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Analytical Geometry
Similarity, Congruence and Proofs
Similarity, Congruence and Proofs
How tall is the Eiffel Tower? How would you measure its height? Did someone actually go up to it
with a tape measure? Did the builders measure it piece by piece when it was being constructed? We
can actually use geometry and algebra to measure the height of anything. You can measure the
height of the Eiffel Tower just by standing next to it! All you would need are a few basic
measurements. In this module we will learn how to use similar shapes to measure things from the
real world. You will also learn how to make your own geometric constructions and be sure that they
are mathematically accurate.
Essential Questions
What does it mean for shapes to be similar?
How do transformations affect similarity?
How can two triangles be proven similar?
How can we use similar and congruent triangles to solve problems?
What makes two triangles congruent?
How can geometric theorems be used to solve problems?
Can you use tools to make accurate geometric constructions?
Module Minute
In order for shapes to be similar they have to share some of the same
measurements. A triangle can be enlarged and still be similar to its original shape
because it has the same angle measurements. Transformations like this can still
preserve a shape's similarity to its original features. If we know that two shapes are
similar we can use that information to find missing values in one shape or the other.
We can also use congruence theorems to show that two triangles are congruent.
Different congruence theorems we can use would be if the two triangles have the
same side lengths or if they have two sides with the angle between them congruent or even two
angles with the side between them congruent.
Key Words
Alternate Interior Angles ­ Angles within two lines but are on opposite sides of a transversal.
Complementary Angles ­ Two angles whose sum is 90˚.
Congruent ­ The same shape and the same size.
Equilateral ­Equal sides and equal angles.
Intersection ­ The point where lines cross over one another.
Point ­ A position in space.
Ray ­ A line that has a starting point but no endpoint.
Scale Factor ­ A ratio between two sets of measurements.
Similar Figures ­ Having the same shape but not necessarily the same size.
Supplementary Angles ­ Two angles whose sum is 180˚.
Transformation ­ A change or position or size, including reflection, translation, rotation, or enlargement.
A handout of these key words and definitions is also available in the sidebar.
What To Expect
Similarity in the Coordinate Plane Investigation
Shadow Math Assignment
Similarity Discussion
Transformations and Similarity Quiz
Prove Geometric Theorems Quiz
Company Logo Project
Similarity, Congruence, and Proofs Test
To view the standards from this unit, please download the handout from the sidebar.
Similarity and Transformations
Investigation Assignment: Similarity in the Coordinate Plane
Complete the Similarity in the Coordinate Plane Task found in the sidebar. This is an introductory task that will help you discover some
of the learning objectives of this module. Submit your answers. (You do not need to submit your graph paper.)
When two polygons are similar, they share many characteristics. Once we know what these characteristics are, we can use this information to find
missing pieces of each shape.
View the following video to learn more:
Example 1
The following triangles are similar .
Since the triangles are similar we can determine the scale factor and use that to find the missing sides.
Scale Factor? To find the scale figure, create a ratio with corresponding sides. Since the sides labeled 12 and 4 are corresponding we can compare them in a
ratio.
12 ÷ 4 = 3 By dividing, we can find the scale factor, 3.
To find the missing values, we can either apply the scale factor or solve using ratios. To find y, we will use the scale factor.
The side labeled with 8 corresponds to the side labeled with y.
Since 8 is in the smaller triangle, we can multiply it by the scale factor to find y.
Let's find x by solving with ratios.
Set two sets of corresponding sides equal to each other.
Then solve by cross multiplying:
Example 2
When polygons are similar, corresponding angles are congruent and corresponding sides are proportional.
In the above picture, the corresponding angles are congruent .
<H≌<M, <G≌<L, <F≌<K, <J≌<N. (This is read, angle H is congruent to angle M.)
The corresponding sides can be written as a statement of proportionality.
Once it is determined that two polygons are similar, a similarity statement can be written: FGHJ ~ KLMN.
Now let's try one!
Trapezoid ABCD is similar to trapezoid PQRS.
List all the pairs of congruent angles, and write the ratios of the corresponding sides in a statement of proportionality.
Click on the tabs to see how to complete this problem.
Similarity Proofs
AA Similarity Theorem
If all angles of one triangle are congruent to all angles of another, then the triangles are similar.
