This chapter opens with a set of explorations designed to introduce you to new geometric topics that you will explore further. You will learn about the special properties right triangles as well as find their perimeters and areas. You will explore the relationships of the sides and angles of a triangle. You will also develop a deeper understanding of the Pythagorean theorem. The chapter ends with an exploration of logic and conditional statements.

3.1

What is the Side Relationship With the Angle?

Pg. 3Triangle Inequality

3.1 – What is the Side Relationship With the Angle?Triangle Inequality

Today you are going to discover how the angles in a triangle relate to the sides of a triangle. You are also going to discover when three given side lengths don’t form a triangle.

3.1 – TRIANGLE LENGTHSa. Examine at the triangle below. Measure the side lengths with a ruler and write it on the triangle. Then list the side lengths from smallest to largest. Then list the angles from smallest to largest.

2cm

5cm

6.8cm 25°

10°145°

Sides: _______________________ 

Angles: ________________________

,AB ,BC AC

C , A, B

b. What is the relationship between the largest side and the largest angle? What about the smallest side and the smallest angle?

2cm

5cm

6.8cm 25°

10°145°

b. What is the relationship between the largest side and the largest angle? What about the smallest side and the smallest angle?

2cm

5cm

6.8cm 25°

10°145°

They are opposite from each other

c. Imagine that became smaller. Which side length would change?

2cm

5cm

6.8cm 25°

10°145°

AC

B

d. Imagine that became larger. Which side length would change?

2cm

5cm

6.8cm 25°

10°145°

AB

C

e. What is always the longest side in a right triangle?

2cm

5cm

6.8cm 25°

10°145°

hypotenuse

Opposite Sides

3.2 – ORDERING UPList the sides and angles in order from smallest to largest. Find the missing angles to help you.

Sides: ________________

Angles: _______________

,BC ,AC AB

,A ,B C

Sides: _________________

Angles: ________________

,ST ,RT SR

R, S, T

AB BC, AC

45° A =C, B

,AB ,AC BC60°

C, B, A

3.3 – CONSTRUCTING TRIANGLESConsider the segments below. Construct a triangle with the given side lengths.

3.5 in

3.4 – CONSTRUCTING TRIANGLESConsider the segments below. Construct a triangle with the given side lengths.

3 in

3.5 – CONSTRUCTING TRIANGLESConsider the segments below. Construct a triangle with the given side lengths.

2.5 in

3.6 – IS IT POSSIBLE?a. Use the manipulative provided by your teacher to investigate what is happening in the previous problem. Can a triangle be made with any three side lengths? If not, what condition(s) would make it impossible to build a triangle? Try building triangles with the side lengths provided by your teacher.

Trial #1: ______________

Trial #2: ______________

Trial #3: ______________ 

• Make an equilateral triangle

yes

• Make an isosceles triangle

yes

• Make a scalene triangle

yes

Trial #4: ______________

Trial #5: ______________

Trial #6: ______________ 

• Make a triangle with sides of green, yellow, and blue (8.66cm, 10cm, 12.24cm)

yes

• Make a triangle with sides of orange, purple, and red (5cm, 7.07cm, 14.14cm)

no

• Make a triangle with sides of 2 orange and a yellow (5cm, 5cm, 10cm)

no

b. For those triangles that could not be built, what happened? Why were they impossible?

Two sides need to be long enough to reach

Triangle Inequality Theorem: 

a + b > c

a + c > b

b + c > a

The sum of two sides needs to be greater than the third

http://hotmath.com/util/hm_flash_movie_full.html?movie=/hotmath_help/gizmos/triangleInequality.swf

Triangle inequality

3.7 – TRIANGLE IMPOSSIBLEIs it possible to construct a triangle with the given side lengths? If not, explain why not. a.3, 4, 5 b. 1, 4, 6

c. 17, 17, 33 d. 7, 45, 52

7 > 5yes

5 > 6no

34 > 33yes

52 > 52no

3.8 – MAXIMUM AND MINIMUM LENGTHS Examine the pictures of the triangles below. There is a range of values that will complete a triangle. The fact that there are restrictions on the side of a triangle is referred to as the Triangle Inequality Theorem. Determine the minimum and maximum values that will make a triangle. What value does it have to be above? What value does it have to be below?

x + 13 > 19 x + 19 > 13 13 + 19 > xx > 6 x > -6 32 > x

x < 32

More than 6 Less than 32

6 < x < 32

15 – 14 < x < 15 + 14

1 < x < 29

16 – 13 < x < 16 + 13

3 < x < 29

3.9 – MAXIMUM AND MINIMUM LENGTHSDescribe the possible lengths of the third side of the triangle given the lengths of the other two sides. a. 5m, 17m b. 8in, 12in c. 10ft, 40ft

17 – 5 < x < 17 + 5

12 < x < 22

a. 5m, 17m

12 – 8 < x < 12 + 8

4 < x < 20

b. 8in, 12in

40 – 10 <x < 40 + 10

30 < x < 50

c. 10ft, 40ft

3.10 – SMALLEST SIDEUse the information to determine what is the smallest whole number the following can be:

12 – 10 <x < 12 + 10

2 < x < 22

25 – 15 <x < 25 + 15

10 < x < 40

3.11 – POSSIBLE SIDE LENGTHDetermine a possible length of the missing side of the triangle.

7 – 5 < x < 7 + 5

2 < x < 12

17 – 10 < x < 17 + 10

7 < x < 27

3.12 – PERIMETERA student draws a triangle with a perimeter of 12in. The student says that the longest side measures 7in. How do you know that the student is incorrect?

7in

P = 12in+5in

5 < 7