2. MA T H EMATICS C L ASS
• Polygons (convex or non-convex)
• Diagonal (number of diagonals in a
polygon)
• Angles of a triangle
• Angles of a Polygon
Welcome to class!
Today's Agenda
3. The terms polygon was derived from two Greek words, polus
meaning many and gonos meaning angled.
A polygon is composed of at least three coplanar points. When one
connects the points, the segments formed should intersect only at
the endpoints. Two segments that have a common endpoint should
not be collinear.
POLYGONS NOT POLYGONS
4. Sides
Segments formed
by the points
Consecutive
angles
Angles of a
polygon that a
have a common
side
Consecutive
vertices
Vertices that are
contained in one
side of a polygon
Consecutive sides
Sides that have a
common
endpoint
A point at which two or
more lines meet
vertices
Example: Name a polygon by naming all its vertices in
consecutive order.
Polygon ABCD
Polygon BCDA
A B
C
D
5. CONVEX
A polygon is a convex if and only if
each side lies on the edge of
a half-plane containing the
rest of the polygon
A
B C
D C
A
B
D
In quadrilateral PQRS,
all diagonals are
inside the polygon
In polygon ABCD ,
diagonal BD is outside
the polygon
Convex Non-convex
6. Determine wheter the following are polygons or not. If it is
polygon tell wheter it is convex or non-convex.
Polygon, non-convex Not a polygon Polygon, convex
7. A polygon is classified by the number of side that it has.
The Greek equivalent of the number is used followed by a suffix-gon.
Number of Sides Name of Polygons
3
4
5
6
7
8
9
10
n
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Dodecagon
n-gon
8. A line segment connecting two
non-consecutive vertices of a
polygon
Diagonal
2
Polygon Number of Diagonals
0
5
9. THEOREM 6-1:
The number of diagonals in
a convex polygon with
n sides is given by the
formula n(n-3)
2
Give the number of diagonals
formed by the following polygons
1. decagon
n=10, diagonals= n(n-3)= 10(10-3)
35
=
2 2
2.heptagon
n=7, diagonals= n(n-3)
= 7(7-3) = 14
2 2
11. A polygon is regular if and only if it is both
equilateral and equiangular
A polygon is equilateral if and only if all of its
sides are congruent to each other.
A polygon is equiangular if and only if all of its
angles are congruent to one another
Example:
13. Given: ABC
Prove: m
THEOREM 6-2: The sum of the measures
of the angles of a triangle is 180°
1+m 2+ 3 =180° 1
2
3
4 5
A
B
C
D
STATEMENTS
1. BD || AC
2. ∠1 and ∠4 are alternate interior angles
3. ∠1 ≅ ∠4
4. m ∠1 = m∠4
5. ∠5 and ∠3 are alternate interior angles
6. ∠3 ≅ ∠5
7. m∠3 = m∠5
8. m∠2 + m∠4 + m∠5 = 180°
9. m∠1 + m∠2 + m∠3 = m∠2 + m∠4 + m∠5
10. m∠1 + m∠2 + m∠3 = 180°
REASONS
1. Parallel Postulate
2. Definition of alternate interior angles
3. Theorem 5-1
4. Definition of congruence angles
5. Definition of alternate interior angles
6. Theorem 5-1
7. Definition of congruence angles
8. Definition of supplementary angles
9. Substitution
10. Steps 4, 7, 8
14. Given: ABC
Prove: m
THEOREM 6-2: The sum of the measures
of the angles of a triangle is 180°
1+m 2+ 3 =180° 1
2
3
4
5
A
B
C
D
B D
A C
1
4
3
5
15. 3x+4x+5x=180°
12x=180°
x=15°
Look at these
Math Problems
• The angles of a triangle are in a ratio 3:4:5. Find the
measure of all the angles.
Angles: 3x,4x,5x
3(15)=45°, 4(15)=60°, 5(15)=75°
16. Look at this
Math Problems
2. The largest angle of a triangle is 5 times the smallest
angle. The other angle is five more than the smallest.