∆ABC
∆DEF
SSS Similarity Theorem
If all 3 sets of corresponding sides of 2 triangles are proportional, then the triangles are similar.
To see if these two triangles are similar, we will set up a proportionality statement between the sides
and see if they have the same scale factor.
This shows us that the sides are proportional and the scale factor is .
SAS Similarity Theorem
If two pairs of corresponding sides are proportional and the angles between them are congruent, then
the triangles are similar.
In the above diagram, <ABC
<DEF
because the two angles are vertical angles. We know that vertical angles are congruent.
Since we know that the two triangles have one pair of congruent angles, let's write a proportionality
statement to see if the sides are proportional.
From the proportionality statement we see that the 2 pairs of sides adjacent to the congruent angles
are proportional.
This shows that ∆ABC
∆DEF by SAS similarity postulate and that the scale factor is .
Right Triangles
A right triangle can be divided into two right triangles. If the altitude (height) of a right triangle is drawn, then the two triangles formed are similar to
the original triangle and to each other.
Example 1
The following diagram has three similar right triangles. Do you see them? Roll over the image below to see all three similar triangles.
∆CBD ~ ∆ABC
∆ACD ~ ∆ABC
∆CBD ~ ∆ACD
Example 2
A ramp has a cross section that is a right triangle. The diagram shows the appropriate dimensions of the cross section. Find the height h of the
ramp.
In this problem, we know that the triangles are similar since the three triangles are created by the altitude of a right triangle. In order to find the
missing value, h, we need to set up a proportion. Use the chart to help you set up the proportion.
Once we fill the table in with the information that we know, we can look to see what sides we can compare in a ratio. Notice that we can compare
the short leg and the hypotenuse of the big and medium triangles.
Shadow Math Assignment
Complete the Shadow Math Task found in the tool bar. Submit your completed assignment.
Similarity Discussion
Why does the AA similarity Theorem only need to identify the measurement of 2 congruent angles? Why is it not necessary to find the
measurement of the third angle and show that it is also congruent to the corresponding third angle of a similar triangle?
Post your answer to this discussion.
Transformations and Similarity Proofs Quiz
It is now time to complete the "Transformations and Similarity Proofs" quiz. You will have a limited amount of time to complete your
quiz; please plan accordingly.
Congruence
Watch this video to see the difference between similar and congruent triangles.
In order for triangles to be congruent they must have the congruent corresponding sides. All of the corresponding sides and angles must have the
same measurements. In two congruent figures, all the parts of one figure are congruent to the corresponding parts of the other figure.
(Corresponding means the sides or angles that are in the same positions if you were to align the triangles the same way.)
There are three congruence postulates that can be used to prove that triangles are congruent.
Side­Side­Side (SSS) Congruence
Postulate
Side­Angle­Side (SAS) Congruence
Postulate
Angle­Side­Angle (ASA) Congruence
Postulate
If two sides and the included angle of one triangle
are congruent to two sides and the included angle of If two angles and the included side of one triangle
If three sides of one triangle are congruent to
a second triangle, then the two triangles are
are congruent to two angles and the included
three sides of a second triangle, then the two
congruent.
side of a second triangle, then the two triangles
triangles are congruent.
are congruent.
Example 1
Roll over the image below for further explanation:
From your exploration above, you should have found the following:
, given by hash mark on picture
, given by hash mark on picture
, congruent to itself. Also known as the reflexive property.
Example 2
Use the diagram to name the included angle between the given pair of sides.
The included angle is the angle that is created by the two given sides.
Example 3
State the third congruence that must be given to prove using the indicated postulate.
To complete this problem, mark the given statements on your diagram and look to see what is missing.
Since we have to use SAS Congruence Postulate, we can see that we need the included angle between the given congruent sides.
This shows us that the missing congruence is < J < D.
Example 4
In addition to the three congruence postulates, there are two theorems that can be used to determine whether two triangles are congruent.
Recall that in a right triangle, the sides adjacent to the right angle are called the legs. The side opposite the right angle is called the hypotenuse of
the right triangle.
The Hypotenuse­Leg Congruence Theorem can be used to prove triangles congruent.
