Angle-Angle Similarity Theorem & Side-Side-Side Similarity Theorem Explained

Ratio	Expression	Simplification
$\frac{A C}{A B}$	$\frac{4 5 9 + 1 2 7 5}{4 5 9} = \frac{1 7 3 4}{4 5 9}$	$\frac{3 4}{9}$
$\frac{A E}{A D}$	$\frac{4 0 5 + 1 1 2 5}{4 0 5} = \frac{1 5 3 0}{4 0 5}$	$\frac{3 4}{9}$
$\frac{E C}{D B}$	$\frac{1 3 6 0}{3 6 0}$	$\frac{3 4}{9}$

Ratio

Expression

Simplification

\frac{A C}{A B}

\frac{4 5 9 + 1 2 7 5}{4 5 9} = \frac{1 7 3 4}{4 5 9}

\frac{3 4}{9}

\frac{A E}{A D}

\frac{4 0 5 + 1 1 2 5}{4 0 5} = \frac{1 5 3 0}{4 0 5}

\frac{3 4}{9}

\frac{E C}{D B}

\frac{1 3 6 0}{3 6 0}

\frac{3 4}{9}

Ratio	Expression	Simplified Form
$\frac{A B}{A E}$	$\frac{7}{5 + 9} = \frac{7}{1 4}$	$\frac{1}{2}$
$\frac{A C}{A D}$	$\frac{5}{7 + 3} = \frac{5}{1 0}$	$\frac{1}{2}$

Ratio

Expression

Simplified Form

\frac{A B}{A E}

\frac{7}{5 + 9} = \frac{7}{1 4}

\frac{1}{2}

\frac{A C}{A D}

\frac{5}{7 + 3} = \frac{5}{1 0}

\frac{1}{2}

Are the following pair of triangles similar? If yes, state by which triangle similarity theorem?

a pair of triangles which are not similar

Examining the diagram, we see that we have two right triangles. Additionally, we can identify one pair of acute angles that are also congruent since the triangles share an angle.

Therefore, by the Angle-Angle Similarity Theorem, these triangles are similar.

For similar figures, the ratio of two corresponding sides is equal to the ratio of two other corresponding sides. Notice that in similar figures, sides with the same relative lengths are corresponding. With this information, we can identify what would the corresponding sides be if the triangles are similar.

Let's write the ratios of corresponding sides. 6/15? =8/20? =11/33 If the triangles are similar, all of the ratios should be equal. Let's check if the ratios give the same quotient. 0.4= 0.4≠ 0.33... * Since the ratio of the longest sides is not equal to the ratio of the other two sides, these cannot be similar triangles.

Two triangles are similar if at least two pairs of angles are congruent. Since none of the given angles are congruent, we know that the triangles cannot be similar.

Which of the triangles $B,$ $C,$ and $D$ are similar to $A ?$

Examining the three triangles, we notice that for A and B we have two pairs of congruent angles. This means we can claim similarity by the Angle-Angle Similarity Theorem.

In △ C, we only have information about the side lengths. Since we also know the length of every side in △ A, we can investigate if the triangles are similar by the Side-Side-Side Similarity Theorem. In similar figures, sides with the same relative lengths are corresponding. With this information, we can write and evaluate the ratios of corresponding sides. 2/2.6? =3/4? =3.9/5.4 [0.9em] ⇓ [0.2em] 0.76923...≠0.75≠0.7222 ... * Since the ratios are not equal, we know that the triangles are not similar.

As for D, we have one pair of congruent angles. Since we have also been given the lengths of the sides that have an included angle, we can investigate if D and A are similar by the Side-Angle-Side Similarity Theorem. Again, since sides with the same relative length are corresponding, we can set up ratios to evaluate them. 4.68/3.9? =2.4/2 [0.9em] ⇓ [0.2em] 1.2 =1.2... ✓ The sides have matching ratios. Therefore, D is similar to A.

Consider the following triangles $A B C$ and $A D E .$

Two similar triangles where one is inside the other

Are the triangles similar?

Find the length of the side labeled

x .

Examining the diagram, we see that △ ABC and △ ADE have a pair of congruent angles. We also see that they share an angle at A.

Since the triangles have two pairs of congruent angles, we can claim similarity by the Angle-Angle Similarity Theorem.

To find the length of DE, we can use the similarity between the triangles. Let's separate the triangles to help us identify corresponding sides.

The ratio of corresponding sides in similar shapes is always the same. Let's use this to write an equation. x/10=24/16 Let's solve this equation for x.

As we can see, DE is 15 units long.

Consider $△ A B C .$

Is the following triangle similar to $△ A B C ?$

Any equilateral triangle is also equiangular, meaning all of its angles are congruent. Since △ DEF has three angles of 60^(∘), we know that it is equiangular and, therefore, equilateral. Since △ ABC has three congruent sides, it is equilateral and, therefore, equiangular.

Now we can claim similarity by either Side-Side-Side Similarity Theorem or by the Angle-Angle Similarity Theorem.

Notice that all sides have a length of x. Like the previous section, this also fits the description of an equilateral triangle. Therefore, △ GHI must also be similar to △ ABC.

All equilateral triangles are similar because all of these triangles have the same three pairs of angles. Therefore, as long as we know that two triangles are equilateral or equiangular, we know they are similar.

The Conditions for Triangle Similarity

Catch-Up and Review

Using Similarity of Triangles to Solve Problems

Investigating Triangles With Two Pairs of Congruent Angles

Angle-Angle Similarity Theorem

Proof

Solving Problems Using Angle-Angle-Similarity Theorem

Hint

Solution

Practice Solving Problems Using Similar Triangles

Side-Side-Side Similarity Theorem

Proof

Solving Problems Using Similar Triangles

Answer

Hint

Solution

Step 1

Step 2

Step 3

Answering the Question

Side-Angle-Side Similarity Theorem

Proof

Proving Similarity Between Triangles Given Sides

Hint

Solution

Applying Triangle Similarity Theorems to Solve Problems

Hint

Solution

Extra

The Conditions for Triangle Similarity

Recommended exercises

	11 Theory slides
	11 Exercises - Grade E - A
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