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Mathematics 20
Module 3
Lesson 16
Mathematics 20
The Geometry of Triangles
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Lesson 16
Mathematics 20
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Assignment 16
The Geometry of Triangles
Introduction
Geometry includes the study of the properties of shapes. The study of geometry in this
course is mainly about triangles.
Another basic concept in geometry is the study of arranging concepts in a logical order
through definitions of properties and assumptions. This lesson will start with this concept
by introducing the Congruence Postulates. Through deductive reasoning, you will also be
introduced to the idea of two column proofs.
The study of geometry has been happening for years. Such professionals as architects,
engineers and drafters use the concepts of geometry every day.
This lesson will also show you how to construct congruent triangles, both informally and
formally.
Irrational Number Math
Gordon wants to place a 4-meter pole completely inside a box with dimensions 3 m by 3 m
by 1.5 m. Can this be done?
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Assignment 16
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Assignment 16
Objectives
After completing this lesson you will be able to
•
informally and formally construct congruent angles and congruent triangles.
•
determine the properties of congruent triangles.
•
identify and state the corresponding parts of congruent triangles.
•
determine whether triangles are congruent by SSS, SAS, ASA, AAS, HL or LL
properties.
•
prove that two triangles are congruent by supplying the statements and reasons in
a guided deductive proof.
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Assignment 16
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Assignment 16
16.1 Congruent Triangles
A short review of triangles is necessary to begin this lesson.
A triangle can be classified by the relationship between the sides of the triangle or the
angles of the triangle.
Classification by Sides
Equilateral Triangle
Isosceles Triangle
Scalene Triangle
An equilateral triangle has
three congruent sides.
An isosceles triangle has
two congruent sides.
A scalene triangle has no
congruent sides.
Classification by Angles
Acute Triangle
Equiangular Triangle
An acute triangle has three
acute angles.
(Each angle is less than 90 )
An equiangular triangle is
an acute triangle with all
three angles congruent.
Right Triangle
A right triangle has exactly
one right angle.
Obtuse Triangle
An obtuse triangle has
exactly one obtuse angle.
(greater than 90 0 and less
than 180 0 ).
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Assignment 16
Two figures that have the exact same size and shape are congruent.
Activity 16.11
In each of the following puzzles, determine which two are congruent. If necessary, trace
the figures and slide them over one another. The figures may be flipped or turned in order
to be a match.
The symbol for congruence is  and is read "is congruent to".
If two segments have the same measure or length, they are congruent segments.
AB = 3 cm
CD = 3 cm
AB = CD therefore AB  CD .
Notation:
Notice that AB represents a number but AB represents the set of points
which makes the line. It is incorrect to write AB  CD or AB  CD
Congruent segments are segments which have the same measure.
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Assignment 16
If two angles have the same measure, they are congruent angles.
m RST = 20 
m XYZ = 20 
RST
 XYZ
Congruent angles are angles that have the same measure.
You will notice that in order to place triangle ABC on triangle XYZ, you must line up
vertices A and X. This means that these two vertices are corresponding vertices. The
angles at these vertices also correspond.
Three pairs of sides and three pairs of angles correspond.
A  X
B  Y
C  Z
AB  XY
BC  Y Z
AC  X Z
The same can be shown by the way the triangles are labelled. The corresponding vertices
are arranged in the same order.
ABC  XYZ
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Assignment 16
Congruent Triangles
If ABC is congruent to JKL , then the corresponding angles and the
corresponding sides of the two triangles are congruent.
The congruent corresponding parts are marked by a single
line, double lines and triple lines.
Corresponding angles are:
A  J
•
B  K
•
C  L
•
•
•
•
Corresponding sides are:
AB  JK
BC  KL
AC  JL
Also note that in the same way, congruence can be established in polygons. Two polygons
with the same number of sides have corresponding sides and angles. This correspondence
is determined by the way that the two polygons are labelled.
pentagon EFGHI  pentagon MNOPQ .
This is true if the corresponding angles and the corresponding sides are congruent.
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Activity 16.12
On each of the coordinate planes plot the required points to make the indicated
triangles congruent.
1.
2.
ABC  LMN 1
ABC  LMN
DEF  XYZ
3.
4.
DOG  CAT 2
EFG  OPQ
DOG  CAT
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Example 1
Name all of the congruent parts of the following two triangles.
Solution:
Angles:
•
•
Z  H
A  N
•
Q  B
Sides: •
ZA  HN
AQ  NB
•
•
ZQ  HB
What can you say about the two triangles?
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This shows that the definition of congruence is reversible.
If two triangles are congruent,
then the pairs of corresponding angles and
sides are congruent.
