Mathematics 20
Module 3
Lesson 17
Mathematics 20
Congruent Triangles
149
Lesson 17
Mathematics 20
150
Assignment 17
Congruent Triangles
Introduction
This lesson continues with the proof of congruent triangles. Many definitions and
properties will be reviewed and used when doing two-column proofs. The thought process
is very important and each proof is different. It is important to take the time to think
about a problem first and determine the method that you will use to solve it. Most of the
definitions and properties were covered in Math 10, and therefore the explanation and
diagrams are not in a lot of detail. If you need further explanation, find an alternate
reference, or the Math 10 course for more detail.
The proofs will expand on your knowledge and include properties associated with
corresponding parts of congruent triangles also being congruent.
Constructions of congruent triangles in this lesson will include special triangles such as
isosceles, equilateral and right.
Irrational Number Math
Find the value of 2 +
repeats indefinitely.
Mathematics 20
2+
2+
2+
151
..... . The dots indicate that the pattern
Assignment 17
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Assignment 17
Objectives
After completing this lesson, you will be able to
•
prove triangles congruent by SSS, SAS, AAS, ASA, HL or LL in two-column
deductive proof or paragraph form.
•
understand the definitions associated with proofs.
•
prove corresponding parts of congruent triangles are congruent.
•
construct congruent right triangles.
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Assignment 17
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Assignment 17
17.1 Review of Definitions and Properties
In Lesson 16, you started proving that two triangles were congruent using a two-column
proof. The congruence postulates that you used were:
•
•
•
•
•
•
Side-Side-Side Postulate (SSS Postulate)
Side-Angle-Side Postulate (SAS Postulate)
Angle-Side-Angle Postulate (ASA Postulate)
Angle-Angle-Side Postulate (AAS Postulate)
Hypotenuse-Leg Theorem (HL Theorem)
Leg-Leg Theorem (LL Theorem)
This section will review a number of definitions and properties that will be used in the
two-column proofs where you will be asked to prove that two triangles are congruent.
Reasons for a statement in a proof will be boxed.
At the beginning of a proof you are always given some information. You may use this
information in the proof by simply stating that the reason is given.
Given
When given information stating that the measures of two segments or two angles are
equal, by definition of congruent segments or angles, it can be deducted that these are also
congruent.
If d (A,B) = d (C,D) then
If m A = m B
then
AB  CD
A  B
Congruent segments are segments which have the same measure.
Congruent angles are angles which have the same measure.
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Assignment 17
There are three properties stated by the equivalence relations. The reflexive, symmetric
and transitive properties can be applied to segments, angles, triangles, or any polygon.
Congruence is an equivalence relation on the set of segments and the set of angles. The
following definitions will relate to this
Property
Number
Equality
Reflexive
a=a
Segment
Congruence
Angle
Congruence
AB  AB
ABC  ABC
Symmetric
If a = b,
then b = a
If AB  CD ,
then CD  AB
If ABC  XYZ
then XYZ  ABC
Transitive
If a = b and
b = c, then
a=c
If AB  CD and
CD  EF , then
AB  EF
If ABC  XYZ
and XYZ  RST
than ABC  RST
.
Reflexive Property
Symmetric Property
Transitive Property
A right angle is an angle with a measure of 900.
ABC  XYZ
Any two right angles are congruent.
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Assignment 17
Two lines that meet or intersect to form right angles are called perpendicular lines.
RS  XY
Perpendicular lines meet to form right angles.
The midpoint of a segment is the point that divides the segment into two congruent
segments.
C M  DM
Definition of midpoint.
The bisector of a segment is a line, segment, ray or plane that intersects a segment at its
midpoint or bisects the segment into two congruent segments.
RS bisects CD at M
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CM  DM
157
Assignment 17
The bisector of an angle bisects an angle into two congruent angles.
YM bisects XYZ
XYM  ZYM
Definition of segment bisector.
Definition of angle bisector.
The perpendicular bisector of a segment is a line, ray or segment that is:
•
perpendicular to the segment.
•
intersects the segment at its midpoint.
RS  XY
RS bisects XY at M
• RMX , RMY ,
• XM  YM
YMS , SMX are
right angles
• RMX  RMY  YMS  SMX
Definition of a perpendicular bisector of a segment.
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Assignment 17
An altitude of a triangle is the perpendicular segment from a vertex to the line containing
the opposite side.
AM  BC
Definition of altitude of a triangle.
A median of a triangle is a segment from a vertex to the midpoint of the opposite side.
YD  ZD
Definition of median of a triangle.
Complementary angles are two angles whose measures have the sum of 900.
•
•
RST and TSU are complementary angles.
m RST + m TSU = 90 
Definition of complementary angles.
