Lesson 4

1 / 33

# Lesson 4 - PowerPoint PPT Presentation

Lesson 4. Triangle Basics. Definition. A triangle is a three-sided figure formed by joining three line segments together at their endpoints. A triangle has three sides . A triangle has three vertices (plural of vertex). A triangle has three angles . 3. 2. 1. Naming a Triangle.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Lesson 4' - raphael

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Lesson 4

Triangle Basics

Definition
• A triangle is a three-sided figure formed by joining three line segments together at their endpoints.
• A triangle has three sides.
• A triangle has three vertices (plural of vertex).
• A triangle has three angles.

3

2

1

Naming a Triangle
• Consider the triangle shown whose vertices are the points A, B, and C.
• We name this triangle by writing a triangle symbol followed by the names of the three vertices (in any order).

C

Name

A

B

The Angles of a Triangle
• The sum of the measures of the three angles of any triangle is
• Let’s see why this is true.
• Given a triangle, draw a line through one of its vertices parallel to the opposite side.
• Note that because these angles form a straight angle.
• Also notice that angles 1 and 4 have the same measure because they are alternate interior angles and the same goes for angles 2 and 5.
• So, replacing angle 1 for angle 4 and angle 2 for angle 5 gives

4

5

3

2

1

Example
• In
• What is

D

C

B

A

Example
• In the figure, is a right angle and

bisects

• If then what is

25

50

90

?

65

40

Angles of a Right Triangle
• Suppose is a right triangle with a right angle at C.
• Then angles A and B are complementary.
• The reason for this is that

B

A

C

Exterior Angles
• An exterior angle of a triangle is an angle, such as angle 1 in the figure, that is formed by a side of the triangle and an extension of a side.
• Note that the measure of the exterior angle 1 is the sum of the measures of the two remote interior angles 3 and 4. To see why this is true, note that

4

2

1

3

Classifying Triangles by Angles
• An acute triangle is a triangle with three acute angles.
• A right triangle is a triangle with one right angle.
• An obtuse triangle is a triangle with one obtuse angle.

acute triangle

right triangle

obtuse triangle

A

C

B

Right Triangles
• In a right triangle, we often mark the right angle as in the figure.
• The side opposite the right angle is called the hypotenuse.
• The other two sides are called the legs.

hypotenuse

leg

leg

Classifying Triangles by Sides
• A triangle with three congruent sides is called equilateral.
• A triangle with two congruent sides is called isosceles.
• A triangle with no congruent sides is called scalene.

scalene

isosceles

equilateral

Angles and Sides
• If two sides of a triangle are congruent…
• then the two angles opposite them are congruent.
• If two angles of a triangle are congruent…
• then the two sides opposite them are congruent.
Equilateral Triangles
• Since all three sides of an equilateral triangle are congruent, all three angles must be congruent too.
• If we let represent the measure of each angle, then
Isosceles Triangles
• Suppose is isosceles where
• Then, A is called the vertex of the isosceles triangle, and is called the base.
• The congruent angles B and C are called the base angles and angle A is called the vertex angle.

B

A

C

Example
• is isosceles with base
• If is twice then what is
• Let denote the measure of
• Then

A

x

2x

2x

B

C

Example
• In the figure,

and

• Find
• Since is isosceles,

the base angles are congruent. So,

25

A

130

D

25

50

110

B

20

C

Inequalities in a Triangle
• In any triangle, if one angle is smaller than another, then the side opposite the smaller angle is shorter than the side opposite the larger angle.
• Also, in any triangle, if one side is shorter than another, then the angle opposite the shorter side is smaller than the angle opposite the longer side.
Example
• Rank the sides of the triangle below from smallest to largest.
• First note that
• So,

C

B

A

Medians
• A median in a triangle is a line segment drawn from a vertex to the midpoint of the opposite side.
• An amazing fact about the three medians in a triangle is that they

all intersect in a common

point. We call this

point the centroid

of the triangle.

Another fact about medians is that the distance along a median from the vertex to the centroid is twice the distance from the centroid to the midpoint.

2x

x

Example
• In the medians are drawn, and the centroid is point G.
• Suppose
• Find

A

N

4.5

C

G

P

4

7

M

B

Midlines
• A midline in a triangle is a line segment connecting the midpoints of two sides.
• There are two important facts about a midline to remember:

midline

x

2x

C

D

E

A

Example
• In D and E are the midpoints of respectively.
• If and then find and

B

A

c

b

a

C

B

The Pythagorean Theorem
• Suppose is a right triangle with right angle at C.
• The Pythagorean Theorem states that
• Here’s another way to state the theorem: label the lengths of the sides as shown. Then

leg

leg

hypotenuse

• In words, the Pythagorean Theorem states that the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse, or:

A

C

B

Example
• Suppose is a right triangle with right angle at C.

45

45

45-45-90 Triangles
• A 45-45-90 triangle is a triangle whose angles measure
• It is a right triangle and it is isosceles.
• If the legs measure then the hypotenuse measures
• This ratio of the sides is memorized, and if one side of a 45-45-90 triangle is known, then the other two can be obtained from this memorized ratio.
Example
• In is a right angle and
• If then find
• First notice that too since the angles must add up to
• Then this is a 45-45-90 triangle and so:

B

6

?

45

C

A

C

B

A

30-60-90 Triangles
• A 30-60-90 triangle is one in which the angles measure
• The ratio of the sides is always as given in the figure, which means:
• The side opposite the angle is half the length of the hypotenuse.
• The side opposite the angle is times the side opposite the angle.

B

C

A

Example
• In
• If find
• First note that, since the three angles must add up to
• So this is a 30-60-90 triangle.
The Converse of the Pythagorean Theorem
• Suppose is any triangle where
• Then this triangle is a right triangle with a right angle at C.
• In other words, if the sides of a triangle measure a, b, and c, and

then the triangle is a right triangle where the hypotenuse measures c.

25

24

7

Example
• Show that the triangle in the figure with side measures as shown is a right triangle.