If the pairs of corresponding angles
and sides of a triangle are congruent,
Example 2
then the triangles are congruent.
If FGH  TUV , draw a diagram and mark the diagram to show the
congruent parts.
Solution:
Example 3
From the following corresponding congruent parts, state the triangles that
are congruent.
a)
b)
AB  DE , BC  EJ , CA  JD , A  D, B  E, C  J
WB  RF , BY  FP , YW  PR , W  R, B  F, Y  P
Solution:
a)
b)
ABC  DEJ
WBY  RFP
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Exercise 16.1
1.
In the diagram, UVW  PQR .
State which sides are congruent.
2.
If RUN  CAB , state which sides are congruent.
3.
Correctly write the symbols which state that the two triangles are congruent.
4.
a.
b.
Is ABC  DEF the same as BCA  EFD ? Justify your answer.
Is ABC  DEF the same as ABC  FED ? Justify your answer.
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5.
For each diagram correctly write the symbols which state that certain triangles are
congruent. In some cases there are more than one set of congruent triangles.
a.
c.
b.
d.
e)
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16.2
Exploring Congruent Triangles
Sometimes it is helpful to describe the parts of a triangle in terms of their relative
positions.
•
•
•
•
AB
AB
A
A
is opposite C .
is included between A and B .
is opposite BC .
is included between AB and AC .
Besides the properties or basic terms and definitions associated with geometry, there are
also certain postulates and theorems that are used.
A postulate is a statement that is accepted as being true without proof.
A theorem is a statement that must be proved before it is accepted as being true.
It is not always necessary to have all three sides and all three angles congruent to say
that two triangles are congruent.
The minimum requirements for the proving that two triangles are congruent are stated in
the congruence postulates.
Side-Side-Side (SSS) Congruence Postulate
If three sides of one triangle are congruent, respectively, to three sides of a second
triangle, then the two triangles are congruent.
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Side-Angle-Side (SAS) Congruence Postulate
If two sides and the included angle of one triangle are congruent, respectively, to
two sides and the included angle of a second triangle, then the two triangles are
congruent.
Angle-Side-Angle (ASA) Congruence Postulate
If two angles and the included side of one triangle are congruent, respectively, to
two angles and the included side of a second triangle, then the two triangles are
congruent.
Angle-Angle-Side (AAS) Congruence Postulate
If two angles and the non-included side of one triangle are congruent to two angles
and the non-included side of a second triangle, then the two triangles are congruent.
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Example 1
For each of the pairs of triangles, state the congruence postulate, if any, that
will prove the triangles congruent.
Solution:
a)
b)
ASA Postulate - The congruent side is included between the congruent angles.
SAS Postulate - Two sides and the included angle of one triangle are congruent,
respectively, to the two sides and the included angle of a second triangle.
SSS Postulate - Two sides and the common side are congruent.
None - There is not an AAA Postulate.
c)
d)
Activity 16.2
Why is there not an AAA Postulate?
Use a protractor and ruler for the following constructions.
Triangle One
•
•
•
•
•
Draw a line 4 cm long. Label the line RS.
Construct an angle of measure 900 at point R.
Construct an angle of measure 300 at point S.
The point where the two lines join will be called T.
You now have RST with angles measuring 30, 60 and 90 degree.
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Triangle Two
•
•
•
•
•
Draw a line 6 cm long. Label the line GH.
Construct an angle of measure 900 at point G.
Construct an angle of measure 300 at point H.
The point where the two lines join will be called I.
You now have GHI with angles measuring 30, 60 and 90 degree.
What can you say about the angles in these two triangles?
What can you say about the two triangles?
Is RST  GHI ?
(Remember the definition of congruence triangles)
Example 2
Each pair of triangles below has two corresponding sides or angles marked as
congruent. Indicate the additional information that is needed to enable you
to apply the specified congruence postulates.
a)
For ASA
b)
For SAS
c)
For SAS
d)
For SSS
e)
For AAS
f)
For ASA
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Solution:
a)
b)
c)
d)
e)
f)
M  R
NO  ST
C  F
AB  DE
K  Z
T  Y
An angle is needed on the other side of the included side.
A side is needed on the other side of the included angle.
The angle has to be included between the two sides.
A third side is necessary.
The side is not included between the two angles.
The side is included between the two angles.
Equivalence relations play a role in understanding the properties of congruences.
The following chart will show you the properties of reflexivity, symmetry and transitivity.
This course will utilize the properties of segment congruence and angle congruence.
