CK-12 Geometry
Page 27
they have the same number of sides
for each angle in either polygon there is a corresponding angle in the other polygon that is congruent
the lengths of all corresponding sides in the polygons are proportional
Reminder: Just as we did with congruent figures, we name similar polygons according to corresponding parts. The symbol is used to represent “is similar to.” Some people call this “the congruent sign without the equals part.”
Example 1
Suppose Based on this statement, which angles are congruent and which sides are proportional? Write true congruence statements and proportions.
and
Remember that there are many equivalent ways to write a proportion. The answer above is not the only set of true proportions you can create based on the given similarity statement. Can you think of others?
Example 2
Given:
What are the values of and in the diagram below?
Set up a proportion to solve for :
Now set up a proportion to solve for :
Finally, since is an angle, we are looking for
Example 3
is a rectangle with length and width .
is a rectangle with length and width .
A. Are corresponding angles in the rectangles congruent?
Yes. Since both are rectangles, all the angles in both are congruent right angles.
B. Are the lengths of the sides of the rectangles proportional?
No. The ratio of the lengths is . The ratio of the widths is . Therefore, the lengths of the sides are not proportional.
C. Are the rectangles similar?
No. Corresponding angles are congruent, but lengths of corresponding sides are not proportional.
Example 4
Prove that all squares are similar.
Our proof is a “paragraph” proof in bullet form, rather than a two-column proof:
Given two squares.
All the angles of both squares are right angles, so all angles of both squares are congruent—and this includes corresponding angles.
Let the length of each side of one square be , and the length of each side of the other square be . Then the ratio of the length of any side of the first square to the length of any side of the second square is . So the lengths of the sides are proportional.
The squares satisfy the definition of similar polygons: congruent angles and proportional side lengths - so they are similar
Scale Factors
If two polygons are similar, we know that the lengths of corresponding sides are proportional. If is the length of a side in one polygon, and is the length of the corresponding side in the other polygon, then the ratio is called the scale factor relating the first polygon to the second. Another way to say this is:
The length of every side of the first polygon is times the length of the corresponding side of the other polygon.
Example 5
Look at the diagram below, where and are similar rectangles.
A. What is the scale factor?
Since , then and are corresponding sides. Since is a rectangle, you know that
The scale factor is the ratio of the lengths of any two corresponding sides.
So the scale factor (relating to ) is . We now know that the length of each side of is the length of the corresponding side in .
Comment: We can turn this relationship around “backwards” and talk about the scale factor relating to . This scale factor is just which is the reciprocal of the scale factor relating to
B. What is the ratio of the perimeters of the rectangles?
is a by rectangle. Its perimeter is .
is a by rectangle. Its perimeter is .
The ratio of the perimeters of to is .
Comment: You see from this example that the ratio of the perimeters of the rectangles is the same as the scale factor. This relationship for the perimeters holds true in general for any similar polygons.
Ratio of Perimeters of Similar Polygons
Let’s prove the theorem that was suggested by example 5.
Ratio of the Perimeters of Similar Polygons:
If and are two similar polygons, each with sides and the scale factor of the polygons is , then the ratio of the perimeters of the polygons is .
Given: and are two similar polygons, each with sides
The scale factor of the polygons is
Prove: The ratio of the perimeters of the polygons is
Statement Reason
1. and are similar polygons, each with sides 1. Given
2. The scale factor of the polygons is 2. Given
3. Let and be the lengths of corresponding sides of and 3. Given (polygons have sides each)
4. 4. Definition of scale factor
5. Perimeter of 5. Definition of perimeter
6. 6. Substitution
7. 7. Distributive Property
8. , the perimeter of 8. Definition of perimeter
Comment: The ratio of the perimeters of any two similar polygons is the same as the scale factor. In fact, the ratio of any two corresponding linear measures in similar figures is the same as the scale factor. This applies to corresponding sides, perimeters, diagonals, medians, midsegments, altitudes, etc.
As we’ll see in an upcoming lesson, this is definitely not true for the areas of similar polygons. The ratio of the areas of similar polygons (that are not congruent) is not the same as the scale factor.
Example 6
. The perimeter of is
What is the perimeter of ?
The scale factor relating to is . According to the Ratio of the Perimeter's Theorem, the perimeter of is of the perimeter of . Thus, the perimeter of is .
Lesson Summary
Similar has a very specific meaning in geometry. Polygons are similar if and only if the lengths of their sides are proportional and corresponding angles are congruent. This is same shape translated into geometric terms.
