Triangle
See how to find the angles/sides/area of a triangle.
16 examples and their solutions.
Types of Triangles
Angles
Acute Triangle | Right Triangle | Obtuse Triangle |
---|---|---|
3 acute angles (0 < m∠ < 90) | 1 right angle (m∠ = 90) | 1 obtuse angle (90 < m∠ < 180) |
Sides
Scalene Triangle | Isosceles Triangle | Equilateral Triangle |
---|---|---|
No congruent sides | 2 congruent sides | 3 congruent sides |
Interior Angles of a Triangle
Formula
m∠1 + m∠2 + m∠3 = 180
Example
60 + 3x + 30 + 7x + 10 = 180
10x + 100 = 180
10x = 80
x = 8
Close
Exterior Angle of a Triangle
Formula
m∠1 = m∠2 + m∠3
Example
10x + 7 = 60 + 47
10x + 7 = 107
10x = 100
x = 10
Close
Example
30 + 7x + 3 = 68
7x + 33 = 68
7x = 35
x = 5
Close
Triangle Inequality Theorem
Theorem
a + b > c
a + b: Sum of the shorter sides
c: Longest side
Example
2, 3, 4
Can be the sides of a triangle?
Solution Can be the sides of a triangle?
2 + 3 > 4
5 > 4( o )
Can be the sides of a triangle - [1]
5 > 4( o )
Can be the sides of a triangle - [1]
[1]
5 > 4
This is true.
So [2, 3, 4] can be the sides of a triangle.
This is true.
So [2, 3, 4] can be the sides of a triangle.
Close
Example
2, 7, 9
Can be the sides of a triangle?
Solution Can be the sides of a triangle?
2 + 7 > 9
9 > 9( x )
Cannot be the sides of a triangle - [1]
9 > 9( x )
Cannot be the sides of a triangle - [1]
[1]
9 > 9
This is false.
So [2, 7, 9] cannot be the sides of a triangle.
This is false.
So [2, 7, 9] cannot be the sides of a triangle.
Close
Example
3, 4, 8
Can be the sides of a triangle?
Solution Can be the sides of a triangle?
3 + 4 > 8
7 > 8( x )
Cannot be the sides of a triangle
7 > 8( x )
Cannot be the sides of a triangle
Close
Example
3, 5, 5
Can be the sides of a triangle?
Solution Can be the sides of a triangle?
3 + 5 > 5
8 > 5( o )
Can be the sides of a triangle
8 > 5( o )
Can be the sides of a triangle
Close
Relationship between Sides and Angles of a Triangle
Example
Compare m∠A, m∠B, m∠C.
Solution 8 > 7 > 5
m∠A > m∠B > m∠C
[1]
Larger angle ─[opposite]→ Longer side
Close
Example
Compare AB, AC, BC.
Solution 66 > 62 > 52
AB > BC > AC
[1]
Longer side ─[opposite]→ Larger angle
Close
Example
Compare AB, AC, BC.
Solution 67 = 67 > 46
AC = AB > BC
Close
Isosceles Triangle
Definition
that has two congruent sides (= legs).
The non-congruent side is the base.
The two angles
that are adjacent to the base
are congruent.
Example
3x + 25 = 55
3x = 30
x = 10
Close
Example
2⋅(4x + 3) + 3x - 2 = 180 - [1]
8x + 6 + 3x - 2 = 180
11x + 4 = 180
11x = 176
x = 16
Close
Example
x + x = 70
2x = 70
x = 35
[1]
Left triangle: Isosceles triangle
→ Blue angles: x°
Right triangle: Isosceles triangle
→ Purple angles: 70°
→ Blue angles: x°
Right triangle: Isosceles triangle
→ Purple angles: 70°
[2]
Close
Area of a Triangle
Formula
A = 12bh
Example
A = 12⋅6⋅4
= 3⋅4
= 12
Close
Example
A = 12⋅11⋅5
= 552
Close
Example
A = 12⋅8⋅10
= 4⋅10
= 40
Close