4.1 Triangles and Angles. Definition of a triangle A triangle is three segments joined at three noncollinear end points.

4.1 Triangles and Angles4.1 Triangles and Angles

Definition of a triangle

A triangle is three segments joined at three noncollinear end points.

vertex

vertex

vertex

side side

side

sidesadjacent

Types of Triangles by Sides

3 Sides congruent → Equilateral

Types of Triangles by Sides

2 Sides congruent → IsoscelesPart of the Isosceles

Triangleleg leg

base

Types of Triangles by Sides

No Sides congruent → Scalene

Types of Triangles by Sides

3 Sides congruent → Equilateral

2 Sides congruent → Isosceles

No Sides congruent → Scalene

Types of Triangle by Angles

All Angles less than 90 degrees → Acute

Types of Triangle by Angles

One Angle greater than 90 degrees, but less than 180° →

Obtuse

Types of Triangle by Angles

One Angle equal to 90 degrees → Right

How to classify a triangleChoose one from each category

Sides Angles____ Scalene Acute

Isosceles Right

Equilateral Obtuse

Equiangluar

All the angles are Equal

This will ALWAYS be paired up with Equilateral

Parts of the Right Triangle

Across from the right angle is the hypotenuse.

hypotenuse

leg

leg

Interior Angles vs. Exterior Angles

M

a

N b c

P

Interior angles: <a, <b, <c

Exterior angles: <M, <N, <P

Triangle Sum Theorem

The sum of the three interior angles of a triangle is 180º

a

b c

m<a + m<b + m<c = 180°

Triangle Sum Theorem

Solve for x

25

110

x

Example 2

Find the measure of each angle.

2x + 10

x x + 2

Exterior Angle Theorem

The measure of an exterior angle equals the measure of the two nonadjecent interior angles.

b

a

ba

Example 3

Given that ∠ A is 50º and

∠B is 34º, what is the measure of

∠BCD?

What is the measure of ACB?∠

DA

B

C

Solve for x

14

79

x

Corollary for the fact that interior angles add to 180º

The acute angles of a Right triangle are complementary.

x

x90

Example 4A. Given the following triangle,

what is the length of the hypotenuse?

B. What are the length of the legs?

C. If one of the acute angle measures is 32°, what is the other acute angle’s measurement?

13

12

5

Example 6

Find the missing measures

80°

53°

Example 7

Given: ∆ABC with mC = 90°

Prove: mA + mB = 90°Statement Reason

1. mC = 90°

2. mA + mB + mC = 180°

3. mA + mB + 90° = 180°

4. mA + mB = 90°