7-5 Triangles

Course 1

Warm UpWarm Up

Lesson PresentationLesson Presentation

Problem of the DayProblem of the Day

Warm Up

1. What are two angles whose sum is 90°?

2. What are two angles whose sum is 180°?

3. A part of a line between two points is called a _________.4. Two lines that intersect at 90° are

______________.

complementary angles

supplementary angles

segment

perpendicular

Problem of the Day

Find the total number of shaded triangles in each figure.

3 6 10

Problem of the Day

Find the total number of Total triangles in each figure.

5 13 24

Learn to classify triangles and solve problems involving angle and side measures of triangles.

Vocabulary

acute triangleobtuse triangleright trianglescalene triangleisosceles triangleequilateral triangle

Insert Lesson Title Here

A triangle is a closed figure with three line segments and three angles. Triangles can be classified by the measures of their angles. An acute triangle has only acute angles. An obtuse triangle has one obtuse angle. A right triangle has one right angle.

Acute triangle Obtuse triangle Right triangle

To decide whether a triangle is acute, obtuse, or right, you need to know the measures of its angles.

The sum of the measures of the angles in any triangle is 180°. You can see this if you tear the corners from a triangle and arrange them around a point on a line.

By knowing the sum of the measures of the angles in a triangle, you can find unknown angle measures.

Additional Example 1: Application

D

E

F

To classify the triangle, find the measure of D on the trophy.

So the measure of D is 90°. Because DEF has one right angle, the trophy is a right triangle.

Subtract the sum of the known angle measures from 180°

m D = 180° – (38° + 52°)

m D = 180° – 90°

m D = 90°

Sara designed this triangular trophy. The measure of E is 38°, and the measure of F is 52°. Classify the triangle.

Try This: Example 1

D

E

F

To classify the triangle, find the measure of D on the trophy.

So the measure of D is 136°. Because DEF has one obtuse angle, the trophy is an obtuse triangle.

Subtract the sum of the known angle measures from 180°

m D = 180° – (22° + 22°)

m D = 180° – 44°

m D = 136°

Sara designed this triangular trophy. The measure of E is 22°, and the measure of F is 22°. Classify the triangle.

You can use what you know about vertical, adjacent, complementary, and supplementary angles to find the measures of missing angles.

Take two pencils (or pens) and have them intersect them

in front of you like this…

Finger Dance(Geometry)

Finger Dance(Geometry)

Adjacent Angles

Finger Dance(Geometry)

Vertical Angles

Finger Dance(Geometry)

Supplementary Angles

Finger Dance(Geometry)

Vertical Angles

Finger Dance(Geometry)

Supplementary Angles

Finger Dance(Geometry)

Adjacent Angles

Finger Dance(Geometry)

Adjacent Angles

Finger Dance(Geometry)

Supplementary Angles

Finger Dance(Geometry)

Vertical Angles

Finger Dance(Geometry)

Adjacent Angles

Finger Dance(Geometry)

Vertical Angles

Finger Dance(Geometry)

Vertical Angles

Finger Dance(Geometry)

Supplementary Angles

Finger Dance(Geometry)

Vertical Angles

Finger Dance(Geometry)

Vertical Angles

Finger Dance(Geometry)

Supplementary Angles

Finger Dance(Geometry)

Supplementary Angles

Finger Dance(Geometry)

Supplementary Angles

Finger Dance(Geometry)

Supplementary Angles

For today’s warm-up, take out your list of vocabulary words, and TURN THEM OVER SO YOU CAN’T SEE THEM. Then, answer these three questions alone!1)What are complementary angles?2)What are supplementary angles?3)What is the sum of all the angle measurements in a triangle?

Supplementary Angles

? 50

Get a Partner and see if you can make Complimentary angles with

your hands/arms…

Find Complimentary

Angles

Find Complimentary

Angles

Find Complimentary

Angles

When angles have the same measure, they are said to be congruent.

Vertical angles are formed opposite each other when two lines intersect. Vertical angles have the same measure, so they are always congruent.

M N160°

160°

20° 20°R

P Q

Adjacent angles are side by side and have a common vertex and ray. Adjacent angles may or may not be congruent.

M N160°

160°

20° 20°R

P Q

Identify the type of each angle pair shown.

A.

They are vertical angles.

5 6

Identify the type of each angle pair shown.

B.

7 and 8 are side by side and have a common vertex and ray.

They are adjacent angles.

7 8

Identify the type of each angle pair shown.

A.

3 and 4 are side by side and have a common vertex and ray.

They are adjacent angles.

3

4

Identify the type of each angle pair shown.

B.

7 and 8 are opposite each other and are formed by two intersecting lines.

They are vertical angles.

7

8

65° + 25° = 90°

LMN and NMP are complementary.

Complementary angles are two angles whose measures have a sum of 90°.

P

N

M

L

25°65°

Supplementary angles are two angles whose measures have a sum of 180°.

65° + 115° = 180°

GHK and KHJ are supplementary.

J

K

H

115°65°

G

Find each unknown angle measure.

