Chapter 4

Congruent Triangles

4.1 & 4.6 Triangles and Angles

Triangle: a figure formed by three segments joining three noncollinear points.

Classification by SIDES

Classification by ANGLES

Equilateral Acute

Isosceles Equiangular

Scalene Right

Obtuse

Classification by Sides

• Equilateral Triangle–3 congruent sides

• Isosceles Triangle–2 congruent sides

• Scalene Triangle–No congruent sides

Classification by Angles

• Acute–All angles are acute

• Equiangular–All angles are congruent

• Right –One right angle and 2 acute angles

• Obtuse–One obtuse angle and 2 acute angles

Isosceles Triangle Equilateral

Triangle

Scalene Triangle

Classify the following triangles

Classify the following triangles

65°

58° 57°

130°

Acute scalene Right isosceles

Obtuse isosceles

Parts of a Triangle

• A vertex is one of the three points joining sides of a triangle.

• Two sides sharing a common vertex are adjacent sides.

Parts of a right triangle

• Legs: the sides that form the right angle of the triangle

• Hypotenuse: the side opposite the right angle

Leg

Leg

Hypotenuse

Parts of an isosceles triangle

• Legs: the two congruent sides

• Base: the third side

LegLeg

Base

Angle Measures of Triangles

• Interior Angles: The three original angles

• Exterior Angles: The angles adjacent to the interior angles

Interior Angles Exterior Angles

Triangle Sum Theorem

• The sum of the measures of the interior angles of a triangle is 180°.

B A

C

180CmBmAm

Corollary to the Triangle Sum Theorem

• The acute angles of a right triangle are complementary.

x°

2x°

X = 30°x + 2x = 90°

mA + mB = 90

A

B

More Practice

Find the measures of the missing angles:

1

42°

3 2

95°

40°

1

56°

45° 1 2

350°

m1 = 48°

m1 = 50°

m2 = 40°

m3 = 45°

m1 = 79°

m2 = 51°

m3 = 39°

Exterior Angle Theorem

• The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.

Remote

interior

angles Exterior Angle

x + 65 = 2x + 1065°

x° (2x + 10)°

Exterior A

ngle

m1 = mA + mBA

B

1

x = 55

IsoscelesTriangles

• Base Angles: The two angles in an isosceles triangle adjacent to the base

• Vertex Angle: The angle opposite the base

Base Angle Base Angle

Vertex Angle

Base Angles Theorem

• If two sides of a triangle are congruent, then the angles opposite them are congruent.

CBthenACABIf ,

A

C B

Converse to the Base Angles Theorem

• If two angles of a triangle are congruent, then the sides opposite them are congruent.

CAABthenCBIf ,

Corollary to the Base Angles Theorem

• If a triangle is equilateral, then it is equiangular.

Corollary to the Converse of the Base Angles Theorem

• If a triangle is equiangular, then it is equilateral.

Practice Problems

• Find the measure of the missing angles.

50°A

B

C

m B=80°

m C=50°

A

B

C

m A=60°

m B=60°

m C=60°