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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.

slide21
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

example5
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
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

example6

C

D

E

A

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

B

the pythagorean theorem

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
slide26

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:
example7

A

C

B

Example
  • Suppose is a right triangle with right angle at C.
45 45 90 triangles

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.
example8
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

30 60 90 triangles

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.
example9

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
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.

example10

25

24

7

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