# 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