Property
Number
Equality
Reflexive
a=a
Symmetric
Transitive
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Segment
Congruence
Angle
Congruence
AB  AB
ABC  ABC
If a = b,
then b = a
If AB  CD ,
then CD  AB
If ABC  XYZ
then XYZ  ABC
If a = b and
b = c, then
a=c
If AB  CD and
CD  EF , then
AB  EF
If ABC  XYZ
and XYZ  RST
then ABC  RST
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Here are some examples where the reflexive property can be used.
Statement
Reason
AD  AD
Reflexive Property of
Congruence
X  X
Reflexive Property of
Congruence
CD  CD
Reflexive Property of
Congruence
Exercises 16.2
1.
Are the triangles congruent? Justify your answers by stating the congruence
postulate that applies.
a.
b.
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c.
d.
2.
If the triangles are congruent, state the postulate which justifies your
answer.
If the triangles are not congruent, state the additional information needed to
use a certain postulate to make the triangles congruent.
a.
ABD, CBD
b.
AEB, DEC
c.
AEC, DEB
d.
ABC, DCB
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3.
Complete the statements for the given diagram.
ADC  BDC because of the -------- Postulate. It is given that two pairs of
sides are ---------, and for the third side, CD  CD because of the --------Property of congruence of segments.
4.
If ABC  DEF and DEF  GHI , why is AB  GH ?
16.3 Proving Triangles Congruent
Our minds are often called upon to make decisions. Sometimes these decisions require a
great deal of thought and certain steps must be followed to reach the outcome.
The thought process in these situations is very important.
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Activity 16.3
Mr. Boyes, Mr. Banadyga, Mr. Crow and Mr. Petersen are gentlemen
living in rural Saskatchewan. They live in Meadow Lake, Kamsack,
Estevan and Ponteix.
Use the table and clues to determine where each of the gentlemen live.
a)
Mr. Boyes is a business partner of the men living in southern
Saskatchewan.
b)
Mr. Banadyga is a cousin of the men living in Kamsack and
Estevan.
c)
Mr. Crow and Mr. Petersen just visited northern Saskatchewan.
d)
Mr. Petersen does not live in a city.
Mr. Boyes
Mr. Banadyga
Mr. Crow
Mr. Petersen
Meadow Lake
Kamsack
Ponteix
Estevan
This has been an example of logical thinking. The most important thing to remember is
that a certain thought process must occur in able for you to solve a problem.
This is the same with proofs in Geometry. Many of the steps in the process of solving
geometric proofs use deductive reasoning.
Deductive reasoning is reasoning that begins with a preliminary statement. Other
statements can be logically developed from this statement through the process of
deduction.
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Deductive reasoning is often in the form "If .... then" where the "if" part is the preliminary
statement and the "then" parts are the logical statements that follow from the preliminary
statement.
•
•
•
If it rains out, then the class will not being going on their field trip.
If a person drives a car in Saskatchewan, then he must wear a seat belt.
If a triangle has two congruent sides, then it is isosceles.
Example 1
Express each of the following statements in the "If ... then" form.
a)
b)
c)
d)
When a  b is a positive number, b is less than a.
A square with a side length of 2 cm has a perimeter of 8 cm.
Any two right angles are congruent.
Two circles which have congruent radii have equal areas.
Solution:
a)
b)
c)
d)
If a  b is a positive number, then b is less than a.
If a square has a side length of 2 cm, then it has a perimeter of 8 cm.
If two angles are right angles, then they are congruent.
If two circles have congruent radii, then they have equal areas.
In this form of deductive reasoning, the "if" statement is the hypothesis, and the "then
statement is the conclusion.
If
Then
Hypothesis
Conclusion
When the hypothesis and the conclusion of a statement are reversed, the converse
statement is produced. This reversal does not always result in a correct statement.
Example 2
Take the converse of the statements in Example 2, and determine whether
the converse is true or false.
a)
b)
c)
d)
Mathematics 20
If a  b is a positive number, then b is less than a.
If a square has a side length of 2 cm, then it has a perimeter of 8 cm.
If two angles are right angles, then they are congruent.
If two circles have congruent radii, then they have equal areas.
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Assignment 16
Solution:
a)
If b is less than a, then a  b is a positive number.
True
b)
If a square has a perimeter of 8 cm, then it has a side length of 2 cm.
True
c)
If two angles are congruent, then they are right angles.
False - Two congruent angles are not necessarily right angles.
d)
If two circles have equal areas, then they have congruent radii.
True
Problem solving steps and your ability to use deductive reasoning will help you to prove
that two triangles are congruent.
The problem solving steps are the same ones that you have been using throughout this
course.
•
Read the problem.
