Triangles
Properties of Triangles
Properties of Triangles
Types of Triangles
Equilateral Triangle
Isosceles triangle
Scalene triangle
3 equal sides
2 equal sides
3 unequal sides
3 equal angles.
2 equal angles (base)
3 unequal angles
Any triangle containing a 90o angle is a right-angled triangle
An isosceles or a scalene triangle may contain a right angle.
Right-angled
Right-angled
isosceles triangles.
scalene triangle.
To determine the angle sum of any Triangle
Take 3 identical copies of
this triangle and put them
together, so:
How can we use
this to help us?
3
1
Angles on a straight
line add to 180o
2
These are the same
angles as in the
triangle!
The angle sum of a triangle = 1800
Types of Triangles
2.
1.
3.
Equilateral Triangle
Isosceles triangle
Scalene triangle
3 equal sides
2 equal sides
3 unequal sides
3 equal angles.
2 equal angles (base)
3 unequal angles
4.
5.
Any triangle containing a
90o angle is a rightangled triangle
The angle sum of a triangle = 1800
Calculating unknown Angles
Example 1
Calculate angle a.
65o
a
Angle a = 180 – (90 + 65)
= 180 – 155 = 25o
Example 2
b
Calculate angles a, b and c
a
c
Since the triangle is equilateral,
angles a, b and c are all 60o
(180/3)
Calculating unknown Angles
Example 3
b
Angle a = 65o (base angles of
an isosceles triangle are
equal).
Angle b = 180 –(65 + 65)
Calculate angle a.
65
a
= 180 – 130 = 50o
o
Example 4
Calculate angles x and y
130o
y
x
180 - 130
2
50
=
 25
2
Angles x and y =
Calculating unknown Angles
Example 5
Calculate angles a and b.
a
b
180 - 90
2
90
=
 45
2
Angles a and b =
Example 6
Calculate angle a
27o
15o
Angle a = 180 – (15 + 27)
= 180 – 42 = 138o
a
Properties of Triangles
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Pythagorean Theorem
Quiz on Pythagorean
Theorem
Go to Socrative student
Enter the room: RATONEL
Enter your name following
the format
Section Name
Example, 9E Mark
Trigonometric
Ratio
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Right Triangle Trigonometry
Trigonometry is based upon ratios of the sides of
right triangles.
The six trigonometric functions of a right triangle,
with an acute angle , are defined by ratios of two sides
of the triangle.
hyp
opp
θ
The sides of the right triangle are:
 the side opposite the acute angle ,
 the side adjacent to the acute angle ,
 and the hypotenuse of the right triangle.
adj
Right Triangle Trigonometry
The hypotenuse is the longest side and is always
opposite the right angle.
The opposite and adjacent sides refer to another angle,
other than the 90o.
A
A
Trigonometric Ratios
hyp
opp
θ
adj
The trigonometric functions are:
sine, cosine, tangent, cotangent, secant, and cosecant.
opp
sin  =
cos  = adj
tan  = opp
hyp
hyp
adj
csc  =
hyp
opp
sec  = hyp
adj
cot  = adj
opp
S OH C AH T OA
C HO S HA C AO
Finding an angle from a triangle
To find a missing angle from a right-angled triangle we
need to know two of the sides of the triangle.
We can then choose the appropriate ratio, sin, cos or tan
and use the calculator to identify the angle from the
decimal value of the ratio.
1.
Find angle C
14 cm
C
6 cm
a) Identify/label the names of
the sides.
b) Choose the ratio that
contains BOTH of the
letters.
We have been given the
adjacent and hypotenuse so
we use COSINE:
1.
h
14 cm
a
6 cm
adjacent
hypotenuse
Cos A =
C
a
h
Cos C = 6
14
Cos A =
Cos C = 0.4286
C = cos-1 (0.4286)
C = 64.6o
2. Find angle x
Given adj and opp
need to use tan:
x
3 cm a
Tan A = opposite
adjacent
o
8 cm
Tan A =
o
a
Tan x =
8
3
Tan x = 2.6667
x = tan-1 (2.6667)
x = 69.4o
Finding a side from a triangle
3.
7 cm
We have been given
the adj and hyp so we
use COSINE:
adjacent
Cos A =
hypotenuse
30o
k
Cos A = a
h
Cos 30 = k
7
Cos 30 x 7 = k
6.1 cm = k
4.
We have been given the opp
and adj so we use TAN:
50o
4 cm
Tan A =
r
Tan A =
Tan 50 =
Tan 50 x 4 = r
4.8 cm = r
o
a
r
4
Quiz on
Trigonometric Ratio
Go to Socrative student
Enter the room: RATONEL
Enter your name following the
format
Section Name
Example, 9E Mark