Geometry (A): READ: Similarity Shortcuts Reading

Similarity by AA, SSS, and SAS

Learning Objectives

Use SSS and SAS to determine whether triangles are similar.
Apply SSS and SAS to solve problems about similar triangles.
Determine whether triangles are similar.
Understand AAA and AA rules for similar triangles.
Solve problems about similar triangles.

Introduction

You have an understanding of what similar polygons are and how to recognize them. Because triangles are the most basic building block on which other polygons can be based, we now focus specifically on similar triangles. We’ll find that there’s a surprisingly simple rule for triangles to be similar.

Angles in Similar Triangles

Tech Note - Geometry Software

Use your geometry software to experiment with triangles. Try this:

Set up two triangles, $\triangle ABC$ and $\triangle MNP$ .
Measure the angles of both triangles.
Move the vertices until the measures of the corresponding angles are the same in both triangles.
Compute the ratios of the lengths of the sides

$\frac{AB} {MN} \frac{BC} {NP} \frac{AC} {MP}.$

Repeat steps 1-4 with different triangles. Observe what happens in step 4 each time. Record your observations.

What did you see during your experiment? You might have noticed this: When you adjust triangles to make their angles congruent, you automatically make the sides proportional (the ratios in step 4 are the same). Once we have triangles with congruent angles and sides with proportional lengths, we know that the triangles are similar.

Conclusion: If the angles of a triangle are congruent to the corresponding angles of another triangle, then the triangles are similar. This is a handy rule for similar triangles—a rule based on just the angles of the triangles. We call this the AAA rule.

Caution: The AAA rule is a rule for triangles only. We already know that other pairs of polygons can have all corresponding angles congruent even though the polygons are not similar.

Example 1

The following is false statement: If the corresponding angles of two polygons are congruent, then the polygons are similar.

What is a counterexample to the false statement above?

Draw two polygons that are not similar, but which do have all corresponding angles congruent.

Rectangles such as the ones below make good examples.

Note: All rectangles have congruent (right) angles. However, we saw in an earlier lesson that rectangles can have different shapes—long and narrow vs. stubby and square-ish. In formal terms, these rectangles have congruent angles, but their side lengths are obviously not proportional. The rectangles are not similar. Congruent angles are not enough to ensure similarity for rectangles.

The AA Rule for Similar Triangles

Some artists and designers apply the principle that “less is more.” This idea has a place in geometry as well. Some geometry scholars feel that it is more satisfying to prove something with the least possible information. Similar triangles are a good example of this principle.

The AAA rule was developed for similar triangles earlier. Let’s take another look at this rule, and see if we can reduce it to “less” rather than “more.”

Suppose that triangles $\triangle ABC$ and $\triangle MNP$ have two pairs of congruent angles, say

$\angle{A} \cong \angle{M}$ and $\angle{B} \cong \angle{N}.$

But we know that if triangles have two pairs of congruent angles, then the third pair of angles are also congruent (by the Triangle Sum Theorem).

Summary: Less is more. The AAA rule for similar triangles reduces to the AA triangle similarity postulate.

The AA Triangle Similarity Postulate:

If two pairs of corresponding angles in two triangles are congruent, then the triangles are similar.

Example 2

Look at the diagram below.

A. Are the triangles similar? Explain your answer.

Yes. They both have congruent right angles, and they both have a $35^ \circ$ angle. The triangles are similar by AA.

B. Write a similarity statement for the triangles.

$\triangle ABC: \triangle TRS$ or equivalent

C. Name all pairs of congruent angles.

$\angle{A} \cong \angle{T}, \angle{B} \cong \angle{R}, \angle{C} \cong \angle{S}$

D. Write equations stating the proportional side lengths in the triangles.

$\frac{AB} {TR} = \frac{BC} {RS} = \frac{AC} {TS}$ or equivalent

Exploring SSS and SAS for Similar Triangles

We’ll use geometry software and compass-and-straightedge constructions to explore relationships among triangles based on proportional side lengths and congruent angles.

SSS for Similar Triangles

Tech Note - Geometry Software

Use your geometry software to explore triangles with proportional side lengths. Try this.

Set up two triangles, $\triangle ABC$ and $\triangle MNP$ , with each side length of $\triangle MNP$ being $k$ times the length of the corresponding side of $\triangle ABC$ .
Measure the angles of both triangles.
Record the results in a chart like the one below.

: Repeat steps 1-3 for each value of $k$ in the chart. Keep $\triangle ABC$ the same throughout the exploration.

