We name triangles by A three vertices. Example: ABC C The sides - - PowerPoint PPT Presentation

A C B

We name triangles by three vertices. Example: ∆ABC The sides of a triangle are segments. Examples: AB, BC, AC

There are special relationships that we will examine tomorrow regarding each side of a triangle and the angle that is across from the side. Examples: AB is across from C AC is across from B BC is across from A

Classifying Triangles – By Angles

Acute Triangle – A triangle with three acute angles.

60° 70° 50°

Obtuse Triangle – A triangle with one obtuse angle.

Right Triangle – A triangle with one right angle.

105°

Equiangular Triangle – A triangle with three congruent angles.

Classifying Triangles – By Sides

Scalene Triangle – A triangle with no congruent sides. Isosceles Triangle – A triangle with two congruent sides. Equilateral Triangle – A triangle with three congruent sides.

Theorem – the angle measures in a triangle sum to 180.

4 2 5 3 1 A B C D

Given: AB // DC Prove: m1 + m2 + m3 = 180

4 2 5 3 1 A B C D

Given: AB // DC Prove: m1 + m2 + m3 = 180

• 1. AB // DC
• 1. Given
• 6. m1 + m2 + m3 = 180
• 5. m4 + m2 + m5 = 180
• 4. mABC + m5 = 180.
• 3. mABC = m4 + m2
• 2. If lines are parallel, then

alternate interior angles are congruent.

• 2. m1 = m4;

m3 = m5

• 3. Angle Addition Postulate
• 4. Angle Addition Postulate
• 5. Substitution
• 6. Substitution

Algebra Connection 3x + 15 9x - 2

10 9 x 11 

180 10 9 x 11 2 x 9 15 x 3      

180 1 . 12 x 23  

9 . 167 x 23  3 . 7 x 

A B C

mA = 3(7.3) + 15 mA = 36.9 mC = 9(7.3) – 2 mC = 63.7 mB = 11(7.3) – mB = 79.4

10 9

Check: 36.9 + 63.7 + 79.4 = ______ 180

Corollary 1 – There can be at most one right angle or one obtuse angle in a given triangle.

Corollary 2 – The acute angles in a right triangle are complementary. Given: 3 is a right angle. Prove: 1 and 2 are complementary

3 2 1

Given: 3 is a right angle. Prove: 1 and 2 are complementary 3 2 1

• 1. 3 is a right angle
• 6. 1 and 2 are

complementary

• 5. m1 + m2 = 90
• 4. m1 + m2 + 90 = 180
• 3. m1 + m2 + m3 = 180
• 2. m3 = 90
• 1. Given
• 2. Definition of a right angle
• 3. The angle measures in a

triangle sum to 180.

• 4. Substitution
• 5. Subtraction
• 6. Definition of

complementary angles

3

m1 = 135 m2 = m3 = 60 m4 = m5 = m6 =

4 5 6 2 1

45 120 75 105

m3 + m5 = m______ m2 + m3 = m______ m2 + m5 = m______

1 6 4

Theorem – The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles.

m3 + m5 = m______ m2 + m3 = m______ m2 + m5 = m______

1 6 4

3 4 5 6 2 1

Given: DABC Prove: m1 + m2 = m4 3 4 2 1 A B C

• 1. DABC.
• 5. m1 + m2 = m4
• 4. m1 + m2 + m3 = m3 + m4
• 3. m3 + m4 = 180.
• 2. m1 + m2 + m3 = 180.
• 1. Given
• 5. Subtraction

Property

• 4. Substitution
• 3. Angle Addition Postulate
• 2. The angle measures in

a triangle sum to 180.

Algebra Connection 3x - 1 7x - 2 12x - 27 3x – 1 + 7x – 2 = 12x - 27 10x – 3 = 12x - 27 24 = 2x 12 = x 35° 82° 117°

Algebra Connection 130° 4x 2y 2y + 90 = 130 2y = 40 y = 20 4x + 130 = 180 4x = 50 x = 12.5

Homework: p. 97 WE #1, 2, 5-13, 19, 20 draw all diagrams & 3.1-3.3 Quiz Review Part Three