Triangles and Angles Sec 4.1

GOALS:To classify triangles by their angles and sidesTo find missing angle measures in triangles

B

A

Triangle – formed by three line segments that intersect only at their endpoints.

Cangle

Triangle

ABC

Vertex B

The line segments are , , and AB BC AC

Side AC is opposite angle B

Sides AB & CB are adjacent to vertex B

Angle and Side Classifications Triangles can be classified by their sides and angles.

By sides

Scalene - no congruent sides

Equilateral - three congruent sides

Isosceles - at least two congruent sides

Angle and Side Classifications Triangles can be classified by their sides and angles.

By angles

Acute – an acute triangle is acute because it has three acute angles

Obtuse – an obtuse triangle is obtuse because it has one obtuse angle

Right – a right triangle is right because it has one right angle

Equiangular – an equiangular triangle has three congruent sides

Lets classify some triangles

Notice: All triangles have at least two acute angles so they are classified by the measure of the third angle.

Special trianglesRight Triangle Isosceles Triangle

Can you have an isosceles right triangle?(the base would be the hypotenuse)

Leg “b”

Leg “a”

Hypotenuse “c”

Leg Leg

base

Triangle Sum Theorem The sum of the measures of the angles in a triangle is

180 degrees

180m x m y m z y

x

z

If x = 55, and y = 75, then z = 180 – (55+75) = 50

Exterior Angles of a Triangle

Exterior angles are angles that are adjacent to the interior angles in a triangle. There are 6 in total, 3 pairs of congruent angles.

It is customary to only talk about one exterior angle at each vertex.

x

zy

xv

180m x m v

Exterior Angles Theorem The measure of an exterior angle is equal to the sum of

the two remote interior angles.

If x = 35 degrees and y = 45 degrees, then the measure of angle 1 is equal to 80.

x

y1m m x m y

1

Corollary A corollary to a theorem is a statement that can be

proved easily using a theorem.

Corollary to the Triangle Sum TheoremThe acute angles of a right triangle are complementary.

x

y 90m x m y

You Try! Find the missing angle measure and classify each

triangle

y

30

40

2545

z

40 55 55

ax

Examples Find x or y.

(3 11)x

(4 5)x

50

2y

75

x

40