Find the measure of all angles.
smallest angle=x
largest angle=5x
third angle= 5+x
x+5x+5+x=180°
7x+5=180°
7x=180-5
7x= 175°
7
x=25°
x=25°
5(25)= 125°
25+5=30°
7
19. 45° x 133°
29°
C N
A
Z
R
O
D
m∠ZDC= 180 - m∠Z - m∠C
m∠ZDC = 180 – (90 + 45)
m∠ZDC = 45°
∠ZDC
∠NOR
m∠ONR=180 - m∠O - M∠R
m∠ONR=180 – (29 +133)
m∠ONR= 18°
m∠DAN=180 - m∠NDA - m∠DNA
m∠DAN=180 – (45 + 18)
m∠DAN= 117°
∠DAN
18° 45°
=90°
20. THEOREM 6-3: The
measure of each angle
in an equiangular
triangle is 60°
THEOREM 6-4: The acute
angles of a triangle
are complementary.
THEOREM 6-5: A triangle
have at most one
right or one
obtuse angle.
60°
60°
70°
20°
95°
50°
35°
60°
21. THEOREM 6-6: If two angles of a
triangle are congruent to two angles of
another triangle, then the remaining
pair of angles are also congruent.
Given: ZAP and MEK
∠Z≅∠M, ∠A≅∠E
∠P≅∠K
A
P
K
M
Z
E
STATEMENTS REASONS
1.m∠Z ≅ m∠M , m∠A ≅ m∠E
2. m∠Z = m∠M , m∠A = m∠E
3. m∠Z + m∠A + m∠P = 180°
m∠M + m∠E + m∠K = 180°
4. m∠Z + m∠A + m∠P =
m∠M + m∠E + m∠K
5. m∠Z + m∠A + m∠P =
m∠M + m∠E + m∠K
6. m∠P = m∠K
7. m∠P ≅ m∠K
1. Given
2.Definition of Congruence
Angle
3. Triangle Sum Theorem
4. Substitution
5. Substitution
6. Subtraction Property
7. Definition of
Congruence Angle
22. 1.The measure of one of the acute angles of a right triangle is 5
times the measure of the other acute angle. Find the measure of a
larger angle.
smallest acute angle= x
Larger angle= 5x
x+5x = 90°
6x = 90°
X = 15°
Larger angle is
5x= 5(15°)=75°
23. 2. Given: ON bisects ∠DNA
ON AD
⊥
D
N
O
A
Prove: ∠D ≅ ∠A
STATEMENTS REASONS
1. ON bisects ∠DNA
2. ∠DNO ≅ ∠ANO
3. ON ⊥ AD
4. ∠DON and AON are
right angles
5. ∠DON ≅ ∠AON
6. ∠D ≅ ∠A
1. Given
2. Definition of angle
bisector
3. Given
4. Definition of ⊥
5. All right angles are ≅
6. Theorem 6-6
24. An exterior angle of a convex polygon is an angle
that is adjacent and forms a linear pair with an interior
angle of a polygon when any side of a polygon is extended
1
5
12
4
3
6 7
8
10
9
2
11
W
A
M
VERTICES
INTERIOR
ANGLES
EXTERIOR
ANGLES
W
A
M
1
2
3
5,12
9,11
6,8
25. Example
:
∠CBF is an exterior angle at vertex B. Its measure is
180-50=130°. Notice that the sum of the measures
of the interior angles that are not adjacent to
the exterior angle is equal to the measure
of the exterior angle. Their sum is also 130°.
THEOREM 6-7: The measure of an exterior angle at
a vertex of any polygon is the difference of 180° and
the measure of an interior angle at the same vertex.
E B F
D
C
70°
60°
50°
=130°
=120°
110° =
26. THEOREM 6-8: Exterior angle of a triangle theorem (EATT).
The measure of an exterior angle is equal to the sum
of the measures of its remote interior angles.
Remote interior angles of an exterior angle of a
triangle are the two interior angles that are not
adjacent to that exterior angles.
27. Solve for the value of
x.
1.
2.
3.
