1. Rules of Pythagoras All Triangles:
The sum of the three angles equals 180°.
Right Triangles:
One of its angles must be a right angle (90°).
The side opposite of the right angle is called the hypotenuse (c).
The other sides of the triangle are called the legs (a and b).
Hypotenuse (c)
leg 2 (b)
leg 1 (a)

2. Pythagorean Theorem c a b
If it is a right triangle:
a2 + b2 = c2

3. Pythagorean Theorem and its Converseb
If it is a right triangle then a2 + b2 = c2
The converse states that if a2 + b2 = c2 then it must be a right triangle

4. Right Triangle 17 8
Prove that the following triangle is a right triangle
Be Careful! Assign c correctly! 15
If a2 + b2 = c2 then it is a right triangle
= 172
= 289
289 = 289 , it is a right triangle

5. Special Right Triangles
45° - 45° - 90°
In a 45° - 45° - 90° the two legs are equal.
Hence, we can rewrite the Pythagorean Theorem as a2 + a2 = c2.
This means that 2a2 = c2 or c = a 𝟐

6. Special Right Triangles
45° - 45° - 90°
In a 45° - 45° - 90° triangle, both legs are the same and the length of the hypotenuse is 2 times the length of a leg.
Hypotenuse = 𝟐 ∙ leg

7. Special Right Triangles
45° - 45° - 90° x 40
What is the length of the hypotenuse?
Hypo = x = 𝟐 ∙ leg = 40 𝟐

8. Special Right Triangles
45° - 45° - 90°
100 x
What is the length of a leg?
If hypotenuse = 𝟐 ∙ leg
100 = 𝟐 ∙ x
x = 𝟏𝟎𝟎 𝟐
Remove the radical from the denominator.
x = 𝟏𝟎𝟎 𝟐 ∙ 𝟐 𝟐 = 𝟏𝟎𝟎 𝟐 𝟐
x = 50 𝟐

9. Special Right Triangles
30° - 60° - 90°
In a 30° - 60° - 90° there is a shorter leg and a longer leg. These triangles also have special features.
Hypotenuse = 2∙ Shorter Leg
Longer Leg = 𝟑 ∙ Shorter Leg

10. Special Right Triangles
30° - 60° - 90°
60° 80 x
30° y
What is the length of x and y?
x is shorter side (opposite 30°)
hypotenuse = 2 ∙ shorter leg
80 = 2∙ x
x = 40
y is shorter side (opposite 60°)
Longer Leg = 𝟑 ∙ Shorter Leg
y = 𝟒𝟎 𝟑

11. Acute Triangle c a b
A triangle with three acute (less than 90°) angles
In non-right triangles a2 + b2 will not equal c2
Still use the Pythagorean Theorem to determine if the triangle is acute or obtuse
If c2 < a2 + b2 then the triangle is acute

12. Acute Triangle 10
Prove that the following triangle is an acute triangle 7 8
If it was a right triangle then a2 + b2 = c2
≠ 102
≠ 100
113 > or a2 + b2 > c2
Since 113 > it is an acute triangle

13. Obtuse Triangle c a b
A triangle with one obtuse (greater than 90°) angle
In non-right triangles a2 + b2 will not equal c2
Still use the Pythagorean Theorem to determine if the triangle is acute or obtuse
If c2 > a2 + b2 then the triangle is obtuse

14. Obtuse Triangle 10
Prove that the following triangle is an obtuse triangle 6 7
If it was a right triangle then a2 + b2 = c2
≠ 102
≠ 100
100 > or c2 > a2 + b2
Since > it is an obtuse triangle

15. Practice
Find x:

16. If c is the measure of the hypotenuse, find each missing side:
Practice
If c is the measure of the hypotenuse, find each missing side:
1. a = 12, b = 9, c = ?
c = 15
2. a = 8, b = ?, c = 21
b = 19.4

17. Find the missing measure in each right triangle:
Practice
Find the missing measure in each right triangle: 1. 2.
c = 12.6
x = 21

18. In the following right triangles determine the value of x:
Practice
In the following right triangles determine the value of x: 1. 2. 12
(x + 4) 10 x 2x
(x - 2)
x = 7.3
x = 5.4

19. Practice
The sides of a triangle are listed below, determine whether the triangle is obtuse, acute or right.
1. 8, 9, 13
obtuse
2. 7, 12, 13
acute