Similar Triangles

Similar Triangles

Get out your notebook! It is time to take some notes!

How to Find if Triangles are Similar

Two triangles are similar if they have:

1) All their angles equal

2) Corresponding sides are in the same ratio

But, just like with congruent triangles, we don't need to know all three sides and all three angles, two or three out of the six is usually enough.

There are three ways to find if two triangles are similar: 

AA, SAS and SSS
Just like with congruent triangles A stands for Angle and S stands for Side
The BIG difference is that SIDES ARE NOT CONGRUENT BUT PROPORTIONAL.
Proportions are two equal ratios, comparing two numbers.

 $\frac{a}{b}=\frac{c}{d}$  read: a is to b as c is to d.  

AA - Angle Angle

• If two triangles have two of their angles equal, the triangles are similar.

• If two of their angles are equal, then the third angle must also be equal, because of the Third Angle Theorem.

• So AA could also be called AAA but lets keep it simple! :)

SAS Similarity - "side, angle, side"

• The ratio between two sides is the same as the ratio between another two sides, the sides are proportional

• The included angles are equal

• Does this sound familiar? SAS Congruency theorem

• If two triangles have two pairs of sides in the same ratio and the included angles are also equal, then the triangles are similar.

Another way to look at it:

 $\frac{21}{14}=\frac{15}{10}$  if we reduce them both we get  $\frac{3}{2}$  
Another way to check (or solve) if they are proportional is to cross multiply.
 $21\ \times10\ =\ 15\ \times14$  they both equal 210.

SSS Similarity - "side, side, side"

• If two triangles have three pairs of sides in the same ratio, then the triangles are similar

• Does this sound familiar? SSS triangle congruency.

• Remember a ratio is a comparison of two numbers.

• A ratio can be written like 4:5, or as a fraction  $\frac{4}{5}$ or simply 4 to 5

Another way to look at it

The short sides had a ratio of  $\frac{4}{5}$  

The  medium sides had a ratio of  $\frac{6}{7.5}=\frac{12}{15}=\frac{4}{5}$  when reduced

The long sides had a ratio of  $\frac{8}{10}=\frac{4}{5}$  when reduced
They all have a common ratio.

Maybe you remember solving for proportions

•  $\frac{3}{8}=\frac{x}{12}$  

• You cross multiply and solve for x

• 3 x 12 = 8x

• 36 = 8x

•  $x=\frac{36}{8}=4.5$  

Let's review and try some!

Are these triangles similar? If so, is it by AA, all the angles are equal.

Or is it by SAS, where 2 sides are proportional and the included angle is equal.

Or is it by SSS, where all the sides are proportional.

Maybe they are NOT similar.

Multiple Choice

If two triangles are similar, the corresponding sides are ______________.

• equal

• congruent

• proportional

• none of these

Multiple Choice

If two triangles are similar their angles are______.

• proportional

• congruent

• supplementary

• complementary

Multiple Choice

Determine whether the triangles are similar by AA~, SSS~, SAS~, or not similar.

• SSS

• AA

• SAS

• not similar

Multiple Select

Are the triangles similar?

• Yes

• No

Multiple Choice

State the postulate that proves these triangles are similar.

• SSS

• AA

• SAS

• Not similar

Multiple Choice

State the postulate that proves these triangles are similar.

• SSS
• SAS
• AA
• Not similar

Multiple Choice

Are the triangles similar?

• Yes
• No

Multiple Choice

State the postulate that proves these triangles are similar.

• SSS
• AA
• SAS
• Not similar

Multiple Choice

Which is NOT true about similar triangles.

• The angles in the triangles are congruent to each other.
• The sides are proportional to each other.
• The angles in each triangle add up to 180^o.
• The triangles must have at least one side that is the same length.

When triangles ARE similar, sometimes we want to find the missing side.

To do this we set up a proportion and solve.

Multiple Choice

 $\frac{AB}{AM}=\frac{?}{AD}$  Complete each proportion.

• BC

• AC

• MD

• MC

Multiple Choice

 $\frac{CB}{DM}=\frac{AC}{?}$  Complete each proportion.

• BM

• MA

• BC

• AD

Multiple Choice

If the two triangles are similar, what is the measurement of x?

• 2 ft

• 3 ft

• 4 ft

• 6 ft

When Two Triangles Are Similar

We write

  $\Delta ABC\sim\Delta DEF$  

That means that the angles are congruent

 $\angle A\cong\angle D,\ \angle B\cong\angle E\ and\ \angle C\cong\angle F$  
it also means the sides are proportional
 $\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$  

Multiple Choice

Remember:  $\frac{JK}{GK}=\frac{KL}{KH}=\frac{JL}{GH}$  

• 22.5

• 10

• 9

• 7.5

Multiple Choice

Find side length x. Set up your proportion and solve

• 30

• 25

• 15

• 40

Multiple Choice

Find side length x.

• 16

• 18

• 2.4

• 15

SCALE FACTOR

The scale factor, usually called k, is basically the comparison of the sides.

Multiple Choice

What is the scale factor from B to A? (B:A in simplest form)

• 1:3

• 3

• 2:3

• 3:2

Poll

How do you feel about Similar Triangles?

• It totally makes sense

• I feel pretty good about it

• It's a lot to take in but with practice I can do it.

• I don't quite get it yet. I'll probably do this again or come to office hours for more help.