Triangle Data
	$AB$	$BC$	$AC$	$m\angle{A}$	$m\angle{B}$	$m\angle{C}$

	$MN$	$NP$	$MP$	$m\angle{M}$	$m\angle{N}$	$m\angle{P}$
$k = 2$
$k = 5$
$k = 0.6$

First, you know that all three side lengths in the two triangles are proportional. That’s what it means for each side in $\triangle MNP$ to be $k$ times the length of the corresponding side in $\triangle ABC$ .

You probably notice what happens with the angle measures in $\triangle MNP$ . Each time you made a new triangle $MNP$ for the given value of $k$ , the measures of $\angle{M}, \angle{N},$ and $\angle{P}$ were approximately the same as the measures of $\angle{A}, \angle{B},$ and $\angle{C}$ . Like before when we experimented with the AA and AAA relationships, there is something “automatic” that happens. If the lengths of the sides of the triangles are proportional, that “automatically” makes the angles in the two triangles congruent too. Of course, once we know that the angles are congruent, we also know that the triangles are similar by AAA or AA.

Hands-On Activity

Materials: Ruler/straightedge, compass, protractor, graph or plain paper.

Directions: Work with a partner in this activity. Each partner will use tools to draw a triangle.

Each partner can work on a sheet of graph paper or on plain paper. Make drawings as accurate as possible. Note that it doesn’t matter what unit of length you use.

Partner 1: Draw a 6-8-10 triangle.
Partner 2: Draw a 9-12-15 triangle.
Partner 1: Measure the angles of your triangle.
Partner 2: Measure the angles of your triangle.
Partners 1 and 2: Compare your results.

What do you notice?

First, you know that all three side lengths in the two triangles are proportional. $\frac{6} {9} = \frac{8} {12} = \frac{10} {15} \left (= \frac{2} {3}\right)$
You also probably noticed that the angles in the two triangles are congruent. You might want to repeat the activity, drawing two triangles with proportional side lengths. You should find, again, that the angles in the triangles are automatically congruent.
Once we know that the angles are congruent, then we know that the triangles are similar by AAA or AA.

SSS for Similar Triangles

Conclusion: If the lengths of the sides of two triangles are proportional, then the triangles are similar. This is known as SSS for similar triangles.

SAS for Similar Triangles

SAS for Similar Triangles

If the lengths of two corresponding sides of two triangles are proportional and the included angles are congruent, then the triangles are similar. This is known as SAS for similar triangles.

Example 1

Cheryl made the diagram below to investigate similar triangles more.

She drew $\triangle ABC$ first, with $AB = 40, AC = 80,$ and $m\angle{A} = 30^\circ$ .

Then Cheryl did the following:

She drew $\overline{MN},$ and made $MN = 60$ .

Then she carefully drew $\overline{MP}$ , making $MP = 120$ and $m\angle{M} = 30^\circ$ .

At this point, Cheryl had drawn two segments ( $\overline{MN}$ and $\overline{MP}$ ) with lengths that are proportional to the lengths of the corresponding sides of $\triangle ABC$ , and she had made the included angle, $\angle{M}$ , congruent to the included angle $(\angle{A})$ in $\triangle ABC$ .

Then Cheryl measured angles. She found that:

$\angle{B} \cong \angle{N}$
$\angle{C} \cong \angle{P}$

What could Cheryl conclude? Here again we have automatic results. The other angles are automatically congruent, and the triangles are similar by AAA or AA. Cheryl’s work supports the SAS for Similar Triangles Postulate.

Similar Triangles Summary

We’ve explored similar triangles extensively in several lessons. Let’s summarize the conditions we’ve found that guarantee that two triangles are similar.

Two triangles are similar if and only if:

the angles in the triangles are congruent.
the lengths of corresponding sides in the polygons are proportional.

AAA: If the angles of a triangle are congruent to the corresponding angles of another triangle, then the triangles are similar.

AA: It two pairs of corresponding angles in two triangles are congruent, then the triangles are similar.

SSS for Similar Triangles: If the lengths of the sides of two triangles are proportional, then the triangles are similar.

SAS for Similar Triangles: If the lengths of two corresponding sides of two triangles are proportional and the included angles are congruent, then the triangles are similar.

Indirect Measurement

A traditional application of similar triangles is to measure lengths indirectly. The length to be measured would be some feature that was not easily accessible to a person. This length might be:

the width of a river
the height of a tall object
the distance across a lake, canyon, etc.

To measure indirectly, a person would set up a pair of similar triangles. The triangles would have three known side lengths and the unknown length. Once it is clear that the triangles are similar, the unknown length can be calculated using proportions.

Example 3

Flo wants to measure the height of a windmill. She held a $6\;\mathrm{foot}$ vertical pipe with its base touching the level ground, and the pipe’s shadow was $10\;\mathrm{feet}$ long. At the same time, the shadow of the tower was $85\;\mathrm{feet}$ long. How tall is the tower?

Draw a diagram.