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Assignment 17
Supplementary angles are two angles whose measures have the sum of 1800.
A pair of supplementary angles also can also be referred to as a linear pair.
A linear pair is when two angles together form a straight line.
•
•
•
RST and TSU are supplementary angles.
m RST + m TSU = 180 
RST and TSU are a linear pair.
Definition of supplementary angles.
If two angles are supplementary, then the angles form a linear pair.
Conversely,
If two angles form a linear pair, then they are supplementary.
There are other reasons for proofs that have been developed from the definition of
supplementary angles.
Supplements of congruent angles are congruent.
If A  B ,
then C  D .
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Assignment 17
If two angles are congruent and supplementary, then each is a right angle.
If A  B ,
then A and B are right angles.
Vertically opposite angles are congruent.
•
•
•
1 and 3 are vertically opposite angles.
2 and 4 are vertically opposite angles.
1  3
2  4
Definition of vertical angles.
An isosceles triangle is a triangle with two congruent sides.
JK  JL
Definition of isosceles triangle.
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Assignment 17
Two properties of an isosceles triangle are used in proofs.
In the same triangle, or in congruent
triangles, angles opposite congruent sides are
congruent.
In the same triangle, or in congruent
triangles, sides opposite congruent angles are
congruent.
Parallel lines have many properties that are used in proving triangles congruent.
This diagram shows the various parts of parallel lines with AB
transversal.
CD and
t being the
If two parallel lines are cut by a transversal, then the alternate
interior angles are congruent.
•
•
3  6
4  5
If two parallel lines are cut by a transversal, then the
corresponding angles are congruent.
•
•
1  5
3  7
•
•
2  6
4  8
If two parallel lines are cut by a transversal, then the same side
interior angles are supplementary.
•
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3  5  180
162
Assignment 17
•
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4  6  180
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Assignment 17
Exercise 17.1
1.
State what is given in each case.
a.
b.
c.
Prove that a median of an equilateral triangle is perpendicular to the side of
the triangle.
d.
e.
If each leg of one right triangle is congruent to a leg of another right triangle,
then the triangles are congruent.
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Assignment 17
2.
In this section, (Section 17.1), the statements enclosed in boxes may be used as
reasons to support a statement that is used in a proof.
Example:
Statement about
the diagram
Reason to support
the statement
1.
BE  CE
AE  DE
1.
Given
2.
BEA  CED
2.
Definition of vertical
angles.
3.
AEB  DEC
3.
SAS Postulate
Write the reasons for these statements.
a.
ABC  DEF
Reason ______________________
ABC  DEF
Reason ______________________
b.
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Assignment 17
c.
AB  CB
DAE  BAE
d.
Reason _______________________
Reason _______________________
Given: A  B, B  C
A  C
C  D
FA  FB
e.
Reason ______________________
Reason ______________________
Reason ______________________
DC is the perpendicular bisector of AB .
AC  CB
DCA  DCB
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Reason ______________________
Reason ______________________
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Assignment 17
f.
Given: BD is an altitude.
BD  AC
BDA, BDC are right
angles
BDA  BDC
g.
Reason ______________________
Reason ______________________
Reason ______________________
Given: Lines a and b are parallel and lines c and d are parallel .
1   2
 2  3
1  3
Reason ______________________
Reason ______________________
Reason ______________________
17.2 Proving Triangles Congruent
The congruence postulates outline what is needed in order to prove that two triangles are
congruent. The information that you are given will determine which congruence postulate
that you use.
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Assignment 17
Use the outline for problem solving that was shown in Lesson 16. The process will seem
much simpler if you take a logical approach to each proof.
•
•
•
•
Read the problem.
Develop a plan.
Carry out the plan.
Look back.
Some of the proofs will show how this pattern has been used. In later proofs, it will be
assumed that the pattern has been followed.
Example 1
Given:
ABC with CM bisecting ACB
and AC  BC
Prove:
ACM  BCM
Solution:
Read the problem.
Given:
ABC with CM bisecting ACB
and AC  BC
Prove:
ACM  BCM
Develop a plan.
Mark the congruent parts on the two triangles.
A bisector bisects an angle.
AC  BC
ACM  BCM
Common side.
CM  CM
Determine the congruence postulate that can be applied.
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(Side)
(Angle)
(Side)
SAS Postulate
Assignment 17
Carry out the plan.
1.
2.
3.
4.
5.
Statement
Reason
AC  BC
CM bisects ACB
ACM  BCM
CM  CM
ACM  BCM
1.
2.
3.
4.
5.
Given
Given
Definition of angle bisector
Reflexive Property
SAS 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 SAS Postulate.
Example 2
Given:
Figure with AB  BD , ED  BD
C is the midpoint of BD
Prove:
ABC  EDC
Solution:
Read the problem.
