    # Which similarity theorem supports when two angles of one triangle are congruent to two angles of another triangle?

Two triangles are similar if they have:

Show
• all their angles equal
• corresponding sides are in the same ratio

But 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:

## AA

AA stands for "angle, angle" and means that the triangles have two of their angles equal.

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

So AA could also be called AAA (because when two angles are equal, all three angles must be equal).

## SAS

SAS stands for "side, angle, side" and means that we have two triangles where:

• the ratio between two sides is the same as the ratio between another two sides
• and we we also know the included angles are equal.

If two triangles have two pairs of sides in the same ratio and the included angles are also equal, then the triangles are similar. In this example we can see that:

• one pair of sides is in the ratio of 21 : 14 = 3 : 2
• another pair of sides is in the ratio of 15 : 10 = 3 : 2
• there is a matching angle of 75° in between them

So there is enough information to tell us that the two triangles are similar.

### Using Trigonometry

We could also use Trigonometry to calculate the other two sides using the Law of Cosines:

In Triangle ABC:

• a2 = b2 + c2 - 2bc cos A
• a2 = 212 + 152 - 2 × 21 × 15 × Cos75°
• a2 = 441 + 225 - 630 × 0.2588...
• a2 = 666 - 163.055...
• a2 = 502.944...
• So a = √502.94 = 22.426...

In Triangle XYZ:

• x2 = y2 + z2 - 2yz cos X
• x2 = 142 + 102 - 2 × 14 × 10 × Cos75°
• x2 = 196 + 100 - 280 × 0.2588...
• x2 = 296 - 72.469...
• x2 = 223.530...
• So x = √223.530... = 14.950...

Now let us check the ratio of those two sides:

a : x = 22.426... : 14.950... = 3 : 2

the same ratio as before!

Note: we can also use the Law of Sines to show that the other two angles are equal.

## SSS

SSS stands for "side, side, side" and means that we have two triangles with all three pairs of corresponding sides in the same ratio.

If two triangles have three pairs of sides in the same ratio, then the triangles are similar. In this example, the ratios of sides are:

• a : x = 6 : 7.5 = 12 : 15 = 4 : 5
• b : y = 8 : 10 = 4 : 5
• c : z = 4 : 5

These ratios are all equal, so the two triangles are similar.

### Using Trigonometry

Using Trigonometry we can show that the two triangles have equal angles by using the Law of Cosines in each triangle:

In Triangle ABC:

• cos A = (b2 + c2 - a2)/2bc
• cos A = (82 + 42 - 62)/(2× 8 × 4)
• cos A = (64 + 16 - 36)/64
• cos A = 44/64
• cos A = 0.6875
• So Angle A = 46.6°

In Triangle XYZ:

• cos X = (y2 + z2 - x2)/2yz
• cos X = (102 + 52 - 7.52)/(2× 10 × 5)
• cos X = (100 + 25 - 56.25)/100
• cos X = 68.75/100
• cos X = 0.6875
• So Angle X = 46.6°

So angles A and X are equal!

Similarly we can show that angles B and Y are equal, and angles C and Z are equal.

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Triangles are similar if two pairs of sides are proportional and the included angles are congruent.

By definition, two triangles are similar if all their corresponding angles are congruent and their corresponding sides are proportional. It is not necessary to check all angles and sides in order to tell if two triangles are similar. In fact, if you know only that two pairs of sides are proportional and their included angles are congruent, that is enough information to know that the triangles are similar. This is called the SAS Similarity Theorem.

SAS Similarity Theorem: If two sides in one triangle are proportional to two sides in another triangle and the included angle in both are congruent, then the two triangles are similar. Figure $$\PageIndex{1}$$

If $$\dfrac{AB}{XY}=\dfrac{AC}{XZ}$$ and $$\angle A\cong \angle X$$, then $$\Delta ABC\sim \Delta XYZ$$.

What if you were given a pair of triangles, the lengths of two of their sides, and the measure of the angle between those two sides? How could you use this information to determine if the two triangles are similar?