23°
40°
x°
84°
7x° 5x°
(5x-15)° (10x-30)°
(4x)°
x=40+23=63
x=63°
7x+5x=84
12x=84
x = 7°
4x+5x-15 =10x-30
9x-10x = -30+15
-x = -15
x = 15°
28. 1.m∠5 m∠1
2. m∠2 m∠4
3. m∠7 m∠5
4. m∠6 m∠4
5. m∠4 m∠3
THEOREM 6-9: The measure of an exterior angle of
a triangle is greater than any of the remote interior angles
For the following figure, fill in the blank with one of
the following symbols, <, >, or = to make the
statement is true
1
2 3 7
4
5
6
=
<
>
=
>
29. 1. m∠5 m∠1
2. m∠2 m∠4
3. m∠7 m∠5
4. m∠6 m∠4
5. m∠4 m∠3
Answers:
=
<
<
=
<
the angles are vertical angles
∠2 is remote interior angle of ∠4
∠7 is an exterior angle and ∠1 is a
remote interior angle and m ∠1m≅ ∠5
vertical angles
4 is an exterior angle and 3 is a
remote interior angle
30. = ; =
Polygons Number of triangles
the polygon can be
divided
Sum of Angles
ANGLES OF A POLYGON
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
n-gon
1
2
3
4
5
n-2
180°
2 (180°) = 360°
3 (180°) = 540°
4 (180°) = 720°
5 (180°) = 900°
(n - 2) (180°)
31. THEOREM 6-10: The sum of the interior angles
of any polygon having n sides is 180°(n-2)
THEOREM 6-11: The measure of one angle of a
regular polygon is given by 180(n-2) / n
Examples:
1.Give the sum of the interior angles of a polygon with the
following number of sides. a.15 b. 12
Applying Theorem 6-10:
a. Sum = 180°(n-2) = 180°(15-2) = 2340°
b. Sum = 180°(n-2) = 180°(12-2) = 1800°
32. 2. Given the sum of the measures of the interior
angles of a polygon, give the number of sides.
a. 1080° b. 1980°
a. Sum = 180(n-2)= 1080°
n-2= 1080°
180°
=6
n = 6+2= 8
b. Sum = 180(n-2)= 1980°
n-2= 1980°
180°
=11
n = 11+2= 13
33. 3. Determine the measure of one of the interior
angles of the following regular polygons.
a. octagon b. decagon
Applying Theorem 6-11:
180°(n-2)
n
=
180°(8-2)
=
8
135°
a.
180°(n-2)
n
=
180°(10-2)
=144°
b.
10
34. 4. Given the measure of one interior angle of a regular
polygon, give the number of sides.
a. 120°
a. 180°(n-2)
n
= 120°
180°(n-2) = 120°n
180n-360 = 120°n
180n - 120n = 360
n = 6
Performing the converse of Theorem 6-11
35. 5. Find the measure of the unknown angles, given
the type of polygon and the measure of the rest.
a. Quadrilateral: 54°, 86°, 70°
b. Hexagon: 122°, 123°, 124°, 124°, 125°, 126°
a. Measure of remaining angle =
360°- (54°+86°+70°) = 150°
b. Measure of remaining angle =
720°- (122°+123°+124°+125°+126°) = 100°
36. Theorem 6-12: The sum of the different measures
of the exterior angles of any polygon is 360°
Theorem 6-13: All the exterior angles of a
regular polygon are congruent.
Theorem 6-14: The measure of one exterior
angle of a regular n-sided polygon is 360° / n
37. Examples:
1. What is the sum of the different measures
of the exterior angles of a polygon with 99 sides?
2. What is the measure of the exterior angle
of a regular octagon?
Ans. whatever number of sides, the sum of
the exterior angles is always 360°
Ans. The measure is 360/n = 360 / 8 =45°
38. Ans. The sum of the remaining angles is
360° - 240° = 120° . The three remaining angles
are congruent, so they have the same measurement.
So, the measure of each is 120° / 3 = 40°
3. Three of the exterior angles of a hexagon have
a sum of 240°. The remaining exterior angles are
congruent to each other. Determine the
measure of the remaining angles.
39. And we're done for
the day!
THANK YOU FOR LISTENING!!!
MA T H EMATICS C L ASS
Rorelay Entero BSED 1-B
