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.

A BcAA B

C

c

b

a

A

-three sides:-three angles: -three vertices:

Theorem

The sum of the measures of the three angles of any triangle is 180 .

For

A B C, ,

A B C

a b c, ,

A B C, ,

A B C : m A m B m C 1 8 0

Classification of Triangles According to Angles

Acute TriangleAll angles are acute .

Right TriangleExactly one angle is right.

Obtuse TriangleExactly one angle is obtuse .

A B

C

AA B

C

A

C A

B

A B

C

Classification of Triangles According to Sides

Scalene TriangleNo two sides are equal in the length .

Isosceles TriangleTwo sides have equal length.

Equilateral TriangleAll sides have equal length .

A BcAA B

C

c

b

a

A

a

bbh

a

aa

60 60

60

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.

1

2

3

4

5

67

C onvex po lygon Nonconvex po lygon

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 Characteristics Representation

Parallelogram 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 Rectangle with-all sides equal in length OR

Rhombus with

–four right angles

Regular quadrilateral

Trapezoid Quadrilateral with

-exactly one pair of parallel sides (basis)

Isosceles trapezoid –if the nonparallel sides are equal in measure

Quadrilateral

Trapezoid(one pa ir o f para lle l s ides)

Parallelogram(two pa irs o f para lle l s ides)

Rectangle Rhombus

Square

Since rectangles, rhombuses and squares are parallelograms, the properties of parallelogram are the properties of rectangle, a rhombus and a square.

Polygon Number of sides Number of triangles Sum of the angles measures

4 2

5 3

6 4

. . .

. . .

. . .

n (n-2)

The Angles of a PolygonTo 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 .

2 1 8 0 3 6 0

3 1 8 0 5 4 0

4 1 8 0 7 2 0

n 2 1 8 0

Exercises

1. Classify each triangle as scalene, isosceles, or equilateral. Then classify it as acute, right, or obtuse.

a)

5

77

b)

5

9

7

c)

4

53

d)

5

6 7

e)

6

66

acute and isosceles triangle obtuse and scalene triangle right and scalene triangle

acute and scalene triangle acute and equilateral triangle

Exercises

2. Find the measure of angle A.

a) b) c)

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.

C

A

B

67

C A

49

BA

BC

69 59

m A 4 6 m A 4 1 m A 5 2

m A m B

m C m B

1 0

4 03 3 0 1 8 0m B

m A

m B

m C

4 0

5 0

9 0

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) b) c)

1 34B

A5 2

A 7 8

7 2 0

4 6

m B

3 6 0

1 2 8

m A

5 4 0

1 0 2

m A