Note: It is safe to assume that the sun’s rays hit the ground at the same angle. It is also proper to assume that the tower is vertical (perpendicular to the ground).

The diagram shows two similar right triangles. They are similar because each has a right angle, and the angle where the sun’s rays hit the ground is the same for both objects. We can write a proportion with only one unknown, $x$ , the height of the tower.

$\frac{x} {85} &= \frac{6} {10} \\ 10x &= 85 \cdot 6 \\ 10x &= 510 \\ x &= 51$

Thus, the tower is $51\;\mathrm{feet}$ tall.

Note: This is method considered indirect measurement because it would be difficult to directly measure the height of tall tower. Imagine how difficult it would be to hold a tape measure up to a $51-\;\mathrm{foot}-\mathrm{tall}$ tower.

Lesson Summary

The most basic way—because it requires the least input of information—to assure that triangles are similar is to show that they have two pairs of congruent angles. The AA postulate states this: If two triangles have two pairs of congruent angles, then the triangles are similar.

Once triangles are known to be similar, we can write many true proportions involving the lengths of their sides. These proportions were the basis for doing indirect measurement.

Points to Consider

Think about some right triangles for a minute. Suppose two right triangles both have an acute angle that measures $58^\circ$ . Then the ratio $\frac {\text{length of long leg}} {\text{length of short leg}}$ is the same in both triangles. In fact, this ratio, called “the tangent of $58^\circ$ ” is the same in any right triangle with a $58^\circ$ angle. As mentioned earlier, this is the reason for trigonometric functions of a given angle being constant, regardless of the specific triangle involved.

The following questions are for your own review. The answers are listed below for you to check your work and understanding.

Review Questions

Triangle 1 has sides with lengths $3\;\mathrm{inches} , 3\;\mathrm{inches}$ , and $4\;\mathrm{inches}$ .

Triangle 2 has sides with lengths $3\;\mathrm{feet} , 3\;\mathrm{feet}$ , and $4\;\mathrm{feet}$ .

Are Triangle 1 and Triangle 2 congruent? Explain your answer.
Are Triangle 1 and Triangle 2 similar? Explain your answer.
What is the scale factor from Triangle 1 to Triangle 2?
Why do we not study an ASA similarity postulate?

Use the chart below for exercises 4-4e.

Must $\triangle ABC$ and $\triangle MNP$ be similar?

	$m\angle{A}$	$m\angle{B}$	$m\angle{C}$	$AB$	$BC$	$AC$	$m\angle{M}$	$m\angle{N}$	$m\angle{P}$	$MN$	$NP$	$MP$
4a.				$3$	$5$	$6$				$6$	$3$	$5$
4b.	$50^\circ$	$40^\circ$					$100^\circ$	$80^\circ$
4c.				$8$	$4$	$10$				$10$	$6$	$12$
4d.	$63^\circ$			$100$		$150$		$63^\circ$		$20$	$30$
4e.			$100^\circ$		$24$	$15$			$110^\circ$	$32$	$20$
4f.				$30.0$	$20.0$	$32.0$				$22.5$	$15.0$	$24.0$

Hands-On Activity

Materials: Ruler/straightedge, compass, protractor, graph or plain paper.

Directions: Work with a partner in this activity. Each partner will use tools to draw a triangle.

Each partner can work on a sheet of graph paper or on plain paper. Make drawings as accurate as possible. Note that it doesn’t matter what unit of length you use.

Partner 1: Draw $\triangle ABC$ with $AB = 20$ , $m\angle{A}$ = $40^\circ$ , and $AC = 30$

Partner 2: Draw $\triangle MNP$ with $MN = 30$ , $\angle{M}$ = $40^\circ$ , and $MP = 45$ .

A. Are sides $\overline{AB}$ , $\overline{AC}$ , $\overline{MN}$ , and $\overline{MP}$ proportional? $\frac{20} {30} = \frac{30} {45} = \frac{2} {3}$

Partner 1: Measure the other angles of your triangle.

Partner 2: Measure the other angles of your triangle.

Partners 1 and 2: Compare your results.

B. Are the other angles of the two triangles (approximately) congruent?

C. Are the triangles similar? If they are, write a similarity statement and explain how you know that the triangles are similar. $\triangle ABC: \triangle MNP$ .

Review Answers

No. One is much larger than the other.
Yes, SSS. The side lengths are proportional.
$12$
There is no need. With the A and A parts of ASA we have triangles with two congruent angles. The triangles are similar by AA.
4a. Yes

4b. No

4c. No

4d. Yes

4e. No

4f. Yes
1. Yes
2. Yes
3. Yes. All three pairs of angles are congruent, so the triangles are similar by AAA or AA.

Last modified: Monday, June 28, 2010, 6:14 PM