1. Creating Triangles
Concept 41

2. Working with your partner, create each triangle described and fill in the chart below Let a = the shortest side and c = the longest side. #
Sides
Does it make a triangle? a b
a+b
<, >, = c 1
3, 5, 7 2
3, 4, 6 3 4
4, 4, 6 5 6 7 8

3. Using the information from the chart to help you, answer the questions:
1) With a as the shortest side and c as the longest side, if 𝒂+𝒃>𝒄 then what do you know?
 2) With a as the shortest side and c as the longest side, if 𝒂+𝒃=𝒄 then what do you know?   
3) With a as the shortest side and c as the longest side, if 𝒂+𝒃<𝒄 then what do you know?
4) If two sides of a triangle are 6cm and 4cm, could the third side be 7cm? Justify your answer.
5) If two sides of a triangle are 3in. and 7in., what are the possible side lengths for the third side?

4. Use the pieces from the activity to create the following triangles
Use the pieces from the activity to create the following triangles. Once each triangle is created then measure the angles in the triangle. Draw a sketch and label all the information.
1. AB = 3, BC = 6, and CA = LM = 5, MN = 4, NL = 7

5. List the angles and side in order from greatest to least.

7. Square Root Review

8.
∙ 3
2 6∙3
2 2∙3∙3 9
2 2∙9
2∙3 2
6 2

9. ∙ 8
∙ 2
5 2∙8
7 36
5 16
7∙6 42
5∙4 20
∙ 3
2 9
2∙3 6

10.
9 1
3 3
3 1 1 3 ∙ ∙ = =
4 2
40 5
= 4∙10 5
=2 2 =

11. Pythagorean Theorem
Concept 42

12. Finding the length of the missing side in a right triangle
Hypotenuse
Leg
A Greek Philosopher and mathematician (credited with the first official proof of the Pythagorean Theorem)
Leg
RIGHT

13. Here is what each letter represents:
A and b represent lengths of legs, c represents the length of the hypotenuse (longest side)

14. = 𝑐 2
36+36= 𝑐 2
6 m
72= 𝑐 2
72 =𝑐
6 m
6 2 =𝑐

15. 𝑎 = 13 2
𝑎 2 +64=169
𝑎 2 =105
𝑎= 105
𝑎= 105

16. = 𝑐 2
16+121= 𝑐 2
137= 𝑐 2
137 =𝑐

17. = 16 2
64+100=256
164=256

18. 𝑎 2 + 𝑏 2 = 𝑐 2
𝑎 2 + 𝑏 2 = 𝑐 2

19. (−4) = 𝑐 2 -4
16+36= 𝑐 2
52= 𝑐 2 6
52 =𝑐
2 13 =𝑐

20. Determining the type of triangle.
Concept 43

21. Complete the columns of the table using specific lengths.
Pick three side lengths to represent a triangle.
Determine that it is possible to make a triangle. Remember the sum of two sides must be greater than the third.
Then determine what type of triangle you think it looks like. Make sure to get some of each triangle. (measure angles to be sure)
Once you have done this for 12 triangles then you can complete the next rows of the table.
Follow what each of the next three-column state.
On the last column make and inequality or equation, that relates the longest side to the sum of the other two sides.

22. Pick three side lengths. a + b > c Does it make a Triangle?
What type of triangle does it appear to be (acute, right,
or obtuse)?
Longest side
Shortest side
Medium side
Write an inequality or equation about the side length relationship
3, 4, 5
3 + 4 > 5
7 > 5
5 2 = 25
3 2 = 9
4 2 = 16
25 ____
25 = 25
7, 9, 12
7, 8, 9

23. Pick three side lengths. a + b > c Does it make a Triangle?
What type of triangle does it appear to be (acute, right,
or obtuse)?
Longest side
Shortest side
Medium side
Write an inequality or equation about the side length relationship

24. 𝑐 2 = 𝑎 2 + 𝑏 2  _____________ triangle
Make a statement about how to tell the type of triangle from the lengths of the sides. (Look now at the last column and compare it with the type of triangle it is.)
𝑐 2 = 𝑎 2 + 𝑏 2  _____________ triangle
𝑐 2 > 𝑎 2 + 𝑏 2  _____________ triangle
𝑐 2 < 𝑎 2 + 𝑏 2  _____________ triangle
Right triangle
Obtuse triangle
Acute triangle