Given:
Figure with AB  BD , ED  BD
C is the midpoint of BD
Prove:
ABC  EDC
Develop a plan.
Mark the right angles on the two triangles.
D and B
(Angle)
A midpoint bisects a segment.
DC  BC
(Side)
Mark the vertical angles.
DCB and BCA
(Angle)
Determine the congruence postulate that can be applied.
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ASA Postulate
Assignment 17
Carry out the plan.
Statement
Reason
1.
2.
AB  BD , ED  BD
D and B are right angles.
1.
2.
3.
4.
5.
6.
7.
D  B
C is the midpoint of BD
DC  BC
DCB  BCA
ABC  EDC
3.
4.
5.
6.
7.
Given
Perpendicular lines meet to form right
angles
Any two right angles are congruent
Given
Definition of midpoint
Definition of vertical angles
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.
The way in which you label the corresponding parts is very important. It all depends
on how the two triangles correspond so that they are congruent.
Most of the time, it is important to label angles using three letters. Sometimes in a
diagram the angles are labelled with numbers. In this case refer to a certain angle in its
numbered form.
Example 3
Given:
ABC with RC  SC , 5  6
Prove:
BCR  ACS
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Assignment 17
Solution:
Read the problem.
Given:
ABC with RC  SC , 5  6
Prove:
BCR  ACS
If it is easier to see, separate the triangles.
Develop a plan.
Mark the congruent sides on the two triangles.
RC  SC
(Side)
Common angle.
C  C
(Angle)
Mark the congruent angles.
5  6
(Angle)
•
 5 and  6 are not in the triangles that are to be proven, so it is necessary to see if
this information can lead to other corresponding parts being congruent.
•
It does follow that 7 and 8 are supplementary angles of the two congruent
angles and therefore congruent as well.
Determine the congruence postulate that can be applied.
ASA Postulate
Carry out the plan.
Statement
Reason
1.
2.
3.
Given
Given
Definition of a linear pair.
4.
RC  SC
5  6
5 and 7 are supplementary.
6 and 8 are supplementary.
7  8
4.
5.
6.
C  C
BCR  ACS
5.
6.
Supplements of congruent angles are
congruent.
Reflexive property
ASA Postulate
1.
2.
3.
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Assignment 17
Example 4
Given:
Figure ABCD with
AD  DC , CB  AB ,
DC BA
Prove:
ADC  CBA
Solution:
Read the problem.
Given:
Figure ABCD with
AD  DC , CB  AB ,
DC
BA
Prove:
ADC  CBA
Develop a plan.
Mark on the diagram everything that is given.
•
•
•
The common side is congruent.
Two right angles are formed by the perpendicular lines and are therefore congruent.
From the two segments that are parallel, the alternate interior angles are
congruent.
Determine the congruence postulate that can be applied.
AAS Postulate
The side is not included between the two congruent angles.
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Assignment 17
Carry out the plan.
Statement
Reason
1.
2.
AD  DC , CB  AB
ADC and CBA are right angles
1.
2.
3.
ADC  CBA
3.
4.
5.
DC BA
ACD  CAB
4.
5.
6.
7.
CA  AC
ADC  CBA
6.
7.
Example 5
Given:
Figure ABCD with
AD  DC , CB  AB ,
CD  AB
Prove:
ADC  CBA
Given
Perpendicular lines meet to
form right angles.
Any two right angles are
congruent.
Given
If two lines are parallel, the
alternate interior angles are
congruent.
Reflexive property
AAS Postulate
Solution:
Read the problem.
Given:
Figure ABCD with
AD  DC , CB  AB ,
CD  AB
Prove:
ADC  CBA
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Assignment 17
Separating the triangles,
Develop a plan.
Mark on the diagram everything that is given.
•
•
•
Two right angles are formed by the perpendicular lines and are therefore congruent.
This makes these triangles right triangles.
The common side is congruent. This is also the hypotenuse that is congruent.
Two segments are congruent. These are the legs of the right triangle that are
congruent.
Determine the congruence postulate that can be applied.
HL Theorem
This theorem can only be applied when the triangles involved are right triangles.
Carry out the plan.
Statement
Reason
1.
2.
AD  DC , CB  AB
ADC and CBA are right angles
1.
2.
3.
4.
5.
6.
ADC and CBA are right triangles
CD  AB
CA  AC
ADC  CBA
3.
4.
5.
6.
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Given
Perpendicular lines meet to
form right angles
Definition of a right triangle
Given
Reflexive property
HL Theorem
Assignment 17
As you begin the exercises, here are some points to remember.
•
•
•
•
Determine which triangles you want to prove congruent. Draw a diagram to
separate the triangles, if that gives you a better idea of which parts are
corresponding.
