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### 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

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

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

- 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.

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

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

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

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:

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

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.

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.

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