GEO05-1 Angles Of Triangles

McGraw Hill |

Angles of Triangles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

Content Objective
Students will prove and apply the Triangle Angle-Sum Theorem,
Exterior Angle Theorem, and Triangle Angle-Sum Theorem
Corollaries.

Language Objectives

• Students use less than and greater than to explain the

differences between triangles and their angle relationships.

5-1 Angles of Triangles

Math Response

What is the slope of the line through (6,9) and (7,1)?

Type answer here

Correct Answer: -8

Deg^°

Rad

Multiple Choice

Which of these lines is perpendicular to the line represented by the equation y = -3x + 4 ?

• y = 5x - 4

• y = 1/3x + 1

• y = -3x - 4

• y = -1/3x + 6

Math Response

What is the exact distance between points (1, 2) and (7, 3)? Use a radical symbol if the answer isn't a perfect square root.

Type answer here

Correct Answer: \sqrt{37}

Deg^°

Rad

McGraw Hill |

Angles of Triangles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

Given: △ABC
Prove: m∠A + m∠B + m∠C = 180°

Proof:

Learn
Interior Angles of Triangles

Step 3

Step 2

Step 1

Step 4

Theorem 5.1: Triangle Angle-Sum Theorem
The sum of the measures of the interior angles of a triangle is 180°.

Match

Find the measure of each
numbered angle.

• m∠1

• m∠2

• m∠3

123

52

29

• 123

• 52

• 29

McGraw Hill |

Angles of Triangles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

Learn
Exterior Angles of Triangles

exterior
angles

An exterior angle of a triangle is an
angle formed by one side of the triangle
and the extension of an adjacent side. A
triangle has three exterior angles. ∠4 is
an exterior angle of △ABC.

remote
interior
angles

Each exterior angle of a triangle has two
remote interior angles that are not
adjacent to the exterior angle. ∠1 and ∠3
are the remote interior angles for ∠4.

McGraw Hill |

Angles of Triangles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

Learn
Exterior Angles of Triangles

Theorem 5.2: 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.

Given: △ABC

Prove: m∠A + m∠B = m∠1

(continued on the next slide)

McGraw Hill |

Angles of Triangles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

Example 2
Use the Exterior Angle Theorem

ARCHITECTURE Find the measure of
∠DAB in the front face of the building.

McGraw Hill |

Angles of Triangles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

Example 2
Use the Exterior Angle Theorem

m∠DAB = m∠ABC + m∠BCA

Exterior Angle
Theorem

(12x + 7)° = (6x − 4)° + 65°

Substitution

x = 9

Solve.

m∠DAB = [12(9) + 7]° or 115°

Math Response

Find the measure of ∠XYZ.

Type answer here

Correct Answer: 90

Deg^°

Rad

McGraw Hill |

Angles of Triangles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

Example 2
Use the Exterior Angle Theorem

Check

PUZZLES Find the measure of ∠XYZ.

90°

McGraw Hill |

Angles of Triangles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

Learn
Triangle Angle-Sum Corollaries

A corollary is a theorem with a proof that follows as a direct result
of another theorem. As with a theorem, a corollary can be used as a
reason in a proof. The corollaries below follow directly from the
Triangle Angle-Sum Theorem. (basically postulates)

Corollary 5.1
The acute angles of a right triangle are complementary.

Corollary 5.2
There can be at most one right or obtuse angle in a triangle.

McGraw Hill |

Angles of Triangles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

Example 3
Find Angle Measures in Right Triangles

Find each measure.

a. m∠BCD

McGraw Hill |

Angles of Triangles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

Example 3
Find Angle Measures in Right Triangles

m∠BCD = [2(28) − 2]° or 54°

m∠BCD + m∠DBC = 90°

Corollary 5.1

(x + 8)° + (2x − 2)° = 90°

Substitution

x = 28

Solve.

Math Response

Find m∠BAF.

Type answer here

Correct Answer: 30

Deg^°

Rad