Example $$\PageIndex{1}$$

Determine if the following triangles are similar. If so, write the similarity theorem and statement. Figure $$\PageIndex{2}$$

Solution

We can see that $$\angle B\cong \angle F$$ and these are both included angles. We just have to check that the sides around the angles are proportional.

\begin{aligned} \dfrac{AB}{DF} &=\dfrac{12}{8}=\dfrac{3}{2} \\ \dfrac{BC}{FE}&=\dfrac{24}{16}=\dfrac{3}{2} \end{aligned}

Since the ratios are the same $$\Delta ABC\sim \Delta DFE$$ by the SAS Similarity Theorem.

Example $$\PageIndex{2}$$

Determine if the following triangles are similar. If so, write the similarity theorem and statement. Figure $$\PageIndex{3}$$

Solution

The triangles are not similar because the angle is not the included angle for both triangles.

Example $$\PageIndex{3}$$

Are the two triangles similar? How do you know? Figure $$\PageIndex{4}$$

Solution

We know that $$\angle B\cong \angle Z$$ because they are both right angles and $$\dfrac{10}{15}=\dfrac{24}{36}$$. So, $$\dfrac{AB}{XZ}=\dfrac{BC}{ZY}$$ and $$\Delta ABC\sim \Delta XZY$$ by SAS.

Example $$\PageIndex{4}$$

Are there any similar triangles in the figure? How do you know? Figure $$\PageIndex{5}$$

Solution

$$\angle A$$ is shared by $$\Delta EAB$$ and $$\Delta DAC$$, so it is congruent to itself. Let’s see if $$\dfrac{AE}{AD}=\dfrac{AB}{AC}$$.

\begin{aligned} \dfrac{9}{9+3}&=\dfrac{12}{12+5} \\ \dfrac{9}{12}&=\dfrac{3}{4}\neq \dfrac{12}{17}\qquad \text{ The two triangles are not similar. }\end{aligned}

Example $$\PageIndex{5}$$

From Example 4, what should $$BC$$ equal for $$\Delta EAB\sim \Delta DAC$$?

Solution

The proportion we ended up with was $$\dfrac{9}{12}=\dfrac{3}{4}\neq \dfrac{12}{17}$$. AC needs to equal 16, so that $$\dfrac{12}{16}=dfrac{3}{4}$$. $$AC=AB+BC$$ and $$16=12+BC$$. $$BC$$ should equal 4.

Fill in the blanks.

1. If two sides in one triangle are _________________ to two sides in another and the ________________ angles are _________________, then the triangles are ______________.

Determine if the following triangles are similar. If so, write the similarity theorem and statement.

1. Figure $$\PageIndex{6}$$

Find the value of the missing variable(s) that makes the two triangles similar.

1. Figure $$\PageIndex{7}$$
2. Figure $$\PageIndex{8}$$
3. Figure $$\PageIndex{9}$$

Determine if the triangles are similar. If so, write the similarity theorem and statement.

1. $$\Delta ABC$$ is a right triangle with legs that measure 3 and 4. $$\Delta DEF$$ is a right triangle with legs that measure 6 and 8.
2. $$\Delta GHI$$ is a right triangle with a leg that measures 12 and a hypotenuse that measures 13. $$\Delta JKL$$ is a right triangle with legs that measure 1 and 2.
3. Figure $$\PageIndex{10}$$
4. Figure $$\PageIndex{11}$$
5. Figure $$\PageIndex{12}$$
6. Figure $$\PageIndex{13}$$
7. $$\overline{AC}=3$$

$$\overline{DF}=6$$ Figure $$\PageIndex{14}$$
1. Figure $$\PageIndex{15}$$
2. Figure $$\PageIndex{16}$$
3. Figure \(\PageIndex{17}\

Term Definition
AA Similarity Postulate If two angles in one triangle are congruent to two angles in another triangle, then the two triangles are similar.
Congruent Congruent figures are identical in size, shape and measure.
Dilation To reduce or enlarge a figure according to a scale factor is a dilation.
SAS SAS means side, angle, side, and refers to the fact that two sides and the included angle of a triangle are known.
SAS Similarity Theorem The SAS Similarity Theorem states that if two sides in one triangle are proportional to two sides in another triangle and the included angle in both are congruent, then the two triangles are similar.
Similarity Transformation A similarity transformation is one or more rigid transformations followed by a dilation.

Video: Congruent and Similar Triangles

Activities: SAS Similarity Discussion Questions

Study Aids: Polygon Similarity Study Guide

Practice: SAS Similarity

Real World: Triangle Similarity   