
6
n
Mr. Chin-Sung Lin
Interior Angles
3 4 5 and 6 are interior angles
k
1
3
5
7
8
2
4
m
6
n
Mr. Chin-Sung Lin
Exterior Angles
1 2 7 and 8 are exterior angles
k
1
3
5
7
8
2
4
m
6
n
Mr. Chin-Sung Lin
Same-Side Interior Angles
3 and 5 are same-side interior angles
4 and 6 are same-side interior angles
k
1
3
5
7
8
2
4
m
6
n
Mr. Chin-Sung Lin
Alternate Interior Angles
3 and 6 are alternate interior
angles
4 and 5 are alternate interior
k
angles
1
2
3
5
7
8
4
m
6
n
Mr. Chin-Sung Lin
Alternate Exterior Angles
1 and 8 are alternate exterior angles
2 and 7 are alternate exterior angles
k
1
3
5
7
8
2
4
m
6
n
Mr. Chin-Sung Lin
Corresponding Angles
1 and 5, 2 and 6, 3 and 7,
4 and 8 are corresponding angles
k
1
3
5
7
8
2
4
m
6
n
Mr. Chin-Sung Lin
Angles of Transversal and Lines
Review:
k
1
3
5
7
8
2
4
m
6
n
Mr. Chin-Sung Lin
Angles Formed by
Parallel Lines
Mr. Chin-Sung Lin
Parallel Lines
Coplanar lines that have no points in
common, or have all points in common
and, therefore, coincide
Mr. Chin-Sung Lin
Angles Formed by Parallel Lines
Eight angles are formed by the
transversal and the two parallel lines
k
1
3
5
7
2
4
m
6
8
n
Mr. Chin-Sung Lin
Parallel Lines and Transversal
 Corresponding Angles Postulate
 Alternate Interior Angles Theorem
 Same-Side Interior Angles Theorem
Mr. Chin-Sung Lin
Parallel Lines and Transversal
 Converse of Corresponding Angles
Postulate
 Converse of Alternate Interior Angles
Theorem
 Converse of Same-Side Interior Angles
Theorem
Mr. Chin-Sung Lin
Corresponding Angles Postulate
If two parallel lines are cut by a
transversal, then corresponding
angles are congruent
If m || n,
1  5,
2  6,
3  7, and
4  8
k
1
3
5
7
2
4
m
6
8
n
Mr. Chin-Sung Lin
Alternate Interior Angles Theorem
If two parallel lines are cut by a
transversal, then alternate interior
angles are congruent
k
1
If m || n,
3  6, and
4  5
3
5
7
2
4
m
6
8
n
Mr. Chin-Sung Lin
Alternate Interior Angles Theorem
1
3
5
7 8
2
k
4
m
6
n
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
Mr. Chin-Sung Lin
Alternate Interior Angles Theorem
1
3
5
7 8
Statements
2
4
k
m
6
n
Reasons
1. m || n
1. Given
2. 3  7 and 4  8
2. Corresponding angles
3. 6  7 and 5  8
3. Vertical angles
4. 3  6 and 4  5
4. Substitution property
Mr. Chin-Sung Lin
Same-Side Interior Angles Theorem
If two parallel lines are cut by a
transversal, then the pairs of sameside interior angles are supplementary
If m || n,
3 and 5,
4 and 6
are supplementary
k
1
3
5
7
2
4
m
6
8
n
Mr. Chin-Sung Lin
Same-Side Interior Angles
Theorem
1
3
5
7 8
2
k
4
m
6
n
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
Mr. Chin-Sung Lin
Same-Side Interior Angles Theorem
1
3
5
7 8
Statements
2
4
k
m
6
n
Reasons
1. m || n
1. Given
2. 3  7 and 4  8
2. Corresponding angles
3. 6 and 8, and 5 and 7
3. Supplementary angles
are supplementary
4. 6 and 4, and 5 and 3
are supplementary
4. Substitution property
Mr. Chin-Sung Lin
