Download Trigonometry Notes on The Geometry of Triangles.

Trigonometry
Notes on The Geometry of Triangles.
Fundamental Definitions: Triangles
A triangle is a 2
dimensional shape
made up of 3 line
segments at each
endpoint.
Thus, the following are not triangles:
We can denote a triangle with vertices A, B and C, the corners, as ΔABC . The order of the vertices doesn’t matter.
Properties of Triangles: The Sum of the Angles in a Triangle
The sum of the 3
angles in any triangle
adds up to a straight
angle, 180˚= π .
θ
β
α
θ + α + β = 180° = π
Why is this true? Look at this figure.
The 2 diagonal sides are transversals for the 2 parallel lines, giving angles with the same the same measurement on top. Since the 3
angles have to form a straight angle, θ + α + β = 180° = π .
SCC:Rickman
Notes on The Geometry of Triangles.
Page #1 of 4
Example #1: If a triangle has 2 angles measuring the following, find the 3rd angle. a) 123˚ and 14˚, b)
π
3π
and
, and c) 102˚ and 85˚.
3
5
►
a)
θ + 123° + 14° = 180°
θ + 137° = 180°
θ = 180° − 137°
θ = 43°
Thus, the 3rd angle is 43˚.
b)
π 3π
θ+ +
=π
3 5
5π 9 π
θ+
+
=π
15 15
14π 15π
θ+
=
15
15
π
θ=
15
π
The 3rd angle is
.
15
c)
θ + 102° + 85° = 180°
θ + 187° = 180°
θ = 180° − 187°
θ = -7°
Since, an angle in a triangle
would be a traditional angle,
and thus not negative, there is
no triangle with angles of
102˚ and 85˚.
□
Classification of Triangles: Name
SCC:Rickman
Classification of Triangles
Definition Using θ
Acute
All 3 angles are acute
Obtuse
1 obtuse angle with 2 acute angles
Right
1 right angle with 2 acute angles
Equilateral
or
Equiangular
All 3 sides have the same length
and
All 3 angles are the same size
Isosceles
2 sides have the same length
and
2 angles are the same size
Scalene
Sides have 3 difference lengths
and
Angles are 3 different sizes
Notes on The Geometry of Triangles.
Example Graph
Page #2 of 4
Example #2: If a triangle has the 2 given angles, classify the triangle as one of the above. a)
5π
2π
3π
3π
and
, b)
and
,
8
7
11
14
c) 87˚ and 73˚, d) 28˚ and 28˚ and e) 60˚ and 60˚.
►
a)
Because 5 is larger than half of
8,
5π π
>
8 2
, and it’s an obtuse triangle.
5π 3π
θ = π−
−
8 11
88π − 55π − 24
θ=
88
9π
θ=
88
Thus, it is also a scalene
triangle.
□
b)
2π 3π
θ = π−
−
7 14
14π 4π 3π
θ=
−
−
14 14 14
7π π
θ=
=
14 2
c)
θ = 180° − 87° − 73°
θ = 20°
Since 20°,87°, 73° < 90° ,
it’s an acute scalene
triangle.
d)
θ = 180° − 28° − 28°
θ = 124° > 90°
Thus, it’s an obtuse
isosceles triangle.
e)
θ = 180° − 60° − 60°
θ = 60°
Thus, it’s an acute
equilateral triangle.
It’s a right
scalene triangle.
Example #3: If a right triangle has an angle with the following measurement, find the other angle. a) 42˚, and b)
3π
.
5
►
a)
θ = 180° − 90° − 42°
θ = 48°
Thus, the 3rd angle is 48˚.
b)
π 3π
−
2 5
10π 5π 2π
θ=
−
−
10 10 10
3π
θ=
10
θ = π−
Hence, the 3rd angle is
3π
.
10
□
Similar Triangles: Similar Triangles
2 triangles are similar,
if and only if,
they have the same 3
measurements for their
angles.
SCC:Rickman
Notes on The Geometry of Triangles.
Page #3 of 4
Property of the Sides of Similar Triangles
The corresponding
sides of a similar
triangles have the same
ratio.
a1 b1 c1
=
=
a2 b2 c2
Example #4: Are the given triangles similar? If so find the length of the missing sides.
Given:
BD parallel to CE
Length of
Length of
Length of
Length of
Find:
Lengths of
CE = 15in.
BD = 12in.
AE = 20in.
AC = 16in.
AB and AD .
►
Since BD is parallel to CE ,
measurement of ∠ABD = measurement of ∠ACE ,
and measurement of ∠ADB = measurement of ∠AEC .
Also, the triangles share ∠CAE . Thus, the angles are the same, and ΔABD and ΔACE are similar triangles.
Thus,
15in 5 x = length of AB y = length of AD
ratio =
=
20 5
16 5
12in 4
=
=
y 4
x 4
⎛4⎞
⎛4⎞
16 ⎜ ⎟ = x
20 ⎜ ⎟ = y
5
⎝ ⎠
⎝5⎠
64
16 = y
=x
5
64
in and 16in.
Thus, the missing sides have lengths of
5
□
SCC:Rickman
Notes on The Geometry of Triangles.
Page #4 of 4