71° + a = 90°–71° –71°

a = 19°

The sum of the measures is 90°.

a

71°

A. The angles are complementary.

Find each unknown angle measure.

125° + b = 180°–125° –125°

b = 55°

The sum of the measures is 180°.

b125°

B. The angles are supplementary.

Find each unknown angle measure.

c = 82° Vertical angles are congruent.

c 82°

C. The angles are vertical angles.

P

S R

Q

T

68°

55°

Do you see any “straight angles” in this figure? Where?

What are the “straight angles” called? SUPLEMENTARY

Do you see any complementary angles?

What measure do complementary angles have?

90 degrees

180 degrees

What measure do supplementary angles have?

Additional Example 2A: Using Properties of Angles to Label Triangles

Use the diagram to find the measure of each indicated angle.

QTR and STR are supplementary angles, so the sum of m QTR and m STR is 180°.

m QTR = 180° – 68°

= 112°

P

S R

Q

T

68°

55°

A. QTR

Additional Example 2B: Using Properties of Angles to Label Triangles

QRT and SRT are complementary angles, so the sum of m QRT and m SRT is 90°.m SRT = 180° – (68° + 55°) = 180° – 123°

= 57°

m QRT = 90° – 57°

= 33°

P

S R

Q

T

68°

55°

B. QRT

Try This: Example 2A

Use the diagram to find the measure of each indicated angle.

MNO and PNO are supplementary angles, so the sum of m MNO and m PNO is 180°.

m MNO = 180° – 44°

= 136°

L

P O

M

N

44°

60°

A. MNO

Try This: Example 2B

MON and PON are complementary angles, so the sum of m MON and m PON is 90°.

m PON = 180° – (44° + 60°) = 180° – 104°

= 76°

m MON = 90° – 76°

= 14°

L

P O

M

N

44°

60°

B. MON

CLASSIFYING TRIANGLES

BY ANGLE MEASURES

1. Acute – has all acute angles

2. Obtuse – has an obtuse angle

3. Right – has a right angles (90 degrees)

CLASSIFYING TRIANGLES

BY LENGTH OF SIDES

1.Scalene Triangles – All sides are different2. Isosceles Triangle – two sides are the same3. Equilateral Triangle- All sides are congruent

BY ANGLE MEASURES

1. Acute – has all acute angles

2. Obtuse – has an obtuse angle

3. Right – has a right angles (90 degrees)

Triangles can be classified by the lengths of their sides. A scalene triangle has no congruent sides. An isosceles triangle has at least two congruent sides. An equilateral triangle has three congruent sides.

Additional Example 3: Classifying Triangles by Lengths of Sides

Classify the triangle. The sum of the lengths of the sides is 19.5 in.

c = 6.5

6.5 in.

N L c

M

c + (6.5 + 6.5) = 19.5

c + 13 = 19.5

c + 13 – 13 = 19.5 – 13

6.5 in.

Side c is 6.5 inches long. Because LMN has three congruent sides, it is equilateral.

Try This: Example 3

Classify the triangle. The sum of the lengths of the sides is 21.6 in.

d = 7.2

7.2 in.

C A d

B

d + (7.2 + 7.2) = 21.6

d + 14.4 = 21.6 d + 14.4 – 14.4 = 21.6 – 14.4

7.2 in.

Side d is 7.2 inches long. Because ABC has three congruent sides, it is equilateral.

WARM-UP

If the angles can form a triangle, classify the triangle as acute, obtuse, or right.

1. 37°, 53°, 90° 2. 65°, 110°, 25°

3. 61°, 78°, 41° 4. 115°, 25°, 40°

The lengths of three sides of a triangle are

given. Classify the triangle.

5. 12, 16, 25 6. 10, 10, 15

not a triangleright

Insert Lesson Title Here

acute obtuse

scalene isosceles

Making Sense of It

L

P O

M

N

40°

30°60°

80°100°

50°

100°

80°

5

How many squares do you see?

14

How many squares do you see?

How many squares do you see?

How many squares do you see?

How many squares do you see?

How many squares do you see?

30

How many squares do you see?

How many squares do you see?

How many squares do you see?

Try this next one as a team!

How many squares do you see?

How many squares do you see?

How many squares do you see?

How many squares do you see?

37

For today’s warm-up, take out your list of vocabulary words, and TURN THEM OVER SO YOU CAN’T SEE THEM. Then, answer these three questions alone!1)What are complementary angles?2)What are supplementary angles?3)What is the sum of all the angle measurements in a triangle?

L

P O

M

N

44°

60°

Can you see Complimentary Angles?

L

P O

M

N

44°

60°

Can you seeSupplementary Angles?

L

P O

M

N

44°

60°

Can you see Triangles?

L

P O

M

N

44°

60°

Can you see Vertical Angles?

L

P O

M

N

44°

60°

L

P O

M

N

44°

60°76°

14°

136°

30°136°

44°

L

P O

M

N

42°

58°

L

P O

M

N

42°

58°80°

10°

138°

32°

138°

42°

L

P O

M

N

40°

60°

L

P O

M

N

40°

60°80°10°

140°

30°

140°

40°