Triangles and Polygons
Triangles
A triangle is a closed figure with three sides and three angles.
A geometric figure is said to be closed when one can start at the point, trace the entire
figure and end at the starting point without lifting the pencil from the paper.
C
 ABC
-three sides: a, b, c
-three angles:  A,  B,  C
-three vertices: A, B, C
b
a
A
c
B
Theorem
The sum of the measures of the three angles of any triangle is 180 .
For  ABC: m A  m B  m C  180
Classification of Triangles According to Angles
C
Acute Triangle
All angles are acute .
A
B
B
Right Triangle
Exactly one angle is right.
C
A
C
Obtuse Triangle
Exactly one angle is obtuse .
A
B
Classification of Triangles According to Sides
C
Scalene Triangle
No two sides are equal in the length .
b
a
A
Isosceles Triangle
Two sides have equal length.
c
b
B
b
h
a
60
Equilateral Triangle
All sides have equal length .
a
a
60
60
a
Polygons
A polygon is a closed figure with more then three sides.
A convex polygon is a polygon such that no line containing a side of
the polygon also contains a point in the interior of the polygon.
7
6
5
1
Conve x polygon
4
2
3
Nonconve x polygon
Polygon may be classified by the number of sides.
Number of Sides
Polygon
3
Triangle
4
Quadrilateral
5
Pentagon
6
Hexagon
7
Heptagon
8
Octagon
9
Nonagon
10
Decagon
12
Dodecagon
20
Icosagon
A regular polygons a polygon in which all sides have the same length and all
angles are equal in measure.
The equilateral triangle is the regular polygon with three sides.
Types of Quadrilaterals
Name
Parallelogram
Characteristics
Quadrilateral with
-both pairs of opposite sides are parallel
-have the same measure
-the opposite angles are congruent
-diagonals bisect each other
Rectangle
Parallelogram with
-four right angles
-the diagonals are equal in measure
Rhombus
Parallelogram with
-all sides equal in length
-the diagonals are perpendicular to each
other
Square
Trapezoid
Rectangle with
-all sides equal in length OR
Rhombus with
–four right angles
Regular quadrilateral
Quadrilateral with
-exactly one pair of parallel sides (basis)
Isosceles trapezoid –if the nonparallel
sides are equal in measure
Representation
Quadrilate ral
Trape zo id
(one pa ir of pa ra lle l s ide s )
Paralle lo g ram
(two pa irs of pa ra lle l s ide s )
Re c tang le
Rho mbus
S quare
Since rectangles, rhombuses and squares are parallelograms, the properties of
parallelogram are the properties of rectangle, a rhombus and a square.
The Angles of a Polygon
To find the sum of the measures of the angles of four or more sides, first draw all
possible diagonals from one vertex of the polygon. Then use the fact that the sum of
the measures of three angles of triangle is 180 .
Polygon
Number of sides Number of triangles Sum of the angles measures
4
2
2  180  360
5
3
3  180  540
6
.
.
.
n
4
.
.
.
(n-2)
4  180  720
.
.
.
 n  2  180
Exercises
1. Classify each triangle as scalene, isosceles, or equilateral.
Then classify it as acute, right, or obtuse.
a)
c)
b)
5
3
9
7
7
7
4
5
5
acute and isosceles triangle
obtuse and scalene triangle
d)
right and scalene triangle
e)
7
6
5
acute and scalene triangle
6
6
6
acute and equilateral triangle
Exercises
2. Find the measure of angle A.
A
A
B
a)
c)
b)
49
67
69
C
B
C
B
C
m A  46
m A  41
m A  52


3. In triangle ABC, m A is 10 less than m B and m C is 40
greater than m  B. Find the measure of each angle.
m A  m B  10
m C  m B  40
59
A
3m B  30  180
m A  40
m B  50
m C  90
Exercises
3. Find the sum of measures of interior angles for the figure given.
Then find the measure of angle A and angle B.
a)
c)
b)
A
A
B
134
720
m B  46
52
360
m A  128
540
m A  102
78