Hypotenuse­Leg (HL) Congruence Theorem: If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a
second right triangle, then the two triangles are congruent.
Triangles ABC and DEF are congruent by the HL Theorem.
Example 5
Is enough information given to prove that the triangles ABC and DCB are congruent? If so, which postulate or theorem is used?
ANSWER : Yes, the two triangles are congruent by the HL Congruence Theorem.
The other theorem that can be used to tell prove triangles congruent is the Angle­Angle­Side Congruence Theorem.
Angle­Angle­Side (AAS) Congruence Theorem : If two angles and a non­included side of one triangle are congruent to two angles and the
corresponding non­included side of a second triangle, then the two triangles are congruent.
Triangles ABC and DEF are congruent by the AAS Theorem.
Example 6
Using the AAS Congruence Theorem and the given information, state the third congruence that is needed to prove that triangles ABC and DEF are
congruent.
Given: ∠A ∠D, ∠B ∠E, _____ ______?
Self­Check :
What postulate or theorem can used to prove that triangle ABF is congruent to triangle CBD? AAS, HL, SSS, or SSA?
SOLUTION
Watch the video showcase below to learn more about congurence.
Prove Geometric Theorems
A proof is a logical argument that shows a statement is true. One common proof that we will use is a two­column proof. A two­column proof has
numbered statements and corresponding reasons that show an argument in logical order.
Since we have learned some basics about congruence in our previous lesson, we can use that to begin writing proofs. Watch this video to see an
example of proving two triangles congruent .
We can also prove congruence with other polygons and shapes, for example, line segments. When we begin doing proofs in our next lesson, you
will need to use line statements about lines and angles.
Remember that a line segment is a portion of a line with two endpoints.
An angle is made up of two line segments, rays, or lines.
Recall that a theorem is a statement that can be proven and a postulate is a rule that is accepted without proof. We will use both of these
terms to justify answers.
Example 1
How would we find the length from A to B?
If = 16, what is the value of x?
To find x, we will use the Segment Addition Postulate.
Segment Addition Postulate:
If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C.
This means that x + 10 = 16, x = 6.
Example 2
The angle addition postulate may be used for some problems.
Angle Addition Postulate:
If P is in the interior of <RST, then the measure of <RST is equal to the sum of the measures of <RSP and <PST.
Properties of Congruence
There are three common properties that can be used when showing congruence. Here are examples of each one.
Sometimes we use substitution:
Substitution: Replacing something with another of equal value.
If x = 2y + 3 and = 2y + 3, then x =
.
Watch the video showcase below to learn more about proving geometric theorems.
Using coordinates to Prove Geometry Theorems Algebraically
Now that we know how to prove triangles congruent we can use the coordinate plane to help us complete algebraic proofs.
There are a few things you want to remember before working in the coordinate plane:
Distance Formula : To find the distance between two points use the distance formula. This can help us determine whether figures have congruent
sides.
The distance between points (x1, y1) and (x2, y2) is:
Slope Formula : The slope of a line can help us determine if sides of a shape are parallel or perpendicular. The slope between points (x1, y1) and
(x2, y2) is:
Parallel lines have the same slope and perpendicular lines have opposite (negative reciprocal slopes).
Example
Use coordinates to determine if the triangles ABC and DEF are congruent.
A(1, 1)
B(4,0)
C(7, 5)
D(4, ­5)
E(6, ­6)
F(9, ­1)
Answer
First, let's plot the points and see what the triangles look like.
We can then use the distance formula to see if the triangles have equal corresponding sides.
Distance of side AB:
Distance of side BC:
Distance of side CA:
Distance of side DE:
Distance of side EF:
Distance of side FD:
AB ≠ DE, BC = EF, and CA ≠ FD, therefore the two triangles are not congruent by SSS Congruence Postulate.
Angle Pair Relationships
Reminder!
Alternate Interior Angles ­ Angles within two lines but are on opposite sides of a transversal.
Complementary Angles ­ Two angles whose sum is 90˚.
Supplementary Angles ­ Two angles whose sum is 180˚.
Adjacent
Linear
Linear Pair
Angles
Pair
Postulate
Two angles that share a common vertex and
side, but have no common interior points.
