4. 4
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.
1
2
3
5. 5
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).
A B
C
∆
Name
ABC
6. 6
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 3 5 180m m m∠ + ∠ + ∠ = °
1 2
34 5
1 3 2 180 .m m m∠ + ∠ + ∠ = °
180 .°
7. 7
Example
• In
• What is
, 95 and 37 .DEF m D m E∆ ∠ = ° ∠ = °
?m F∠
180m D m E m F∠ + ∠ + ∠ = °
95 37 180m F°+ °+ ∠ = °
132 180m F°+ ∠ = °
180 132m F∠ = °− °
48m F∠ = °
8. 8
Example
• In the figure, is a right angle and
bisects
• If then what is
A B C
D
C∠
DB .ADC∠
65 ,m DBC∠ = ° ?m A∠
9065
25
40
In , 180 (65 90 ) 25 .BCD m BDC∆ ∠ = °− °+ ° = °Since is a bisector, 2(25 ) 50 .DB m ADC∠ = ° = °
50
In , 180 (90 50 ) 40 .ACD m A∆ ∠ = °− °+ ° = °
?
9. 9
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
A
B
C
ABC∆
180m A m B m C∠ + ∠ + ∠ = °
90 180m A m B∠ + ∠ + ° = °
90m A m B∠ + ∠ = °
10. 10
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
1 2 3
4
1 180 2 (because of the straight angle)m m∠ = °− ∠
3 4 180 2 (because of the triangle)m m m∠ + ∠ = °− ∠
So, 1 3 4m m m∠ = ∠ + ∠
11. 11
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
12. 12
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.
B C
A
hypotenuse
leg
leg
13. 13
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.
equilateral isosceles scalene
14. 14
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.
15. 15
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
x
180
3 180
60
x x x
x
x
+ + = °
= °
= °
16. 16
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.
ABC∆
.AB AC≅
BC
A
B
C
17. 17
Example
• is isosceles with base
• If is twice then
what is
• Let denote the measure of
• Then
A
B C
ABC∆
.BC
m B∠ ,m A∠
?m A∠
x
.A∠
x
2x 2x
2 .m B x m C∠ = = ∠
Then 2 2 5 180 .x x x x+ + = = °So, 180 /5 36 .m A x∠ = = ° = °
18. 18
Example
• In the figure,
and
• Find
• Since is isosceles,
the base angles are
congruent. So,
A
B
C
D
,AD BD≅
20 ,m C∠ = ° 25 .m A∠ = °
and .m CBD m CDB∠ ∠
20
25
25
110
130
50ABD∆
25 .m ABD∠ = °
Then 130 since the angles of
must add up to 180 .
m ADB
ADB
∠ = °
∆ °
Then 50 since and
are supplementary.
m BDC ADB
BDC
∠ = ° ∠
∠
Then 110 since the angles of
must add up to 180 .
m CBD
BCD
∠ = °
∆ °
19. 19
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.
20. 20
Example
• Rank the sides of the triangle below from
smallest to largest.
• First note that
• So,
A B
C
55° 53°
180 (55 53 ) 72 .m C∠ = °− °+ ° = °
72°
.AC BC AB< <
21. 21
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.
22. 22
• 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
23. 23
Example
• In the medians are
drawn, and the centroid is
point G.
• Suppose
• Find
A
B
CG
M
N
P
ABC∆
12,AM = 7,BM =
and 9.BG =
, , and .GM MC GN
4 7
4.5
Let and 2 . Then
2 12, and so 4.
GM x AG x
x x GM x
= =
+ = = =
Since is a midpoint,
7.
M
MC BM= =
Since is half of ,
0.5(9) 4.5
GN BG
GN = =
24. 24
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
A midline is parallel to one side
of the triangle.
A midline is half the length of
the side to which it is parallel.
x
2x
25. 25
Example
• In D and E are the
midpoints of
respectively.
• If and
then find and
BA
C
D E
,ABC∆
and ,AC BC
7.5DE = 56m ABC∠ = °
AB .m BED∠
is a midline. So,
2 2(7.5) 15.
DE
AB DE= = =
and and are
interior angles on the same side of the
transversal. So, they are supplementary.
So, 180 56 124 .
AB DE ABC BED
m BED
∠ ∠
∠ = °− ° = °
P
26. 26
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
ABC∆
B C
A
2 2 2
BC AC AB+ =
a
bc
2 2 2
a b c+ =
27. 27
• 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:
2 2 2
leg leg hypotenuse+ =
leg
leg
hypotenuse
28. 28
Example
• Suppose is a right
triangle with right angle at C.
B C
A
ABC∆
If 5 and 12 find .BC AC AB= =
2 2 2
2
2
5 12
25 144
169
169 13
AB
AB
AB
AB
+ =
+ =
=
= =
If 5 and 5 find .BC AC AB= =
2 2 2
2
2
5 5
25 25
50
50 25 2 5 2
AB
AB
AB
AB
+ =
+ =
=
= = =g
If 12 and 6 find .AB BC AC= =
2 2 2
2
2
6 12
36 144
144 36 108
108 36 3 6 3
AC
AC
AC
AC
+ =
+ =
= − =
= = =g
29. 29
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.
45 , 45 , and 90 .° ° °
x
2.x
x
x
2x
45
45
30. 30
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:
45
A C
B
,ABC∆ C∠ 45 .m A∠ = °
6,AB = .BC
6 ?
45m B∠ = °
180 .°
: : : : 2.
So, 2 6 and
6 6 2 6 2
3 2
22 2 2
So, 3 2.
AC BC AB x x x
x
x
BC
=
=
= = = =
=
31. 31
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.
A B
C
60°
30°
x
2x 3x
30 , 60 , and 90 .° ° °
30°
60°
3 30°
32. 32
Example
• In
• If find
• First note that, since the three angles
must add up to
• So this is a 30-60-90 triangle.
A C
B
60°
30°
, 60 and 30 .ABC m A m B∆ ∠ = ° ∠ = °
12,BC = .AB
180 ,° 90 .m C∠ = °
: : 2 : : 3
So, 12 3, which gives
12 12 3 12 3
4 3.
33 3 3
So, 2 8 3
AB AC BC x x x
x
x
AB x
=
=
= = = =
= =
33. 33
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.
ABC∆
2 2 2
.AC BC AC+ =
2 2 2
a b c+ =
34. 34
Example
• Show that the triangle in the
figure with side measures as
shown is a right triangle.
7
2425
2 2 2
49 5767 24 25 562+ = =+ =
35. 35
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for Female Education”
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your pens you realize quite how
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