Converse of Corresponding
Angles Postulate
If two lines are cut by a transversal and
corresponding angles are congruent,
then the lines are parallel
If 1  5,
2  6,
3  7 or
4  8,
m || n,
k
1
3
5
7
2
4
m
6
8
n
Mr. Chin-Sung Lin
Converse of Alternate Interior
Angles Theorem
If two lines are cut by a transversal and
alternate interior angles are
congruent, then the lines are parallel
k
1
If 3  6 or
4  5,
m || n
3
5
7
2
4
m
6
8
n
Mr. Chin-Sung Lin
Converse of Alternate Interior
Angles Theorem
1
3
5
7 8
2
k
4
m
6
n
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
Mr. Chin-Sung Lin
Converse of Alternate Interior
Angles Theorem
1
3
5
7 8
Statements
2
4
k
m
6
n
Reasons
1. 3  6 or 4  5
1. Given
2. 6  7 and 5  8
2. Vertical angles
3. 3  7 or 4  8
3. Substitution property
4. m || n
4. Converse of corresponding
angle postulate
Mr. Chin-Sung Lin
Converse of Same-Side Interior
Angles Theorem
If two lines are cut by a transversal and
the pairs of same-side interior angles
are supplementary, then the lines are
parallel
k
1
If 3 and 5, or
4 and 6
are supplementary,
m || n,
3
5
7
2
4
m
6
8
n
Mr. Chin-Sung Lin
Converse of Same-Side Interior
Angles Theorem
1
3
5
7 8
2
k
4
m
6
n
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
Mr. Chin-Sung Lin
Converse of Same-Side Interior
Angles Theorem
1
3
5
7 8
Statements
1. 4 and 6, or 3 and 5
are supplementary
2. 8 and 6, and 7 and 5
are supplementary
3. 4  8 or 3  7
4. m || n
2
4
k
m
6
n
Reasons
1. Given
2. Supplementary angles
3. Supp. angle theorem
4. Converse of corresponding
angle postulate
Mr. Chin-Sung Lin
Methods of Proving Lines Parallel
 Congruent Corresponding Angles
 Congruent Alternate Interior Angles
 Same-Side Interior Angles supplementary
 Both lines are perpendicular to the same
line
Mr. Chin-Sung Lin
Examples
Mr. Chin-Sung Lin
Angles Formed by Parallel Lines
Classify the following angles into alternate
interior angles, same-side interior angles
or corresponding angles
q
4
3 8
7
p
2
1
r
6
5
Mr. Chin-Sung Lin
Angles Formed by Parallel Lines
Alternate interior angles: 3 & 6, 2 & 7
Same-side interior angles: 2 & 3, 6 & 7
Corresponding angles: 1 & 3, 2 & 4, 5 & 7, 6 & 8
q
4
3 8
7
p
2
1
r
6
5
Mr. Chin-Sung Lin
Angles Formed by Parallel Lines
If p and q are parallel, calculate the value of
all the angles
q
r
4
3 125o
7
p
2
1
6
5
Mr. Chin-Sung Lin
Angles Formed by Parallel Lines
Calculate the value of all the angles
q
55o
p
r
125o 125o
55o 55o
125o
125o
55o
Mr. Chin-Sung Lin
Angles Formed by Parallel Lines
If p and q are parallel, calculate the value of x
q
r
4
3 x+50o
7
p
2
1
6
x–10o
Mr. Chin-Sung Lin
Angles Formed by Parallel Lines
If p and q are parallel, calculate the value of x
x + 50o + x – 10o = 180o
2x = 140o
x = 70o
q
r
4
3 x+50o
7
p
2
1
6
x–10o
Mr. Chin-Sung Lin
Angles Formed by Parallel Lines
Given: 1  8
Prove: m || n
k
1
3
5
7
2
4
m
6
8
n
Mr. Chin-Sung Lin
Angles Formed by Parallel Lines
1