•
Develop a plan.
•
Carry out the plan.
•
Look back.
Any time that you are asked to prove that two triangles are congruent, you will be given
information about the two triangles. You will need to take this information and develop a
way to use the information in order to prove the two triangles congruent.
You have learned four ways to prove that two triangles are congruent:
•
SSS Postulate
•
SAS Postulate
•
ASA Postulate
•
AAS Postulate
In each of these cases, you need three congruences of either angles or sides to prove that
the triangles are congruent.
It is important to organize the steps that are necessary in proofs. In this course the
majority of proofs will be done using two columns.
•
•
The first column will include the statements and show the logical steps that will
result in the proof of the triangles being congruent.
The second column will list the reasons why each statement is true.
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Assignment 16
Example 3
ABC and KLM
Given:
AB  KL , A  K, B  L
ABC  KLM
Prove:
Solution:
Read the problem.
Given:
ABC and KLM
AB  KL , A  K, B  L
Prove:
ABC  KLM
Develop a plan.
Mark the congruent parts on the two triangles.
AB  KL
(Side)
A  K
(Angle)
B  L
(Angle)
Determine the congruence postulate that can be applied.
ASA Postulate
Carry out the plan.
1.
2.
3.
4.
Statement
Reason
AB  KL
A  K
B  L
ABC  KLM
1.
2.
3.
4.
Given
Given
Given
ASA Postulate
Look Back
The final statement of the proof will always be what was originally asked to be proven. In
this example, the last step was that the triangles were in fact congruent, and this was
determined by using the ASA Postulate.
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Example 4
Given:
ABC
AB  AC , BD  CD
Prove:
BAD  CAD
Solution:
Read the problem.
Given:
ABC
AB  AC , BD  CD
Prove:
BAD  CAD
Develop a plan.
Mark the congruent parts on the two triangles.
AB  AC
BD  CD
Mark the common side between the two triangles.
AD  AD
Determine the congruence postulate that can be applied.
SSS Postulate
Carry out the plan.
1.
2.
3.
4.
Statement
Reason
AB  AC
BD  CD
AD  AD
BAD  CAD
1.
2.
3.
4.
Given
Given
Reflexivity property
SSS Postulate
Look back.
The final statement showed that the triangles were proven congruent by the SSS
Postulate. Many times, a common angle or side can be the third pair of corresponding
parts when proving triangles congruent.
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Exercise 16.3
1.
Write the converse of each statement and then state whether it is true or not.
a.
b.
c.
d.
e.
f.
g.
If two angles are right angles, then they are congruent.
If two segments have the same measure, then they are congruent.
For real numbers, if a = b , then b = a .
If two angles are vertically opposite angles, then they are congruent.
If two triangles are congruent, then the three corresponding pairs of angles of
the triangles are congruent.
If a triangle is isosceles, then it is equilateral.
Given two supplementary angles, if they are right angles, then they are
congruent.
Write a two column proof for each of the following problems.
Include GIVEN and PROVE statements.
2.
If segments AD and CB bisect each other, prove that ABE  DCE .
3.
Prove that ADC  BDC
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4.
Prove that ABD  ACD .
5.
Prove that ABC  AED .
16.4 Constructing Congruent Angles and Triangles
There are many different ways of constructing angles and triangles informally.
•
•
•
•
paper folding
mira
tracing
ruler and protractor
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Assignment 16
Activity 16.41
Given:
ABC
Construct:
DEF such that ABC  DEF by tracing.
•
•
•
•
Cut out the triangle.
Trace the triangle on another piece of paper.
Label the new triangle.
List the congruent sides.
Activity 16.42
Given:
MNO
Construct: XYZ such that MNO  XYZ by paper folding.
•
•
•
•
•
Choose one arm of the angle.
Fold over the paper using this arm as the edge.
Trace over top of the other arm.
Label the new angle.
List the angle so that it corresponds with the congruence.
There are also formal methods of proof that are more exact. Informal methods may be
easier, but are sometimes not as accurate.
The formal method of proof that is used is a straight edge and compass.
This course will review the construction of congruent angles from Mathematics 10, and
then explore constructing congruent triangles by utilizing the knowledge that has been
developed from the congruence postulates.
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Assignment 16
Constructing Congruent Angles
Example 1
Construct an angle congruent to the following angle A using a straightedge and
compass.
Solution
Step 1
Step 2
Use a straightedge to draw a ray and label its endpoint D.
With the point of the compass at A, draw an arc through both arms, labelling points
B and C, of the original angle. Using the same compass setting or distance, and
with the point of the compass on D, draw an arc crossing the new ray. Label this
point E.
