CK-12 Geometry
Page 28
Similarity by 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.
Introduction
You have been using the AA postulate to work with similar triangles. AA is easy to state and to apply. In addition, there are other similarity postulates that should remind you of some of the congruence postulates. These are the SSS and SAS similarity postulates. These postulates will give us more tools for recognizing similar triangles and solving problems involving them.
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, and , with each side length of being times the length of the corresponding side of .
Measure the angles of both triangles.
Record the results in a chart like the one below.
Repeat steps 1-3 for each value of in the chart. Keep the same throughout the exploration.
Triangle Data
First, you know that all three side lengths in the two triangles are proportional. That’s what it means for each side in to be times the length of the corresponding side in .
You probably notice what happens with the angle measures in . Each time you made a new triangle for the given value of , the measures of and were approximately the same as the measures of and . 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.
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 first, with and .
Then Cheryl did the following:
She drew and made .
Then she carefully drew , making and .
At this point, Cheryl had drawn two segments ( and ) with lengths that are proportional to the lengths of the corresponding sides of , and she had made the included angle, , congruent to the included angle in .
Then Cheryl measured angles. She found that:
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.
Points to Consider
Have you ever made a model rocket? Have you seen a scale drawing? Do you know people who use blueprints? Do you enlarge pictures on your computer or shrink them? These are all examples of similar two-dimensional or three-dimensional objects.
Review Questions
Triangle 1 has sides with lengths , and .
Triangle 2 has sides with lengths , and .
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 and be similar?
4a.
4b.
4c.
4d.
4e.
4f.
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 with , = , and
Partner 2: Draw with , = , and .
A. Are sides , , , and proportional?
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. .
Review Answers
No. One is much larger than the other.
Yes, SSS. The side lengths are proportional.
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
Yes
Yes
Yes. All three pairs of angles are congruent, so the triangles are similar by AAA or AA.
Proportionality Relationships
Learning Objectives
Identify proportional segments when two sides of a triangle are cut by a segment parallel to the third side.
Divide a segment into any given number of congruent parts.
Introduction
We’ll wind up our study of similar triangles in this
section. We will also extend some basic facts about similar triangles to dividing segments.
Dividing Sides of Triangles Proportionally
Think about a midsegment of a triangle. A midsegment is parallel to one side of a triangle, and that it divides the other two sides into congruent halves (because the midsegment connects the midpoints of those two sides). So the midsegment divides those two sides proportionally.
Example 1
Explain the meaning of "the midsegment divides the sides of a triangle proportionally."
Suppose each half of one side of a triangle is long, and each half of the other side is long.
One side is divided in the ratio , the other side in the ratio Both of these ratios are equivalent to and to each other.
We see that a midsegment divides two sides of a triangle proportionally. But what about some other segment?
Tech Note - Geometry Software
Use your geometry software to explore triangles where a line parallel to one side intersects the other two sides. Try this:
1. Set up .
2. Draw a line that is parallel to and that intersects both of the other sides of .
3. Label the intersection point on as ; label the intersection point on as .
Your triangle will look something like this.
parallel to
4. Measure lengths and calculate the following ratios.
______ and ______
5. Compare your results with those of other students.
Different students can start with different triangles. They can draw different lines parallel to . But in each case the two ratios, and , are approximately the same. This is another way to say that the two sides of the triangle are divided proportionally. We can prove this result as a theorem.
Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides into proportional segments.
Proof.
Given: with
Prove:
Statement Reason
1. 1. Given
2. 2. Corresponding angles are congruent
3. 3. AA Similarity Postulate
4. 4. Segment addition postulate
5. 5. Corresponding side lengths in similar triangles are proportional
6. 6. Substitution
7.
7. Algebra
8. 8. Substitution
9. 9. Addition property of equality
Can you see why we wrote the proportion this way, rather than as , which is also a true proportion?
It’s because , but there is no similar way to simplify .
Note: The converse of this theorem is also true. If a line divides two sides of a triangle into proportional segments, then the line is parallel to the third side of the triangle.
Example 2
In the diagram below, .
What is an expression in terms of for the length of ?
According to the Triangle Proportionality Theorem,
There are some very interesting corollaries to the Triangle Proportionality Theorem. One could be called the Lined Notebook Paper Corollary!