Given: 1  8
Prove: m || n
Statements
3
5
2
4
k
m
6
7 8
n
Reasons
1. 1  8
1. Given
2. 1  4 and 8  5
2. Vertical angles
3. 4  5
3. Substitution property
4. m || n
4. Converse of alternate
interior angle theorem
Mr. Chin-Sung Lin
Angles Formed by Parallel Lines
Given: m1 + m6 = 180
m6 + m9 =180
Prove: p || n
k
1
3
5
7
9
10
11 12
2
p
4
6
m
8
n
Mr. Chin-Sung Lin
Angles Formed by Parallel Lines
k
1
Given: m1 + m6 = 180
m6 + m9 =180
Prove: p || n
3
5
7
9
11
Statements
1. m1 + m6  180
m6 + m9  180
2. 1  4 and 6  7
3. m4 + m6  180
m7 + m9  180
4. p || m, m || n
5. p || n
2
6
m
8
10
12
p
4
n
Reasons
1. Given
2. Vertical angles
3. Substitution property
4. Converse of same-side
interior angle theorem
5. Transitive property
Mr. Chin-Sung Lin
Angles Formed by Parallel Lines
Given: Quadrilateral ABCD
~ DA, and BC || DA
BC =
Prove: AB || CD
B
C
A
D
Mr. Chin-Sung Lin
Angles Formed by Parallel Lines
Given: Quadrilateral ABCD
~ DA, and BC || DA
BC =
Prove: AB || CD
B
C
A
D
Mr. Chin-Sung Lin
Triangle Angle-Sum
Theorem
Mr. Chin-Sung Lin
Triangle Angle-Sum Theorem
The sum of the measures of the
interior angles of any triangle is 180
Mr. Chin-Sung Lin
Triangle Angle-Sum Theorem
The sum of the measures of the
interior angles of any triangle is
A
180.
Draw the graph
B
Given: ∆ ABC
Prove: mA + mB + mC = 180
C
Mr. Chin-Sung Lin
Triangle Angle-Sum Theorem
D
A
1
Statements
2
B
C
1.
1.
2.
3.
4.
2.
3.
4.
Reasons
Mr. Chin-Sung Lin
Triangle Angle-Sum Theorem
D
A
1
Statements
1.
2.
3.
4.
B
Let AD be the line through A
and parallel to BC
B  1 and C  2
m1 + mA + m2 = 180
mB + mA + mC = 180
2
C
Reasons
1. Use properties of parallel lines
2. Alt. interior angles theorem
3. Def. of straight angle
4. Substitution property
Mr. Chin-Sung Lin
Corollaries to the
Triangle Angle-Sum
Theorem
Mr. Chin-Sung Lin
Corollary
A corollary is a statement that follows
directly from the theorem
A corollary is a statement that can be
easily proved by applying the
theorem
Mr. Chin-Sung Lin
Corollary 1
If two angles of one triangle are
congruent to two angles of another
triangle, then the third angles of
the triangles are congruent
Mr. Chin-Sung Lin
Proof of Corollary 1
If two angles of one triangle are congruent to
two angles of another triangle, then the
third angles of the triangles are congruent
Given: A  X, and
B  Y
Prove: C  Z
A
Y
B
X
Z
C
Mr. Chin-Sung Lin
Y
B
Proof of Corollary 1
X
A
Statements
Z
C
Reasons
1. A  X, B  Y
1. Given
2. mA + mB + mC = 180
2. Triangle angle sum theorem
mX + mY + mZ = 180
3. mA + mB + mC =
3. Substitution property
mX + mY + mZ
4. mC = mZ or C  Z
4. Subtraction property
Mr. Chin-Sung Lin
Corollary 2
The two acute angles of a right
triangle are complementary
Mr. Chin-Sung Lin
Proof of Corollary 2
The two acute angles of a right triangle are
complementary
Given: mC = 90
Prove: A and B are complementary
B
A
C
Mr. Chin-Sung Lin
B
Proof of Corollary 2
C
A
Statements
Reasons