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Assignment 16
Step 3
Step 4
Step 5
Set your compass to the length of BC . With the point of the compass on E and with
the distance BC , draw an arc through the arc containing E. Label the intersection
of these two arcs F.
With the straightedge, draw the ray that has D as its endpoint and passes through
point F.
FDE is now congruent to CAB .
Constructing Congruent Triangles
Constructing a congruent angle or triangle means that the new angle or triangle
has the exact same measure as the original one.
Example 1
Given:
Segments measuring 3 cm, 4 cm and 5 cm
Construct:
ABC with side lengths measuring these segments.
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Assignment 16
Solution:
Step 1
Use a straight edge to draw a ray and label its endpoint A.
Step 2
Set the compass to the length of the 5 cm segment. With the point of the
compass on point A, draw an arc with a length of
5 cm. Label this point B.
Step 3
Set the compass to the length of the 4 cm segment. With the point of the
compass on point A, draw an arc with a length of
4 cm above AB .
Step 4
Set the compass to the length of the segment which is 3 cm. With the point of
the compass on point B, draw an arc with a length of
3 cm above AB .
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Assignment 16
Label the point C where these two arcs intersect. Join AC and BC .
Step 5
ABC is a triangle with side lengths of 3 cm, 4 cm and 5 cm.
We can construct a congruent triangle that has sides of these lengths because the
SSS Congruence Postulate, which states all triangles with congruent sides will be
congruent.
Example 2
Given:
RST
Construct:
JKL such that RST  JKL using the SAS Postulate.
Solution:
Read the problem.
The measures of two sides and the included angle of RST must be congruent to two
sides and the included angle of JKL .
Develop a plan. (Here is one plan of three)
•
•
•
Construct a segment congruent to RS .
Construct a segment congruent to ST .
Construct the included angle congruent to RST .
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Assignment 16
Carry out the plan.
Construct the angle first.
Step 1
Use a straight edge to draw a ray and label the endpoint K.
Step 2
Construct an angle congruent to RST with the vertex at K. Follow the
instructions from "Constructing Congruent Triangles".
Step 3
With the compass, measure the length of RS . With the point of the compass
at K, draw an arc with the length of RS on the arm of the angle. Label the
point J where the arc crosses the arm.
J
K
Step 4
L
With the compass, measure the length of ST . With the point of the compass
at K, draw an arc with the length of ST on the ray. Label this point L.
J
K
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Assignment 16
Step 5
Join JL .
J
You now have RST  JKL using the SAS Postulate.
K
L
Exercise 16.4
1.
Given A and B construct, with a compass and straight edge, both angles
adjacent to each other to form angle C. With a protractor find the measure of each
angle and the measure of the constructed combined adjacent angles (angle C).
Write a general statement about your observations.
2.
Given A and B construct an angle which is the supplement of the sum of the
measures of angles A and B. Use the same angles as those given in Question 1.
3.
Given two angles and the included side construct the DEF . State the postulate
which assures that there is only one such triangle. Use the same angles as those
given in Question 1.
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Assignment 16
4.
Given two angles and a non included side construct a triangle. State the postulate
which assures that there is only one such triangle.
Hint: The measures of the angles of a triangle must add up to 180 .
First construct the supplement of the sum of the two given triangles.
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Assignment 16
Conclusion
In the introduction you were given a Irrational Number Math problem. The solution to
this problem is:
Yes
2
2
2
3 + 3 + 1. 5 =
22 .25  4 .7
The 4 m pole will fit diagonally.
Note: A summary of acceptable reasons for proofs can be found in the
Appendix on page 381.
Summary
The following is a list of concepts that you have learned in this lesson:
•
Congruent segments are segments which have the same measure.
•
Congruent angles are angles which have the same measure.
•
Two triangles are congruent if the corresponding angles and sides of the two
triangles are congruent.
•
The Congruence Postulates are:
Side-Side-Side (SSS) Congruence Postulate
If three sides of one triangle are congruent, respectively, to three sides of a second
triangle, then the two triangles are congruent.
Side-Angle-Side (SAS) Congruence Postulate
If two sides and the included angle of one triangle are congruent, respectively, to
two sides and the included angle of a second triangle, then the two triangles are
congruent.
Angle-Side-Angle (ASA) Congruence Postulate
If two angles and the included side of one triangle are congruent, respectively, to
two angles and the included side of a second triangle, then the two triangles are
congruent.
Angle-Angle-Side (AAS) Congruence Postulate
If two angles and the non-included side of one triangle are congruent, respectively,
to two angles and the non-included side of a second triangle, then the two triangles
are congruent.