Parallel Lines and Transversals
Example 3
Look at the diagram below. We can make a corollary to the previous theorem.
are labels for lines
are lengths of segments
are parallel but not equally spaced
We’re given that lines , and are parallel. We can see that the parallel lines cut lines and (transversals). A corollary to the Triangle Proportionality Theorem states that the segment lengths on one transversal are proportional to the segment lengths on the other transversal.
Conclusion: and
Example 4
The corollary in example 3 can be broadened to any number of parallel lines that cut any number of transversals. When this happens, all corresponding segments of the transversals are proportional!
The diagram below shows several parallel lines, and , that cut several transversals , and .
lines are all parallel.
Now we have lots of proportional segments.
For example:
, and many more.
This corollary extends to more parallel lines cutting more transversals.
Lined Notebook Paper Corollary
Think about a sheet of lined notebook paper. A sheet has numerous equally spaced horizontal parallel segments; these are the lines a person can write on. And there is a vertical segment running down the left side of the sheet. This is the segment setting the margin, so you don’t write all the way to the edge of the paper.
Now suppose we draw a slanted segment on the sheet of lined paper.
Because the vertical margin segment is divided into congruent parts, then the slanted segment is also divided into congruent segments. This is the Lined Notebook Paper Corollary.
What we’ve done here is to divide the slanted segment into five congruent parts. By placing the slanted segment differently we could divide it into any given number of congruent parts.
History Note
In ancient times, mathematicians were interested in bisecting and trisecting angles and segments. Bisection was no problem. They were able to use basic geometry to bisect angles and segments.
But what about trisection dividing an angle or segment into exactly three congruent parts? This was a real challenge! In fact, ancient Greek geometers proved that an angle cannot be trisected using only compass and straightedge.
With the Lined Notebook Paper Corollary, though, we have an easy way to trisect a given segment.
Example 5
Trisect the segment below.
Draw equally spaced horizontal lines like lined notebook paper. Then place the segment onto the horizontal lines so that its endpoints are on two horizontal lines that are three spaces apart.
slanted segment is same length as segment above picture
endpoints are on the horizontal segments shown
slanted segment is divided into three congruent parts
The horizontal lines now trisect the segment. We could use the same method to divide a segment into any required number of congruent smaller segments.
Lesson Summary
In this lesson we began with the basic facts about similar triangles the definition and the SSS and SAS properties. Then we built on those to create numerous proportional relationships. First we examined proportional sides in triangles, then we extended that concept to dividing segments into proportional parts. We finalized those ideas with a notebook paper property that gave us a way to divide a segment into any given number of equal parts.
Points to Consider
Earlier in this book you studied congruence transformations. These are transformations in which the image is congruent to the original figure. You found that translations (slides), rotations (turns), and reflections (flips) are all congruence transformations. In the next lesson we’ll study similarity transformations transformations in which the image is similar to the original figure. We’ll focus on dilations. These are figures that we zoom in on, or zoom out on. The idea is very similar to blowing up or shrinking a photo before printing it.
Review Questions
Use the diagram below for exercises 1-5.
Given that
Name similar triangles.
Complete the proportion.
Lines , , and are parallel.
What is the value of ?
Lines , , and are parallel, and .
What is the value of ?
What is the value of ?
Explain how to divide a segment into seven congruent segments using the Lined Notebook Paper Corollary.
Review Answers
or equivalent
Place the original segment so that one endpoint is on the top horizontal line. Slant the segment so that the other endpoint is on the seventh horizontal line below the top line. These eight horizontal lines divide the original segment into seven congruent smaller segments.
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sp; Similarity Transformations
Learning Objectives
Draw a dilation of a given figure.
Plot the image of a point when given the center of dilation and scale factor.
Recognize the significance of the scale factor of a dilation.
Introduction
Earlier you studied one group of transformations that “preserve” length. This means that the image of a segment is a congruent segment. These congruence transformations are translations, reflections, and rotations.
In this lesson, you’ll study one more kind of transformation, the dilation. Dilations do not preserve length, meaning the image of a segment can be a segment that is not congruent to the original. You’ll see that the image of a figure in a dilation is a similar, not necessarily congruent, figure.
Dilations
A dilation is like a “blow-up” of a photo to change its size. A dilation may make a figure larger, or smaller, but the same shape as the original. In other words, as you’ll see, a dilation gives us a figure similar to the original.