1. mC = 90
1. Given
2. mA + mB + mC = 180
2. Triangle angle sum theorem
3. mA + mB = 180 - 90 = 90
3. Subtraction property
4. A and B are complementary
4. Def. of complementary angles
Mr. Chin-Sung Lin
Corollary 3
Each acute angle of an isosceles right
triangle measured 45o
Mr. Chin-Sung Lin
Proof of Corollary 3
Each acute angle of an isosceles right triangle
measured 45o
Given: mC = 90, ∆ABC is isosceles
Prove: mA = mB = 45
C
A
B
Mr. Chin-Sung Lin
C
Proof of Corollary 3
A
Statements
B
Reasons
1. mC = 90
1. Given
2. mA + mB + mC = 180
2. Triangle angle sum theorem
3. mA + mB + 90 = 180
3. Substitution property
4. mA + mB = 90
4. Subtraction property
5. mA = mB
5. Base angle theorem
6. 2mA = 2mB = 90
6. Substitution property
7. mA = mB = 45
7. Division property
Mr. Chin-Sung Lin
Corollary 4
Each angle of an equilateral triangle
has measure 60
Mr. Chin-Sung Lin
Proof of Corollary 4
Each angle of an equilateral triangle has
measure 60
Given: Equilateral ∆ ABC
Prove: mA = mB = mC= 60
C
A
B
Mr. Chin-Sung Lin
C
Proof of Corollary 4
A
Statements
B
Reasons
1. Equilateral ∆ ABC
1. Given
2. mA = mB = mC
2. Euailateral triangle theorem
3. mA + mB + mC = 180
3. Triangle angle sum theorem
4. mA + mA + mA = 180
4. Substitution property
mB + mB + mB = 180
mC + mC + mC = 180
5. mA = mB = mC = 60
5. Division property
Mr. Chin-Sung Lin
Corollary 5
Prove: mA + mB + mC + mD= 360
B
A
C
D
Mr. Chin-Sung Lin
Proof of Corollary 5
B
3
1
A
2
Statements
1. Let AC divides quadrilateral
ABCD into two triangles
2. m1 + mB + m3 = 180
m2 + mD + m4 = 180
3. m1 + mB + m3 +
m2 + mD + m4 = 360
4. mA + mB + mC + mD
= 360
C
4
D
Reasons
1. Use properties of triangles
2. Triangle angle-sum theorem
3. Addition property
4. Partition property
Mr. Chin-Sung Lin
Exterior Angle Theorem
Mr. Chin-Sung Lin
Exterior Angle of a Triangle
An exterior angle of a triangle is formed
when one side of a triangle is extended.
The nonstraight angle outside the
triangle, but adjacent to an interior
angle, is an exterior angle of the triangle
C
A
B
Mr. Chin-Sung Lin
Exterior Angle Theorem
The measure of an exterior angle of a
triangle is equal to the sum of the
measures of the two non-adjacent
interior angles
Mr. Chin-Sung Lin
Exterior Angle Theorem
The measure of an exterior angle of a
triangle is equal to the sum of the
measures of the two non-adjacent
interior angles
Given: ∆ ABC
Prove: m1 = mA + mC
A
C
1
B
Mr. Chin-Sung Lin
C
Exterior Angle Theorem
1
A
Statements
B
Reasons
1. m1 + mB = 180
1. Supplementary angles
2. mA + mB + mC = 180
2. Triangle angle-sum theorem
3. mA + mB + mC =
3. Substitution property
m1 + mB
4. m1 = mA + mC
4. Subtraction property
Mr. Chin-Sung Lin
Application Examples
Mr. Chin-Sung Lin
Triangle Angle Sum Theorem
∆ ABC is an isosceles triangle. The vertex angle
, C, exceeds the measure of each base
angle by 30 degrees. Find the degree
measure of each angle of the triangle.