The ratio of the lengths of corresponding sides in similar polygons is called the scale factor. Lengths of other corresponding linear measures, such as perimeter, diagonals, etc. have the same scale factor.
Points to Consider
Scale factors show the relationship between corresponding linear measures in similar polygons. The story is not quite that simple for the relationship between the areas or volumes of similar polygons and polyhedra (three-dimensional figures). We’ll study these relationships in future lessons.
Similar triangles are the basis for the study of trigonometry. The fact that the ratios of the lengths of corresponding sides in right triangles depends only on the measure of an angle, not on the size of the triangle, makes trigonometric functions the property of an angle, as you will study in Chapter 8.
Review Questions
True or false?
All equilateral triangles are similar.
All isosceles triangles are similar.
All rectangles are similar.
All rhombuses are similar.
All squares are similar.
All congruent polygons are similar.
All similar polygons are congruent.
Use the following diagram for exercises 8-11.
Given that rectangle rectangle .
What is the value of each expression?
Given that , what is the scale factor of the triangles?
Use the diagram below for exercises 13-16.
Given:
What is the perimeter of
What is the perimeter of
What is the ratio of the perimeter of to the perimeter of
Prove: . [Write a flow proof.]
is the midpoint of and is the midpoint of in . Name a pair of parallel segments.
Name two pairs of congruent angles.
Write a statement of similarity of two triangles.
If the perimeter of the larger triangle in is , what is the perimeter of the smaller triangle?
If the area of is , what is the area of quadrilateral
Review Answers
True
False
False
False
True
True
False
or
or equivalent
, so the sides are all proportional.
or equivalent
Similarity by AA
Learning Objectives
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, and .
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
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 and have two pairs of congruent angles, say
and
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 angle. The triangles are similar by AA.
B. Write a similarity statement for the triangles.
or equivalent
C. Name all pairs of congruent angles.
D. Write equations stating the proportional side lengths in the triangles.
or equivalent
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 vertical pipe with its base touching the level ground, and the pipe’s shadow was long. At the same time, the shadow of the tower was 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, , the height of the tower.
Thus, the tower is 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 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 . Then the ratio is the same in both triangles. In fact, this ratio, called “the tangent of ” is the same in any right triangle with a angle. As mentioned earlier, this is the reason for trigonometric functions of a given angle being constant, regardless of the specific triangle involved.
Review Questions
Use the diagram below for exercises 1-5.
Given that :
Name two similar triangles.
Explain how you know that the triangles you named in exercise 1 are similar.
Write a true proportion.
Name two triangles that might not be similar.
If and , what is the length of ?
Given that , , and in the diagram below
Write an expression for in terms of .
Prove the following theorem:
If an acute angle of a right triangle is congruent to an acute angle of another right triangle, then the triangles are congruent.
Write a flow proof.
Use the following diagram for exercises 8-12.
In a geometry reality competition, the teams must estimate the width of the river shown in the diagram. Here’s what they did.
Anna, Bela, and Carlos stayed on the upper bank of the river.
Darryl and Eva paddled across to the lower bank of the river.
Carlos placed a marker at .
Darryl placed a marker directly across from Carlos at .
Bela walked back from the bank in a line with the markers at and and placed a marker at .
&
nbsp; Anna walked on a path perpendicular to and placed a marker at .
Eva moved along the lower bank until she was lined up with and , and placed a marker at .
, and are on land, so they can be measured easily. was measured to be .
Name two similar triangles.
Explain how you know that the triangles in exercise 8 are similar.
Write a proportion in which the only unknown measure is .
How wide is the river?
Discuss whether or not the triangles used to answer exercises 8-11 are good models for a river and its banks.
Review Answers
or equivalent
The triangles have two pairs of congruent alternate interior angles and one pair of congruent vertical angles. They are similar by AAA and AA.
Any proportion obtained from
or for example
One acute angle in each triangle is congruent to an acute angle in the other triangle. Also, since they are right triangles, both triangles have a right angle, and these right angles are congruent. The triangles are congruent by AA.
and are congruent right angles. , so and are congruent alternate interior angles. The triangles are similar by AA.
The river is approximately
wide.
This seems to be a good model. The banks are roughly straight enough to be lines. The banks appear to be nearly parallel. If we can accept that parallel straight lines adequately represent the river banks, then the model is a good one.