Mathematics 20
125
Assignment 16
•
Equivalence relations have the properties of reflexivity, symmetry and transitivity.
•
Deductive reasoning is reasoning that begins with a preliminary statement. Other
statements can be logically developed from this statement through the process of
deduction. Deductive reasoning is often in the form "if ... then".
•
Triangles can be constructed through both informal and formal methods. The
formal method of construction is done with a straight edge and compass.
•
The Congruence Postulates can also be used in the construction of congruent
triangles
Mathematics 20
126
Assignment 16
Answers to Exercises
Exercise 16.1
1.
UV  PQ
VW  QR
UW  PR
2.
RU  CA
UN  AB
RN  CB
3.
ABC  RTS
4.
a.
b.
5.
Mathematics 20
Yes. In both cases the same pairs of sides and
angles are congruent.
No. The same pairs of sides and vertices are not
congruent.
For example, in the first case A  D , but in the
second case A  F .
a.
ADC  BDC
b.
AEB  CED
c.
ABD
ABE
BAC
DAE
d.
ADE  BCE
ADB  BCA
AEB  BEA
e.
BDF
BDC
ABE
BFC
BAC
 ACE
 ACD
 CAB
 EAD
 CEF
 CEB
 ACD
 CFB
 CAB
127
Assignment 16
Exercise 16.2
1.
a.
b.
c.
d.
Yes,
No,
Yes,
Yes,
2.
a.
b.
Yes, SSS,
Yes, SAS,
congruent.
Yes, SSS,
Yes, ASA,
c.
d.
SAS
The angle is not included between the two sides.
AAS
ASA
3.
SSS, congruent,
4.
AB  DE
DE  GH
 AB  GH
BD  BD because it is a common side.
BEA  CED because vertically opposite angles are
AC  DB because BC is common to AC and DB .
BC  BC because it is a common side.
reflexive
since ABC  DEF .
since DEF  GHI .
by transitive property of congruent segments.
Exercise 16.3
1.
a.
b.
c.
d.
e.
f.
g.
2.
If two angles are congruent, then they are right angles. False
If two segments are congruent, then they have the same measure, True
For real numbers, if b = a, then a = b . True
If two angles are congruent, then they are vertically opposite. False
If three corresponding pairs of angles of two triangles are congruent, then the
triangles are congruent. False
If a triangle is equilateral, then it is isosceles. True
Given two supplementary angles, if they are congruent, then they are right
triangles. True
Given:
AD and CB bisect each other.
Prove:
ABE  DCE
Proof:
Mathematics 20
1.
Statement
AEB  DEC
2.
3.
CE  BE , AE  DE
ABE  DCE
Reason
1.
Vertically opposite angles are
congruent.
2.
The segments bisect each other.
3.
SAS Postulate
128
Assignment 16
3.
Given:
AD  BD
ADC and BDC
Prove:
ADC  BDC
Proof:
4.
1.
2.
3.
Statement
AD  BD
ADC  BDC
CD  CD
4.
ADC  BDC
Given:
AB  AC
BD  DC
Prove:
ABD  ACD
Proof:
5.
1.
2.
3.
Statement
AB  AC
BD  DC
AD  AD
4.
ABD  ACD
Given:
ABC  AED
BC  ED
Prove:
ABD  AED
Proof:
Mathematics 20
are right angles.
1.
2.
3.
Statement
ABC  AED
BC  ED
BAC  EAD
4.
ABC  AED
Reason
1.
Given
2.
Right angles are congruent.
3.
Reflexive Property of
Congruence
4.
SAS Postulate
Reason
1.
Given
2.
Given
3.
Reflexive Property of
Congruence
4.
SSS Postulate
Reason
1.
Given
2.
Given
3.
Vertically opposite angles
are congruent
4.
AAS Postulate
129
Assignment 16
Exercise 16.4
1.
with compass and straight edge
measuring with protractor
m A  32 
m B  52 
m A  m B  84 
The constructed angle (combined adjacent angles) and the sum of angle A and angle B are
both 84°.
2.
Mathematics 20
130
Assignment 16
3.
Steps
1.
Draw DG
2.
At D construct DE such that DE  AB .
3.
Construct  angles at D and E such that D  A and E  B .
4.
Extend the arms of the angles to intersect at F.
DEF is constructed using the ASA congruence postulate.
4.
Mathematics 20
131
Assignment 16
With the information given we could use the AAS congruence postulate but, we
are not given enough information to confidently construct the triangle. We
must first find the third angle ( C ) and use the ASA congruence postulate
using A and C with included side AC .
See question 3 for detailed steps for ASA.