C
A
B
Mr. Chin-Sung Lin
Triangle Angle Sum Theorem
∆ ABC is an isosceles triangle. The vertex angle
, C, exceeds the measure of each base
angle by 30 degrees. Find the degree
measure of each angle of the triangle
C
(x + 30) + x + x = 180
3x + 30 = 180
3x = 150
A
x = 50
X+30
X
X
B
Mr. Chin-Sung Lin
Triangle Angle-Sum Theorem
Find the values of x and y
5y
4x
4x
y
3y
Mr. Chin-Sung Lin
Triangle Angle-Sum Theorem
Find the values of x and y
5y
5y + y + 4x = 180
4x + 4x + 3y = 180
4x + 6y = 180……(1)
8x + 3y = 180……(2)
4x
y
4x
3y
(1)/2, 2x + 3y = 90..…..(3)
(2) - (3),
6x = 90, and then X = 15……(4)
(4) Substitutes into (3),
2(15) + 3y = 90, 3y = 60, y = 20
So,
X =15
Y = 20
Mr. Chin-Sung Lin
Triangle Angle-Sum Theorem
Given: m2 = mC
Prove: m1 = mB
A
B
E
2
1
D
C
Mr. Chin-Sung Lin
Triangle Angle-Sum Theorem
A
B
Statements
E
2
C
1. m2 = mC
2. m1 + m2 + mA = 180
mB + mC + mA = 180
3. m1 + m2 + mA =
mB + mC + mA
4. m1 + mC + mA =
mB + mC + mA
5. m1 = mB
1
D
Reasons
1. Given
2. Triangle angle-sum theorem
3. Substitution property
4. Substitution property
5. Subtraction property
Mr. Chin-Sung Lin
AAS Postulate
Mr. Chin-Sung Lin
Postulates that Prove Congruent
Triangles
Side-Side-Side Congruence (SSS)
Side-Angle-Side Congruence (SAS)
Angle-Side-Angle Congruence (ASA)
Mr. Chin-Sung Lin
Postulates that Prove Congruent
Triangles
Side-Side-Side Congruence (SSS)
Side-Angle-Side Congruence (SAS)
Angle-Side-Angle Congruence (ASA)
Angle-Angle-Side Congruence (AAS)
Mr. Chin-Sung Lin
Angle-Angle-Side Congruence (AAS)
If two of corresponding angles and a nonincluded side are equal, then the
triangles are congruent
Mr. Chin-Sung Lin
AAS Postulate
Given CA is an angle bisector of DCB, and B =~
D
~
Prove ∆ ACD = ∆ ACB
D
A
C
B
Mr. Chin-Sung Lin
AAS Postulate
Given CA is an angle bisector of DCB, and B =~
D
~
AAS
Prove ∆ ACD = ∆ ACB
D
A
C
B
Mr. Chin-Sung Lin
AAS Postulate Corollary 1
Two right triangles are congruent if their
hypotenuses and one of the acute angles are
congruent
Given ∆ ABC and ∆ DEF are right triangles
AB = DE, A =~ D
~
Prove ∆ ABC = ∆ DEF
A
B
E
C D
F
Mr. Chin-Sung Lin
AAS Postulate Corollary 2
If a point lies on the bisector of an angle, then it is
equidistant from the sides of the angle
Mr. Chin-Sung Lin
Congruent
Right Triangles
(HL Postulate)
Mr. Chin-Sung Lin
Postulates that Prove Congruent
Triangles
Side-Side-Side Congruence (SSS)
Side-Angle-Side Congruence (SAS)
Angle-Side-Angle Congruence (ASA)
Angle-Angle-Side Congruence (AAS)
Mr. Chin-Sung Lin
Postulates that Prove Congruent
Triangles
Side-Side-Side Congruence (SSS)
Side-Angle-Side Congruence (SAS)
Angle-Side-Angle Congruence (ASA)
Angle-Angle-Side Congruence (AAS)
Hypotenuse-Leg Postulate (HL)
Mr. Chin-Sung Lin
Hypotenuse-Leg Postulate
(HL Postulate)
If the hypotenuse and a leg of one right
triangle are congruent to the hypotenuse
and a leg of another right triangle, then
