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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.

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Lesson 4

Lesson 4

Triangle Basics


Definition
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
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 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
Example

  • In

  • What is


Example1

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
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
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
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


Right triangles

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
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
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
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
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


Example2
Example

  • is isosceles with base

  • If is twice then what is

  • Let denote the measure of

  • Then

A

x

2x

2x

B

C


Example3
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
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.


Example4
Example

  • Rank the sides of the triangle below from smallest to largest.

  • First note that

  • So,

C

B

A


Medians
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.


2x

x


Example5
Example median from the vertex to the centroid is twice the distance from the centroid to the midpoint.

  • 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
Midlines median from the vertex to the centroid is twice the distance from the centroid to the midpoint.

  • 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


Example6

C median from the vertex to the centroid is twice the distance from the centroid to the midpoint.

D

E

A

Example

  • In D and E are the midpoints of respectively.

  • If and then find and

B


The pythagorean theorem

A median from the vertex to the centroid is twice the distance from the centroid to the midpoint.

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 median from the vertex to the centroid is twice the distance from the centroid to the midpoint.

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:


Example7

A median from the vertex to the centroid is twice the distance from the centroid to the midpoint.

C

B

Example

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


45 45 90 triangles

45 median from the vertex to the centroid is twice the distance from the centroid to the midpoint.

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.


Example8
Example median from the vertex to the centroid is twice the distance from the centroid to the midpoint.

  • 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


30 60 90 triangles

C median from the vertex to the centroid is twice the distance from the centroid to the midpoint.

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.


Example9

B median from the vertex to the centroid is twice the distance from the centroid to the midpoint.

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
The Converse of the Pythagorean Theorem median from the vertex to the centroid is twice the distance from the centroid to the midpoint.

  • 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.


Example10

25 median from the vertex to the centroid is twice the distance from the centroid to the midpoint.

24

7

Example

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


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