Mathematics 20
132
Assignment 16
Mathematics 20
Module 3
Assignment 16
Mathematics 20
133
Assignment 16
Mathematics 20
134
Assignment 16
Optional insert: Assignment #16 frontal sheet here.
Mathematics 20
135
Assignment 16
Mathematics 20
136
Assignment 16
Assignment 16
Values
(40)
A.
Multiple Choice: Select the best answer for each of the following and place a
() beside it. Your calculations will not be evaluated and need not be shown.
1.
The statement that the given two triangles are congruent is ***.
____
____
____
____
2.
a.
b.
c.
d.
RT
RT
RS
ST
 XY
 XZ
 YZ
 ZX
a.
b.
c.
d.
AB  IH
BC  HI
AB  GH
ABC  GHI
If ABC  DFE and DEF  GHI , then ***.
____
____
____
____
Mathematics 20
 DEF
 EFD
 FED
 FDE
If ABC  FED and DEF  GHI , then ***.
____
____
____
____
4.
ABC
ABC
ABC
ABC
If RST  XYZ, then ***.
____
____
____
____
3.
a.
b.
c.
d.
a.
b.
c.
d.
ABC
ABC
ABC
ABC
 HIG
 HGI
 IHG
 GIH
137
Assignment 16
5.
6.
7.
In ABC ***.
____
a.
____
____
b.
c.
____
d.
For the given triangles ***.
____
a.
____
b.
____
____
c.
d.
a.
b.
c.
d.
m F  5 4 
m D  7 0 
m E  5 6 
m F  5 6 
For ABC to be congruent to DFE it is sufficient to have ***.
____
____
____
____
Mathematics 20
ABC  DEF by
SSS
ABC  DEF by
SAS
ABC  DEF by ASA
the triangles need not be congruent
If m A = 56 , m B = 70 , and ABC  DEF, then ***.
____
____
____
____
8.
A is included between
B and C.
A is opposite B and C.
A is included between
AB and AC
A is opposite BA
a.
b.
c.
d.
AB
BC
AB
E
 DF
 FD
 DE
 C
138
Assignment 16
9.
In the statement, "if two circles are concentric, they have the same
center", the phrase "...they have the same center" is the ***.
____
____
____
____
10.
11.
____
a.
____
b.
____
c.
____
d.
If one angle is acute, then the other angle is obtuse in a
linear pair
If the angles form a linear pair, then they are acute and
obtuse
If one angle is acute and the other angle is obtuse, they
form a linear pair
Given a linear pair, if one angle is obtuse, the other is
acute
In the theorem "If one angle of a linear pair is acute, then the other is
obtuse” the Given statement is ***.
a.
b.
c.
d.
A, B form a linear pair
A, B form a linear pair, A is acute
one angle is acute
one angle is obtuse
The postulate that can be used to prove that the triangles are
congruent is ***.
____
____
____
____
Mathematics 20
congruence
hypothesis
conclusion
converse
The converse of the statement "If one angle of a linear pair is acute,
then the other angle is obtuse." is "***".
____
____
____
____
12.
a.
b.
c.
d.
a.
b.
c.
d.
ASA
SAS
SSA
SSS
139
Assignment 16
13.
The postulate that can be used to prove that the triangles are
congruent is ***.
____
____
____
____
14.
a.
b.
c.
d.
SSS
SAS
ASA
AAS
Given that ACB  DEF , the length of EF is ***.
____
____
____
____
Mathematics 20
ASA
SSS
SAS
AAS
The postulate that can be used to prove that the triangles are
congruent is ***.
____
____
____
____
15.
a.
b.
c.
d.
a.
b.
c.
d.
6 2
72
170
5 34
140
Assignment 16
16.
The least common multiple of 4 x , 8 x 2 , and 12 x 3 is ***.
____
____
____
____
17.
18.
19.
Mathematics 20
a.
b.
c.
d.
4x
24 x 3
384 x 6
48 x 3
The simplified form of
____
a.
____
b.
____
c.
____
d.
1
x  1

2x
is ***.
1  x
1  2x
1
+ 2x
x  1
1
 2
x  1
2x + 1
x  1
The solution to 6 x 2 + 11 x + 3 = 0 is ***.
____
a.
____
b.
____
c.
____
d.
0, 3
2
, 3
3
3
1
 , 
2
3
3 1
,
2 3
The solution to 49 x 2  81 = 0 is ***.
____
a.
____
b.
____
c.
____
d.
7
9
9

7
81
49
49 , 81

141
Assignment 16
20.
The solution to 2 x( x  5) = x 2  8 x  1 is ***.