the two triangles are congruent
A
B
E
C D
F
Mr. Chin-Sung Lin
Side-Side-Angle Case (SSA)
The condition does not guarantee
congruence, because it is possible to
have two incongruent triangles. This is
known as the ambiguous case
Mr. Chin-Sung Lin
HL Postulate – Example
Given BD is the altitude of an isosceles triangle
∆ ABC
~
Prove ∆ ABD = ∆ CBD
B
A
D
C
Mr. Chin-Sung Lin
HL Postulate – Example
Given BD is the altitude of an isosceles triangle
∆ ABC
~
HL
Prove ∆ ABD = ∆ CBD
B
A
D
C
Mr. Chin-Sung Lin
HL Postulate – Example
B
A
Statements
D
1. ∆ ABC is isosceles triangle
C
Reasons
1. Given
BD is the altitude of ∆ ABC
2. mBDA = 90; mBDC = 90
2. Def. of altitude
3. BA  BC
3. Def of isosceles triangle
4. BD  BD
4. Reflexive property
5. ∆ ABD  ∆ CBD
5. HL postulate
Mr. Chin-Sung Lin
Base Angle Theorem
Mr. Chin-Sung Lin
Base Angle Theorem
If two sides of a triangle are congruent, then
the angles opposite these sides are
congruent
(Base angles of an isosceles triangle are
congruent)
Mr. Chin-Sung Lin
Base Angle Theorem
If two sides of a triangle are congruent, then
the angles opposite these sides are
congruent
Draw a diagram like the one below
Given:
Prove:
B
AB  CB
A  C
A
C
Mr. Chin-Sung Lin
B
Base Angle Theorem
A
Statements
D
C
Reasons
1.
1.
2.
3.
4.
5.
6.
2.
3.
4.
5.
6.
Mr. Chin-Sung Lin
B
Base Angle Theorem
A
Statements
1. Draw the angle bisector of
ABC and let D be the point
where it intersects AC
2. ABD  CBD
3. AB  CB
4. BD  BD
5. ∆ ABD = ∆ CBD
6. A  C
D
C
Reasons
1. Any angle of measure less
than 180 has exactly one
bisector
2. Definition of angle bisector
3. Given
4. Reflexive property
5. SAS Postulate
6. CPCTC
Mr. Chin-Sung Lin
Converse of
Base Angle Theorem
Mr. Chin-Sung Lin
Converse of Base Angle Theorem
If two angles of a triangle are congruent,
then the sides opposite these angles are
congruent
Mr. Chin-Sung Lin
Converse of Base Angle Theorem
If two angles of a triangle are congruent,
then the sides opposite these angles are
congruent
Draw a diagram like the one below
Given:
A  C
B
Prove:
AB  CB
A
C
Mr. Chin-Sung Lin
Converse of Base Angle Theorem
B
Statements
A
C
D
1.
1.
2.
3.
4.
5.
6.
2.
3.
4.
5.
6.
Reasons
Mr. Chin-Sung Lin
Converse of Base Angle Theorem
B
Statements
A
D
1. Draw the angle bisector of
ABC and let D be the point
where it intersects AC
2. ABD  CBD
3. A  C
4. BD  BD
5. ∆ ABD = ∆ CBD
6. AB  CB
C
Reasons
1. Any angle of measure less
than 180 has exactly one
bisector
2. Definition of angle bisector
3. Given
4. Reflexive property
5. AAS Postulate
6. CPCTC
Mr. Chin-Sung Lin
Base Angle Theorem - Example
AO  BO and 1  2
AC = BD
Given:
Prove:
A
B
O
C
1
2
D
Mr. Chin-Sung Lin
Base Angle Theorem – Example
AO  BO and 1  2
AC = BD
Given:
Prove:
A
B
O
C
1
2
D
Mr. Chin-Sung Lin
Base Angle Theorem - Example
A
B
O
C
1
2
D
Statements
Reasons
1.
1.
2.
3.
4.
5.
6.
2.
3.
4.
5.
6.