Two adjacent angles are a linear pair if their
noncommon sides are opposite rays.
If two angles form a linear pair,
then they are supplementary.
Vertical Angles
Right Angles
Congruence
Theorem
Congruence
Theorem
Congruent Supplements Theorem
Congruent Complements Theorem
If two angles are supplementary to the
same angle (or to congruent angles) then
they are congruent.
If two angles are complementary to the
same angle (or to congruent angles), then
they are congruent.
Vertical angles are
congruent.
All right angles
are congruent.
Example 1
Use the diagram to find the missing angles.
If m < 3 = 112˚, find m < 1, m < 2 and m < 4. (m represents measure)
m < 1 = m < 3 = 112˚ by Vertical Angles Congruence Theorem.
m < 2 = 180 ­ 112 = 68˚ by Linear Pair Postulate .
m < 4 = m < 2 = 68˚ by Vertical Angles Congruence Theorem .
Prove Geometric Theorems Quiz
It is now time to complete the "Prove Geometric Theorems" quiz. You will have a limited amount of time to complete your quiz; please
plan accordingly.
Geometric Constructions
It is possible to make accurate constructions of geometric figures. We must use certain tools to be sure that our constructions are accurate.
Check out the following video to see how to use a compass to make constructions.
Grab a compass and try working through these activities.
Copy a Segment
Run the simulation below and then practice to see what you learned.
Bisect a Segment
Run the simulation below and then practice to see what you learned.
Bisecting an Angle
Run the simulation below and then practice to see what you learned.
What are the steps for BISECTING AN ANGLE?
Copying an Angle
Run the simulation below and then practice to see what you learned.
What are the steps for COPYING AN ANGLE?
Constructing Perpendicular Lines
Run the simulation below and then practice to see what you learned.
Constructing a Line Parallel to a Given Line
Run the simulation below and then practice to see what you learned.
Constructing Inscribed Polygons
Inscribed polygons are polygons that are inside circles. They are called inscribed because the vertices of the polygons are on the edge of the circle.
The picture shows an equilateral triangle (solid lines) and a regular hexagon (dashed lines) inscribed in a circle.
Both of these polygons can actually be constructed by you!
Notice the dashed lines around the circle. There are 6 of them.
Each of these lines is the length of the radius of the circle. A radius of a circle can fit six times around a circle as shown. When this is done, it
creates an inscribed regular hexagon. Notice, if you connect every other vertex of the hexagon (the solid lines) it creates an inscribed equilateral
triangle. You can try these constructions on your own paper.
An inscribed square can be constructed fairly simply as well. Take a look at the diagram below:
Notice that the vertices of the squares are the endpoints of the circles diameters. If you create perpendicular diameters in a circle and then connect
the endpoints, you will have an inscribed square.
Some video content is reproduced by permission from Math Open Reference.
Module Wrap ­ Up
To be sure that you have completed all the assignments for this module, review the list below. Click on the assignment to revisit the page in the
content where it is located.
Similarity in the Coordinate Plane Investigation
Shadow Math Assignment
Similarity Discussion
Transformations and Similarity Quiz
Prove Geometric Theorems Quiz
Company Logo Project
Similarity, Congruence, and Proofs Test
Now that you have finished the lessons and assignments, engage in the practice below and visit the extra resources in the sidebar. Then, continue
to the next page to complete your final assessment.
Use the proof below to complete the matching activity.
GIVEN: Z is the midpoint of PROVE: Use the proof above to answer..
Standardized Test Preparation
The following problems will allow you to apply what you have learned in this module to how you may see questions asked on a
standardized test. Please follow the directions closely. Remember that you may have to use prior knowledge from previous units in order to
answer the question correctly. If you have any questions or concerns, please contact your instructor.
Final Assessments
Company Logo Project
Complete Company Logo Project found in the sidebar. Submit your completed project.
A rubric for this project is located in the sidebar.
Similarity, Congruence and Proofs Module Test
It is now time to complete the "Similarity, Congruence and Proofs" Test. Once you have completed all self­assessments, assignments,
and the review items and feel confident in your understanding of this material, you may begin. You will have a limited amount of time to
complete your test and once you begin, you will not be allowed to restart your test. Please plan accordingly.