____
____
____
____
Mathematics 20
a.
b.
c.
d.
x=
x=
x=
x=
1
5
0, 5
1
142
Assignment 16
Part B can be answered in the space provided. You also have the option to do
the remaining questions in this assignment on separate lined paper. If you
choose this option, please complete all of the questions on the separate paper.
Evaluation of your solution to each problem will be based on the following.
B.
•
A correct mathematical method for solving the problem is shown.
•
The final answer is accurate and a check of the answer is shown where
asked for by the question.
•
The solution is written in a style that is clear, logical, well organized,
uses proper terms, and states a conclusion.
Complete the two column proof for each of the following problems by
completing a, b, c, and d.
(8)
1.
Prove that the given isosceles triangles are congruent.
Given:
a.
Isosceles ABC and DEF with
C  __________
AB  __________
BC  __________
DE  __________
Prove: b.
___________________________________________________
Proof
1.
2.
3.
4.
5.
Mathematics 20
Statement
B  C ,
E  F
C  F
B  E
BC  EF
ABC  DEF
Reason
1. c. _____________________
2. Given
3. d. _____________________
4. Given
5. ASA Postulate of
Congruence
143
Assignment 16
(8)
2.
If ABC is an isosceles triangle with base BC , and D is any point on
the bisector of A in the interior of ABC , prove that ADB  ADC .
A
C
B
Proof:
3.
Statement
Reason
1.
AB  AC
1. a.
_____________________
2.
BAD  CAD
2. b.
_____________________
3.
AD  AD
3. c.
_____________________
4.
ADB  ADC
4. d.
_____________________
Suppose that AB and CD are perpendicular bisectors of each other at
point E.
Prove that CEB  DEA .
a.
Given:
Proof
Mathematics 20
AB  AC
ADB  ADC
Prove:
D
(16)
Isosceles ABC
B  C
Given:
a _____________________________________________
Statement
Reason
1.
CE  DE
1. b.
_____________________
2.
c. __________________
2.
Definition of segment bisector.
3.
AED  BEC
3. d.
_____________________
4.
CEB  DEA
4.
SAS Postulate
144
Assignment 16
Prove that CEB  DEB .
b.
Given:
Proof
(8)
4.
a _____________________________________________
Statement
Reason
1.
CE  DE
1. a.
_____________________
2.
CEB  DEB
2. b.
_____________________
3.
EB  EB
3. c.
_____________________
4.
CEB  DEB
4. d.
_____________________
Prove that a diagonal of a rectangle divides the rectangle into two
congruent triangles.
A
D
Given:
Rectangle ABCD and
diagonal AC
Prove:
a.
B
Proof
1. b.
Statement
Reason
_____________________
1.
Opposite sides of a rectangle
are congruent.
_____________________
Mathematics 20
_______________________
C
2.
AC  AC
2. c.
_________________________
3.
ABC  CDA
3. d.
_________________________
145
Assignment 16
(10)
C.
1.
Complete the statements in the proof of this theorem.
The HL (hypotenuse leg) theorem for right triangles.
If one pair of corresponding sides are congruent and the hypotenuse of
one triangle is congruent to the hypotenuse of the other triangle, the
right triangles are congruent.
Given:
In right triangles ABC and DEF ,
AC  DF and BC  EF .
Prove:
____________________________
Proof:
Statement
Mathematics 20
Reason
1.
AC  DF
1.
________________
2.
BC  EF
2.
________________
3.
AB 2  BC 2  AC 2
3.
________________
4.
_______________
4.
Pythagorean Theorem
5.
_______________
5.
Subtraction
6.
DE 2  DF 2  EF 2
6.
_________________
7.
DE 2  AC 2  BC 2
7.
_________________
8.
AB 2  DE 2
8.
_________________
9.
________________
9.
Square root of both sides
10.
AB  DE
10.
_________________
11.
_________________
11.
SSS postulate
146
Assignment 16
Write a two column proof for each of the following problems. Include a
diagram and the given and prove statements.
(5)
2.
Prove the following theorem.
The LL (leg leg) theorem for right triangles.
If the corresponding legs of two right triangles are congruent, then the
triangles are congruent.
D
A
C
B
E
Given:
Right triangles ABC and DEF
AB  DE , BC  EF
Prove:
ABC  DEF
F
Proof
Statement
Mathematics 20
Reason
147
Assignment 16
(5)
3.
Make use of the HL or LL theorems to prove the following.
The perpendicular from the vertex to the base of an isosceles triangle
divides the triangle into two congruent triangles.
100
Mathematics 20
148
Assignment 16