Triangle Basics

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Lesson 4

• Triangle Basics

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(No Transcript)
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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.

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2
1
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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
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The Angles of a Triangle

• The sum of the measures of the three angles of
  any triangle is
• Lets 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

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5
3
2
1
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Example

• In
• What is

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Example

• In the figure, is a right angle and
• bisects
• If then what is

25
50
90
65
?
40
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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
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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

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1
2
3
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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
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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
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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
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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.

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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

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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
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Example

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

A
x
2x
2x
B
C
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Example

• In the figure,
• and
• Find
• Since is isosceles,
• the base angles are congruent. So,

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A
130
D
25
50
110
B
20
C
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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.

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Example

• Rank the sides of the triangle below from
  smallest to largest.
• First note that
• So,

C
B
A
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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.

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• 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
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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
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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
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Example

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

B
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The Pythagorean Theorem

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

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• 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

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Example

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

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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.

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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
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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.

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Example

• In
• If find
• First note that, since the three angles must add
  up to
• So this is a 30-60-90 triangle.

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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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Example

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