Lesson 3 Similar Triangles & Other Figures Solidify Understanding

Learning Focus

Determine criteria for triangle similarity.

What is the difference between the common usage of the word similar (e.g., rectangles are more similar to each other than rectangles and triangles are), and the mathematical conventions for the word? What does it mean for two polygons to be similar?

How can I prove (or disprove) that two triangles are similar?

Open Up the Math: Launch, Explore, Discuss

Two figures are said to be congruent if the second can be obtained from the first by a sequence of rotations, reflections, and translations. Previously we found that we only needed three pieces of information to guarantee that two triangles were congruent: SSS, ASA, or SAS.

What about AAA? Are two triangles congruent if all three pairs of corresponding angles are congruent?

In this task we will consider what is true about triangles that are similar, but not congruent.

Definition of Similarity: Two figures are similar if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations.

Mason and Mia are testing out conjectures about similar polygons. Here is a list of their conjectures.

Conjecture 1: All rectangles are similar.

Conjecture 2: All equilateral triangles are similar.

Conjecture 3: All isosceles triangles are similar.

Conjecture 4: All rhombuses are similar.

Conjecture 5: All squares are similar.

1.

Which of these conjectures do you think are true? Why?

Mason is explaining to Mia why he thinks conjecture 1 is true using the diagram.

“All rectangles have four right angles. I can translate and rotate rectangle $A B C D$ until vertex $A$ coincides with vertex $Q$ in rectangle $Q R S T$ . Since $A$ and $Q$ are both right angles, side $A B$ will lie on top of side $Q R$ , and side $A D$ will lie on top of side $Q T$ . I can then dilate rectangle $A B C D$ with point $A$ as the center of dilation, until points $B$ , $C$ , and $D$ coincide with points $R$ , $S$ , and $T$ .”

2.

Does Mason’s explanation convince you that rectangle $A B C D$ is similar to rectangle $Q R S T$ based on the definition of similarity given above? Does his explanation convince you that all rectangles are similar? Why or why not?

Mia is explaining to Mason why she thinks conjecture 2 is true using the diagram.

“All equilateral triangles have three $60 °$ angles. I can translate and rotate $△ A B C$ until vertex $A$ coincides with vertex $Q$ on $△ Q R S$ . Since $A$ and $Q$ are both $60 °$ angles, side $A B$ will lie on top of side $Q R$ , and side $A C$ will lie on top of side $Q S$ . I can then dilate $△ A B C$ with point $A$ as the center of dilation, until points $B$ and $C$ coincide with points $R$ and $S$ .”

3.

Does Mia’s explanation convince you that $△ A B C$ is similar to $△ Q R S$ based on the definition of similarity given above? Does her explanation convince you that all equilateral triangles are similar? Why or why not?

4.

For each of the other three conjectures, write an argument like Mason’s and Mia’s to convince someone that the conjecture is true, or explain why you think it is not always true.

a.

Conjecture 3: All isosceles triangles are similar.

b.

Conjecture 4: All rhombuses are similar.

c.

Conjecture 5: All squares are similar.

Mason has another conjecture: Scaled drawings of polygons are similar figures.

Here is what Mason knows about scaled drawings from previous work:

Corresponding angles of scaled drawings are congruent.
Corresponding sides of scaled drawings are proportional.

Mia proposes they try to justify this conjecture for scaled drawings of triangles. She has suggested the following diagram.

5.

Explain how you can use the definition of similar figures—two figures are similar if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations—to show the following scaled drawings of triangles are similar.

Given: Corresponding angles of $△ A B C$ and $△ R S T$ are congruent, and corresponding sides are proportional by a scale factor $k$ .

6.

How can you extend Mia and Mason’s justification that scaled drawings of triangles are similar to show that scaled drawings of quadrilaterals are similar figures?

Given: Corresponding angles of quadrilateral $A B C D$ and quadrilateral $R S T U$ are congruent, and corresponding sides are proportional by a scale factor $k$ .

While the definition of similarity given at the beginning of the task works for all similar figures, including figures with nonlinear boundaries, an alternative definition of similarity can be given for polygons: Two polygons are similar if all corresponding angles are congruent and all corresponding pairs of sides are proportional.

7.

How does this definition help you find the error in Mason’s thinking about conjecture 1?

8.

How does this definition help confirm Mia’s thinking about conjecture 2?

9.

How might this definition help you think about the other three conjectures?

a.

Conjecture 3: All isosceles triangles are similar.

b.

Conjecture 4: All rhombuses are similar.

c.

Conjecture 5: All squares are similar.

Pause and Reflect

AAA, SAS, and SSS Similarity

From our work with rectangles, it is obvious that knowing that all rectangles have four right angles (an example of AAAA for quadrilaterals) is not enough to claim that all rectangles are similar. What about triangles? In general, are two triangles similar if all three pairs of corresponding angles are congruent?

10.

Explain why the following conjecture is true.

Conjecture: Two triangles are similar if their corresponding angles are congruent.

Use the diagram to support your reasoning. Remember to start by marking what you are given to be true (AAA) in the diagram.

11.

Mia thinks the following conjecture is true. She calls it “AA Similarity for Triangles.” What do you think? Is it true? Why?

Conjecture: Two triangles are similar if they have two pairs of corresponding congruent angles.

12.

Using the diagram given in problem 10, how might you modify your proof that $△ A B C \sim △ D E F$ if you are given the following information about the two triangles:

a.

$∠ A ≅ ∠ D$ , $D E = k \cdot A B, D F = k \cdot A C$ ; that is, $\frac{D E}{A B} = \frac{D F}{A C}$

b.

$D E = k \cdot A B, D F = k \cdot A C$ and $E F = k \cdot B C$ ; that is, $\frac{D E}{A B} = \frac{D F}{A C} = \frac{E F}{B C}$

Ready for More?

Compare and contrast the ways it can be proven that two triangles are congruent to the ways it can be proven that two triangles are similar. Are there other triangle similarity theorems that can be stated and proved?

Takeaways

To prove that two polygons are similar, we need to show that all corresponding angles are congruent, and that all corresponding pairs of sides are proportional. However, if the polygons are triangles, we can show using less information by applying one of the following theorems:

Reflecting on the work with triangle similarity theorems, I learned or was reminded of the following insights about the proof process:

Adding Notation, Vocabulary, and Conventions

In the English language, the word similar means:

In a mathematical context, the word similar means:

In an alternative definition, similarity of polygons means:

Vocabulary

Lesson Summary

In this lesson, we examined what it means to say that two figures are similar geometrically, and we examined conditions under which two triangles will be similar. We wrote and justified several theorems for triangle similarity criteria.

Retrieval

Solve each proportion. Show your work and check your solution.

1.

$\frac{2}{5} = \frac{x}{20}$

2.

$\frac{3}{7} = \frac{18}{b}$

3.

$\frac{4}{x + 2} = \frac{1}{3}$

4.

Create three equivalent ratios for the similar polygons.