Mr. Chin-Sung Lin
Base Angle Theorem - Example
A
B
O
C
Statements
1. 1  2
2. CO  DO
3. AO  BO
4. AOC  BOD
5. ∆ AOC = ∆ BOD
6. AC  BD
1
2
D
Reasons
1. Given
2. Converse of Base Angle
Theorem
3. Given
4. Vertical Angles
5. SAS Postulate
6. CPCTC
Mr. Chin-Sung Lin
Equilateral and
Equiangular
Triangles
Mr. Chin-Sung Lin
Equilateral Triangles
A equilateral triangle is a triangle that
has three congruent sides
B
A
C
Mr. Chin-Sung Lin
Equilateral & Equiangular Triangles
If a triangle is an equilateral triangle, then it
is an equiangular triangle
Mr. Chin-Sung Lin
Polygons
Mr. Chin-Sung Lin
Definition of Polygons
A polygon is a closed plane figure with the
following properties:
1. Formed by three or more line segments
called sides
2. Each side intersects exactly two sides, one
at each endpoint, so that no two sides with
a common end point are colinear
Mr. Chin-Sung Lin
Naming Polygons
Each endpoint of a side is a vertex of the
polygon
A polygon can be named by listing the vertices
in consecutive order
Polygon can be named as ABCDE, CDEAB,
EABCD, or ……
A
B
E
D
C
Mr. Chin-Sung Lin
Convex & Concave Polygons
A polygon is convex if no line that contains a
side of the polygon contains a point in the
interior of the polygon
A polygon that is not convex is called
nonconvex or concave
Interior
Interior
Convex
Concave
Mr. Chin-Sung Lin
Convex & Concave Polygons
A polygon is convex if each of the interior
angles measures less than 180 degrees
A polygon that is concave if at least one
interior angle measures more than 180
degrees
< 1800
Convex
> 1800
Concave
Mr. Chin-Sung Lin
Identifying Polygons
Identify polygons in the following figures and
tell whether the figure is a convex or
concave polygon
Mr. Chin-Sung Lin
Identifying Polygons
Identify polygons in the following figures and
tell whether the figure is a convex or
concave polygon
Concave
Convex
Convex
Concave
Mr. Chin-Sung Lin
Common Polygons
•
Triangle: a polygon that is the union of three line segments
•
Quadrilateral: a polygon that is the union of four line
segments
•
Pentagon: a polygon that is the union of five line segments
•
Hexagon: a polygon that is the union of six line segments
•
Octagon: a polygon that is the union of eight line segments
•
Decagon: a polygon that is the union of ten line segments
•
N-gon: a polygon with n sides
Mr. Chin-Sung Lin
Classifying Polygons -by the Number of Its Sides
No. of sides
Polygons
No. of sides
Polygons
3
Triangle
8
Octagon
4
Quadrilateral
9
Nonagon
5
Pentagon
10
Decagon
6
Hexagon
12
Dodecagon
7
Heptagon
n
n-gon
Mr. Chin-Sung Lin
Classifying Polygons -by the Number of Its Sides
Mr. Chin-Sung Lin
Classifying Polygons -by the Number of Its Sides
Pentagon
Hexagon
Quadrilateral
Triangle
Mr. Chin-Sung Lin
Classifying Polygons -by the Number of Its Sides
Mr. Chin-Sung Lin
Classifying Polygons -by the Number of Its Sides
Decagon
Heptagon
Dodecagon
16-gon
Mr. Chin-Sung Lin
Equilateral / Equiangular / Regular Polygons
A polygon is equilateral if all sides are
congruent
A polygon is equiangular if all interior angles
of the polygon are congruent
A regular polygon is a convex polygon that is
both equilateral and equiangular
Equilateral
Equiangular
Regular Mr. Chin-Sung Lin
Identify Equilateral / Equiangular /
Regular Polygons
