1. Properties of Triangles
A triangle is an enclosed 3 – sided figure.

2. Properties of Triangles
A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles.

3. Properties of Triangles
A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles.
The points at which sides meet are called vertices and are named using capital letters. A B C

4. Properties of Triangles
A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles.
The points at which sides meet are called vertices and are named using capital letters. A
This is now ∆ABC B C

5. Properties of Triangles
A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles.
The points at which sides meet are called vertices and are named using capital letters. A
This is now ∆ABC B C
A triangle also has three angles created by each pair of sides.

6. Properties of Triangles
A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles.
The points at which sides meet are called vertices and are named using capital letters. A
This is now ∆ABC B C
A triangle also has three angles created by each pair of sides.
- sides AC and BC create ∠ACB

7. Properties of Triangles
A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles.
The points at which sides meet are called vertices and are named using capital letters. A
This is now ∆ABC B C
A triangle also has three angles created by each pair of sides.
- sides AC and BC create ∠ACB
- sides AB and BC create ∠ABC

8. Properties of Triangles
A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles.
The points at which sides meet are called vertices and are named using capital letters. A
This is now ∆ABC B C
A triangle also has three angles created by each pair of sides.
- sides AC and BC create ∠ACB
- sides AB and BC create ∠ABC
- sides AB and AC create ∠BAC

9. Properties of Triangles
A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles.
The points at which sides meet are called vertices and are named using capital letters. A
This is now ∆ABC
The three angles in a triangle will ALWAYS add up to 𝟏𝟖𝟎° B C
A triangle also has three angles created by each pair of sides.
- sides AC and BC create ∠ACB
- sides AB and BC create ∠ABC
- sides AB and AC create ∠BAC

10. Properties of Triangles
A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles.
The points at which sides meet are called vertices and are named using capital letters. A
This is now ∆ABC
If ∠ACB =48° and ∠BAC =82°, find ∠ABC
The three angles in a triangle will ALWAYS add up to 𝟏𝟖𝟎° B C
A triangle also has three angles created by each pair of sides.
- sides AC and BC create ∠ACB
- sides AB and BC create ∠ABC
- sides AB and AC create ∠BAC

11. Properties of Triangles
A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles.
The points at which sides meet are called vertices and are named using capital letters. A
This is now ∆ABC
If ∠ACB =48° and ∠BAC =82°, find ∠ABC
∠ACB + ∠BAC + ∠ABC =180°
The three angles in a triangle will ALWAYS add up to 𝟏𝟖𝟎° B C
A triangle also has three angles created by each pair of sides.
- sides AC and BC create ∠ACB
- sides AB and BC create ∠ABC
- sides AB and AC create ∠BAC

12. Properties of Triangles
A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles.
The points at which sides meet are called vertices and are named using capital letters. A
This is now ∆ABC
If ∠ACB =48° and ∠BAC =82°, find ∠ABC
∠ACB + ∠BAC + ∠ABC =180°
48°+82°+ ∠ABC =180°
The three angles in a triangle will ALWAYS add up to 𝟏𝟖𝟎° B C
A triangle also has three angles created by each pair of sides.
- sides AC and BC create ∠ACB
- sides AB and BC create ∠ABC
- sides AB and AC create ∠BAC

13. Properties of Triangles
A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles.
The points at which sides meet are called vertices and are named using capital letters. A
This is now ∆ABC
If ∠ACB =48° and ∠BAC =82°, find ∠ABC
∠ACB + ∠BAC + ∠ABC =180°
48°+82°+ ∠ABC =180°
130°+ ∠ABC =180°
The three angles in a triangle will ALWAYS add up to 𝟏𝟖𝟎° B C
A triangle also has three angles created by each pair of sides.
- sides AC and BC create ∠ACB
- sides AB and BC create ∠ABC
- sides AB and AC create ∠BAC

14. Properties of Triangles
A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles.
The points at which sides meet are called vertices and are named using capital letters. A
This is now ∆ABC
If ∠ACB =48° and ∠BAC =82°, find ∠ABC
∠ACB + ∠BAC + ∠ABC =180°
48°+82°+ ∠ABC =180°
130°+ ∠ABC =180°
∠ABC =180° −130°
The three angles in a triangle will ALWAYS add up to 𝟏𝟖𝟎° B C
A triangle also has three angles created by each pair of sides.
- sides AC and BC create ∠ACB
- sides AB and BC create ∠ABC
- sides AB and AC create ∠BAC

15. Properties of Triangles
A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles.
The points at which sides meet are called vertices and are named using capital letters. A
This is now ∆ABC
If ∠ACB =48° and ∠BAC =82°, find ∠ABC
∠ACB + ∠BAC + ∠ABC =180°
48°+82°+ ∠ABC =180°
130°+ ∠ABC =180°
∠ABC =180° −130°
∠ABC =50°
The three angles in a triangle will ALWAYS add up to 𝟏𝟖𝟎° B C
A triangle also has three angles created by each pair of sides.
- sides AC and BC create ∠ACB
- sides AB and BC create ∠ABC
- sides AB and AC create ∠BAC