25. Examples:
Do the following with math only to determine if they make a triangle, if so what type do they make?
1. 8, 9, , 8, , 14, 18
8 + 9 __ 15
17 > 15
8 + 8 __ 5
16 > 5
8 + 5 ___ 8
13 > 8
__ 18
25.3 > 18
Makes a triangle
Makes a triangle
Does make a triangle
8 2 _____
64 ___
64 ___ 89
15 2 _____
225 ___
225 ___ 145
18 2 _____ ( 8 2 )
324 ___
324 ___ 324
<
> =
Acute triangle
Obtuse triangle
Right triangle

26. Special Right Triangles
Concept 44

27. LEG
Hypotenuse 5 12
3 5
2 6 5
5 2
6 2
6 2
3 5 15 2 2

28. 45°, 45°, 90° Triangles 𝑥 2 x x ÷ 2 ∙ 2 𝑥 2 + 𝑥 2 = ℎ 2 2𝑥 2 = ℎ 2
leg
45°
In a triangle
~ Two angles measure ______ degrees.
~One angle measures ______ degrees.
~ Two sides (the legs) are ___________________.
~ The _____________ is _____ times the length of a _________.
~ The sides are in the ratio:
______, ______, __________.
leg
45°
÷ 2 x
90°
𝑥 2
congruent
Hypotenuse
45°
hypotenuse 2
leg
∙ 2 𝑥 𝑥
𝑥 2
𝑥 2 + 𝑥 2 = ℎ 2
2𝑥 2 = ℎ 2
2𝑥 2 = ℎ 2
𝑥 2 =ℎ

29. Examples: Find the value of each variable in each triangle. 𝑥= =5 𝑥=
=2 3 𝑦= 5 𝑥=
2 3
∙ 2 𝑎= 𝑥=
7 2
= 7∙2 2 𝑦=
7 2 ∙ 2
=7∙2 =7
=14 𝑏=

30. Examples: Find the value of each variable in each triangle.
∙ 2 ∙ 2
10 2 𝑥=
= 10∙ 2 2
=5 2 𝑦=
5 2 𝑥= 9 𝑦=
9∙ 2
=9 2
∙ 2 ∙ 2 𝑥=
= 8∙2 2
5 2 ∙ 2 2 𝑥= 𝑦= =8
= 5∙2 2 𝑦= 8 =5

31. Short Leg
Long Leg
Hypotenuse 6
3 5
8 3
6 3 12
3 15
6 5
3 5
5 2 12
4 3

32. 30°, 60°, 90° Triangles 2𝑥 𝑥 3 𝑥 ÷ 3 ∙2 ÷2 ∙ 3 30° 60° 30° 90°
In a triangle
~ Two acute angles measure ______ and _______degrees.
~One angle measures ______ degrees.
~ The hypotenuse is _________ the length of the __________ leg.
~ The ___________ leg is _____ times the length of the _________ leg.
~ The sides are in the ratio:
_____, _____, ________.
30°
60°
90°
30°
Hypotenuse
Longer leg
twice 2𝑥
shorter
𝑥 3
longer 3
÷ 3 ∙2
shorter ÷2
60°
∙ 3 𝑥
𝑥 3 2𝑥 𝑥
Shorter leg

33. Examples: Find the value of each variable in each triangle.
7 3 ∙ 3 𝑥=
=7∙3
=21 𝑦=
7 3 ∙2 𝑥= =4
=14 3 𝑦=
4∙2 =8
3 2 ∙ 3
12 3 ∙ 3
3 2 𝑥=
12 3 ∙2 𝑦= 𝑥= 𝑦=
=24 3
=12∙3 =
=36

34. Examples: Find the value of each variable in each triangle.
∙ 3 ∙ 3
9 3 𝑥= 𝑦=
3 3 ∙2
5∙ 3 𝑥= =
=6 3 𝑦=
5∙2
=10
=3 3
∙ 3
∙ 3 ∙2 𝑦= 𝑚= 𝑛= ∙2 𝑥=
= 2∙3 3 =
= ∙2 = = =2