Mr. Chin-Sung Lin
Identify Equilateral / Equiangular /
Regular Polygons
Regular
Equilateral
Equiangular
Equilateral
Regular
Mr. Chin-Sung Lin
Interior and Exterior
Angles of Polygons
Mr. Chin-Sung Lin
Consecutive Angles and Vertices
A pair of angles whose vertices are the endpoints
of a common side are called consecutive
angles
The verteices of consecutive angles are called
consecutive vertices or adjacent vertices
A
E
B
Consecutive Angles:
A and
B
Consecutive Angles:
C and
D
Consecutive Vertices: A and B
Non-adjacent Vertices: E and B
D
C
Mr. Chin-Sung Lin
Diagonals of Polygons
A diagonal of a polygon is a line segment whose
endpoints are two non-adjacent vertices
A
Adjacent Vertices of B: A and C
E
B
Non-adjacent Vertices of B: D and E
Diagonals with Endpoint B: BD and BE
D
C
Mr. Chin-Sung Lin
Diagonals of Polygons
A diagonal of a polygon is a line segment whose
endpoints are two non-adjacent vertices
A
Adjacent Vertices of B: A and C
E
B
Non-adjacent Vertices of B: D and E
Diagonals with Endpoint B: BD and BE
D
C
Mr. Chin-Sung Lin
Diagonals of Polygons
All possible diagonals from a vertex
A
B
A
B
F
D
C
A
C
E
A
D
B
G
E
C
B
F
E
D
D
C
Mr. Chin-Sung Lin
Sum of Interior Angles of Polygons
Calculate the sum of intertor angles of each polygon
A
B
A
B
F
D
C
A
C
E
A
D
B
G
E
C
B
F
E
D
D
C
Mr. Chin-Sung Lin
Sum of Interior Angles of Polygons
Calculate the sum of intertor angles of each polygon
A
B
A
B
F
D
2 (1800)
C
A
C
E
A
E
0D
4 (180 )
B
G
C
B
F
E
D
3 (1800)
C
5
(1800)
D
Mr. Chin-Sung Lin
Theorem: Sum of Polygon Interior Angles
The sum of the measures of the interior angles of
a polygon of n sides is 180(n – 2)o
A
B
180 (6 – 2)0 = 720o
F
C
E
D
Mr. Chin-Sung Lin
Theorem: Sum of Polygon Exterior Angles
The sum of the measures of the exterior angles of
a polygon is 360o
Mr. Chin-Sung Lin
Theorem: Sum of Polygon Exterior Angles
The sum of the measures of the exterior angles of
a polygon is 360o
Sum of exterior angles
= 180n – Sum of interior angles
= 180n – 180 (n – 2)
= 180n – 180n + 360
= 360
Mr. Chin-Sung Lin
Theorem: Sum of Polygon Exterior Angles
The sum of the measures of the exterior angles of
a polygon is 360o
A
B
F
C
Sum of exterior angles = 360o
E
D
Mr. Chin-Sung Lin
Example: Interior / Exterior Angles
The measure of an exterior angle of a regular
polygon is 45o
(a) Find the number of sides of the polygon
(b) Find the measure of each interior angle
(c) Find the sum of interior angles
Mr. Chin-Sung Lin
Example: Interior / Exterior Angles
The measure of an exterior angle of a regular
polygon is 45o
(a) Find the number of sides of the polygon
(b) Find the measure of each interior angle
(c) Find the sum of interior angles
(a) 3600/450 = 8 sides
(b) 1800 – 450 = 1350
(c) 1800 (8 – 2) = 1,0800
Mr. Chin-Sung Lin
Example: Interior / Exterior Angles
In quadrilateral ABCD, m A = x, m
m C = x + 22, and m D = 3x
B = 2x – 12,
(a) Find the measure of each interior angle
(b) Find the measure of each exterior angle
Mr. Chin-Sung Lin
Example: Interior / Exterior Angles
In quadrilateral ABCD, m A = x, m
m C = x + 22, and m D = 3x
B = 2x – 12,
(a) Find the measure of each interior angle
(b) Find the measure of each exterior angle
(a) 500, 880, 720, 1500
(b) 1300, 920, 1080, 300
Mr. Chin-Sung Lin
The End
Mr. Chin-Sung Lin