16. Properties of Triangles
EXAMPLE : Find the missing angle in the given triangle. A ?
42° B C

17. Properties of Triangles
EXAMPLE : Find the missing angle in the given triangle.
Solution :
∠𝐴+ ∠𝐵+ ∠𝐶=180° A ?
42° B C

18. Properties of Triangles
EXAMPLE : Find the missing angle in the given triangle.
Solution :
∠𝐴+ ∠𝐵+ ∠𝐶=180°
∠𝐴+90°+42°=180° A ?
42° B C
Shows a 90° angle

19. Properties of Triangles
EXAMPLE : Find the missing angle in the given triangle.
Solution :
∠𝐴+ ∠𝐵+ ∠𝐶=180°
∠𝐴+90°+42°=180°
∠𝐴+132°=180° A ?
42° B C

20. Properties of Triangles
EXAMPLE : Find the missing angle in the given triangle.
Solution :
∠𝐴+ ∠𝐵+ ∠𝐶=180°
∠𝐴+90°+42°=180°
∠𝐴+132°=180°
∠𝐴=180° −132° A ?
42° B C

21. Properties of Triangles
EXAMPLE : Find the missing angle in the given triangle.
Solution :
∠𝐴+ ∠𝐵+ ∠𝐶=180°
∠𝐴+90°+42°=180°
∠𝐴+132°=180°
∠𝐴=180° −132°
∠𝑨=𝟒𝟖° A ?
42° B C

22. Properties of Triangles
A rule to check if 3 line segments create a triangle:
The sum of the smallest two sides MUST be larger than the biggest side.

23. Properties of Triangles
A rule to check if 3 line segments create a triangle:
The sum of the smallest two sides MUST be larger than the biggest side.
EXAMPLE : If AB = 6 , BC = 10, and AC = 15, do these sides create a triangle ?

24. Properties of Triangles
A rule to check if 3 line segments create a triangle:
The sum of the smallest two sides MUST be larger than the biggest side.
EXAMPLE : If AB = 6 , BC = 10, and AC = 15, do these sides create a triangle ?
SOLUTION : YES because > 15

25. Properties of Triangles
A rule to check if 3 line segments create a triangle:
The sum of the smallest two sides MUST be larger than the biggest side.
EXAMPLE : If AB = 6 , BC = 10, and AC = 15, do these sides create a triangle ?
SOLUTION : YES because > 15
EXAMPLE #2 : Do the following line segments create a triangle ?
XY = 34 , XZ = 20 , YZ = 10

26. Properties of Triangles
A rule to check if 3 line segments create a triangle:
The sum of the smallest two sides MUST be larger than the biggest side.
EXAMPLE : If AB = 6 , BC = 10, and AC = 15, do these sides create a triangle ?
SOLUTION : YES because > 15
EXAMPLE #2 : Do the following line segments create a triangle ?
XY = 34 , XZ = 20 , YZ = 10
SOLUTION : NO because < 34

27. Properties of Triangles
Side and angle relationships :
The larger the side, the larger the angle that creates that side.
The larger the angle, the larger the side that contains that angle.

28. Properties of Triangles
Side and angle relationships :
The larger the side, the larger the angle that creates that side.
The larger the angle, the larger the side that contains that angle.
In the given triangle, we have 3 given angles. Order the sides from smallest to largest. X
79°
37°
64° Y Z

29. Properties of Triangles
Side and angle relationships :
The larger the side, the larger the angle that creates that side.
The larger the angle, the larger the side that contains that angle.
In the given triangle, we have 3 given angles. Order the sides from smallest to largest. X
SOLUTION:
Angle order = 37° , 64° , 79°
79°
37°
64° Y Z

30. Properties of Triangles
Side and angle relationships :
The larger the side, the larger the angle that creates that side.
The larger the angle, the larger the side that contains that angle.
In the given triangle, we have 3 given angles. Order the sides from smallest to largest. X
SOLUTION:
Angle order = 37° , 64° , 79°
Sides order = XZ , XY , YZ
79°
37°
64° Y Z

31. Properties of Triangles
Another EXAMPLE :
In the given triangle, we have 3 given sides. Order the angles from smallest to largest. P 26 18 Q R 11

32. Properties of Triangles
Another EXAMPLE :
In the given triangle, we have 3 given sides. Order the angles from smallest to largest. P
SOLUTION:
Side order = 11 , 18 , 26
QR , PQ , PR 26 18 Q R 11

33. Properties of Triangles
Another EXAMPLE :
In the given triangle, we have 3 given sides. Order the angles from smallest to largest. P
SOLUTION:
Side order = 11 , 18 , 26
QR , PQ , PR
Angle order = ∠𝑄𝑃𝑅 , ∠𝑄𝑅𝑃 , ∠𝑃𝑄𝑅 26 18 Q R 11

34. Properties of Triangles
Some special triangles :
Equilateral Triangle : All sides and angles are equal
** the angles all = 60°
60° 10 10
60°
60° 10

35. Properties of Triangles
Some special triangles :
Equilateral Triangle : All sides and angles are equal
** the angles all = 60°
Isosceles Triangle : Two sides and angles are equal
60° 10 10
60°
60° 10 21 21
70°
70°

36. Properties of Triangles
Some special triangles :
Right Triangle : Contains a 90° angle