Condition of Triangles

​MGSE7.G.2 Explore various geometric shapes with given conditions. Focus on creating triangles from three measures of angles and/or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

Multiple Choice

Determine if the 3 numbers can be measures of the sides of a triangle. 9, 8, 10

• Yes

• No

Multiple Choice

Does a triangle with these side lengths exist?

3,7,10

• Yes

• No

Not a triangle

• 1 + 2 = 3

• Is not greater than 3

• Cannot form a triangle

Multiple Choice

Determine if the 3 numbers can be measures of the sides of a triangle. 4, 16, 19

• Yes

• No

Multiple Choice

Can the sides of a triangle have lengths 3, 4, and 9?

• Yes

• No

• I don't know

• Cannot be determined

Multiple Choice

Does a triangle with these
side lengths exist?
(Don't fall for this one.)
0, 11, 12

• Yes

• No

• I don't know.

Multiple Choice

• 5 ft., 8 ft., 15 ft.

• 2 ft., 5 ft., 8 ft.

• 5 ft., 8 ft., 12 ft.

Multiple Choice

Which range could be the measurement of the third side of a triangle if the other two sides were 21cm and 15 cm?

• 7 - 14

• 17-22

• 1-5

• 6-14

Multiple Choice

If two side lengths of a triangle are 10 𝑖𝑛 and 15 𝑖𝑛, which length below could NOT be the third side length?

• 17 𝑖𝑛

• 12 𝑖𝑛

• 24 𝑖𝑛

• 4 𝑖𝑛

Multiple Choice

Given the side lengths of 18mm and 10 mm, what measurement could be the length of the third side of a triangle?

• 7mm

• 4mm

• 28mm

• 24mm

Multiple Select

Which of the following sets of side lengths could produce a triangle. SELECT ALL THAT APPLY.

• 4, 4, 4

• 5, 5, 10

• 3, 6, 9

• 5, 7, 11

• 13, 5, 6

Unique Triangles

• If you are given sides, you cannot do anything to change the triangle.

• Sides are very strict boundaries. You will only be able to make ONE triangle that is unique.

• MAKE SURE YOUR SIDES FOLLOWS THE RULES!

Multiple Choice

Triangle with sides of 7, 8, and 9 cm.

• one unique triangle

• more than one unique triangle

• no triangle

Multiple Choice

Triangle with sides of 1.5, 3, and 1.5 cm.

• one unique triangle

• more than one unique triangle

• no triangle

Multiple Choice

Triangle with angles of 55, 45, 75

• one unique triangle

• more than one unique triangle

• no triangle

Many Triangles: Three Angles (AAA)

• Angles are very flexible! If you are given angles with no side lengths to use, you can make as many triangles as you want!

• You can change the lengths of the sides without changing the angles.

• Make sure all of your angles add up to be 180!

When 3 angles are given, if the angles meet the angle sum theorem (sum of all angles in a triangle = 180), then they will form more than 1 triangle.

Multiple Choice

Triangle with angles of 70, 90, 20

• one unique triangle

• more than one unique triangle

• no triangle

​As you can see, it is possible to make triangles of different sizes even if they have the same angle measurements.

​​Smaller Triangle

​This is called SCALING. Remember that scaling is making something bigger or smaller with the same proportions or the same shape.

Larger Triangle

Multiple Choice

Which of the following cannot be the angle measures of a triangle?

• 110, 43, 27

• 89, 47, 32

• 95, 32, 53

• 100, 30, 50

Dropdown

I ​

Select answer

make a triangle with angles of 25°, 75°, and 80°.

Dropdown

I ​

Select answer

make a triangle with angles of 20°, 70°, and 110°.

Here are the rules about triangles:

• ALL angles need to add up to be 180 degrees! No matter what!

• The 3rd side of a triangle needs to be smaller than the sum of your two shorter sides.

• If you do not meet either of these rules, IT ISN'T A TRIANGLE!

The Sum of the Angles of a Triangle

all 3 angles add to 180 degrees.

Multiple Choice

What is the measure of ∠B?

• 30

• 67

• 45

Multiple Choice

What is the measure of ∠F?

• 180

• 90

• 45

Multiple Choice

What is the measure of ∠I?

• 71

• 50

• 59

Multiple Choice

Find the measure of the indicated angle. 

• 95

• 85

• 35

• 45

​1) All interior angles = 180.

​2) So, we can add all the angles together, even with variables, and set them equal to 180.

​

​x + 10 + 2x + 20 + 2x - 5 = 180

​Combine like terms:

all variables the same. 1x + 2x + 2x = 5x

all constants. 20 + 10 - 5 = 25

​

​Your new equation: 5x + 25 = 180

​ subtract the constant - 25 -25

​ 5x = 155

divide both sides by 5 to isolate the variable.

​ 5x/5 = 155/5

x=31

​Check Your Work:

​Replace all x's with (31) and see if both sides are equal.

​5x + 25 = 180

​5 (31) + 25 = 180

​Use your calculator

​180 = 180

​This proves 31 is the

​answer for x.

Multiple Choice

Solve for x.

• 16

• 11

• 50

• 6

Set up an equation to solve for x

Multiple Choice

Find the missing value of x.

• x = 118^o

• x = 59^o

• x = 69^o

• x = 108^o

Two Angles and One Side (ASA)

When 2 angles and 1 side are given, if the angles have a sum of less than 180 degrees, then they will form a unique triangle.

Two Sides and An Angle (SAS)

When 2 sides and 1 included angle are given, then they will form a unique triangle.

Other Triangles

Multiple Choice

I have a triangle that has sides 3 inches, 5 inches, and 7 inches. I can make...

• One unique triangle

• Many triangles

• This isn't a triangle

Multiple Choice

I have a triangle that has the angles 30, 50, and 50. I can make...

• One unique triangle

• Many triangles

• This isn't a triangle!

Multiple Choice

Determine if which type of triangle equivalence or the following

2 in, 2in, 2in

• Unique

• Many

• No

Multiple Choice

Determine if which type of triangle equivalence or the following

3 cm, 4 cm, 6 cm

• Unique

• Many

• No

Multiple Choice

How many triangles exist with the given specifications?

$$C=48\degree,\ b=12,\ a=25$$  

• No Triangle

• 1 Triangle

• 2 Triangles

• Not enough information

Multiple Choice

How many triangles exist with the given specifications?

$$A=57\degree,\ B=89\degree,\ a=65$$  

• No Triangle

• 1 Triangle

• 2 Triangles

• Not enough information

Multiple Choice

If angle A is 45 degrees, angle B is 45 degrees, and line AB is 5 cm, how many triangles can you create?

• Only 1 unique triangle

• Multiple triangles

• No triangles

Multiple Choice

How many triangles exist with the following angle measures?

100°, 100°, 50°

• No triangle exists with the given angle measures.

• More than one triangle exists with the given angle measures.

• Exactly one unique triangle exists with the given angle measures.

​No Triangle

These are the criteria to determine if it no triangle:​

• When the sum of the angles is NOT 180°​.

• When the sum of two smaller sides are less than the larger side.

​