Use the diagram to mark the congruent parts, or right angles.
Take the information that is given to you and use each piece of information to
determine as much as you can about the congruence of corresponding parts.
The last statement in the two-column proof should be the same as what the
question asked you to prove.
Exercise 17.2
1.
Given: Figure with 1   2 and 3  4 .
Prove: ABD  CBD
2.
Given: Figure with RW  SX and WS  XR .
Prove: RSW  SRX
3.
Given: Figure with CF  DE and FCD  EDC .
Prove: CFD  DEC
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Assignment 17
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Assignment 17
4.
Given: Figure with ABC being an isosceles triangle.
Prove: m 5  m 4  180 
5.
Given:
Prove:
Isosceles triangle ABC with
base BC . BE bisects ABC and
CD bisects ACB .
DBC  ECB
17.3 Proving Corresponding Parts of Congruent
Triangles are Congruent
In the previous section you proved that two triangles were congruent using the different
congruence postulates.
You have also learned that:
If the corresponding parts of two triangles are congruent,
then the triangles are congruent.
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Assignment 17
The converse is also true and will be the main focus of this section.
If two triangles are congruent,
then the corresponding parts of the triangles are congruent.
You will now be asked to prove that either sides or angles of two triangles are congruent.
The following steps will be followed:
•
•
First prove the two corresponding triangles are congruent.
State the corresponding parts (sides or angles) are congruent using the following
reason:
Corresponding parts of congruent triangles are congruent.
•
The abbreviated form CPCTC will be used in this course.
Example 1
Given:
Prove:
Figure with 1  2 and 3  4
AB  CB
First prove the two corresponding triangles are congruent.
Statement
Reason
1.
1  2; 3  4
1.
Given
2.
3.
BD  BD
ABD  CBD
2.
3.
Reflexive Property
ASA postulate
Secondly, the corresponding sides are now congruent.
4.
AB  CB
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178
CPCTC
Assignment 17
Activity 17.3
•
In the following table fill in the third column to state which triangles first need to
be proven congruent so that the corresponding parts can also be congruent.
Diagram
Corresponding
Parts
Congruent
Triangles
3  1
AC  BD
CB  AD
3  2
1  4
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Assignment 17
Example 2
Given:
Isosceles ABC with AB  AC ,
Median AD
Prove:
BAD  CAD
Solution:
Read the problem.
Given:
Isosceles ABC with AB  AC ,
Median AD
Prove:
BAD  CAD
Develop a plan.
Determine the two triangles that contain the two angles that are to be proven congruent.
•
BAD and CAD are the two triangles that first need to be proven congruent.
Mark on the diagram everything that is given.
•
•
•
Two congruent sides are AB  AC .
The common side is congruent.
The median of a triangle goes to the midpoint of BC . It follows that BD  CD .
Determine the congruence postulate that can be applied.
SSS Postulate
Because the two triangles are proven congruent, the corresponding parts of these triangles
are then also congruent.
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Assignment 17
Carry out the plan.
1.
2.
3.
4.
5.
6.
Statement
Reason
AB  AC
AD is the median
BD  CD
AD  AD
BAD  CAD
BAD  CAD
1.
2.
3.
4.
5.
6.
Given
Given
Definition of median
Reflexive property
SSS Postulate
Corresponding parts of congruent
triangles are congruent.
The point to remember when proving that corresponding parts are congruent
is to first prove the triangles congruent that contain the corresponding parts.
Exercise 17.3
1.
Theorem: If two sides of a triangle are congruent, then the angles opposite those
sides are congruent.
Plan 1:
Consider the same triangle in two different ways as ABC and ACB .
Prove these two triangles are congruent. The corresponding parts will
then be congruent.
Plan 2:
2.
Using ABC , construct the bisector of CAB and let it intersect
BC at D.
State and prove the converse of the theorem above in question 1.
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Assignment 17
3.
Given: Figure with S as the midpoint of WB and AS  XS . Prove that
AB  XW .
4.
Given: ABC with AC  BC , AD  BE
Prove: DEC is isosceles.
Hint: Make use of the theorem in Question 1 above.
5.
Given: Figure with CF  DE and FCD  EDC .
Prove: CDG is an isosceles triangle.
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Assignment 17
6.
If JEN  ANE , prove JE // NA .
7.
If ABCD is a rhombus (all sides congruent) prove that a diagonal bisects the angles
with vertices at the end points of the diagonal.
17.4 Constructing Special Triangles
In Lesson 16 you constructed two triangles congruent using the congruence postulates and
the procedure necessary for constructions.
This section will take the process of constructing congruent triangles one step farther by
showing you how to construct special triangles.
These triangles will include:
•
isosceles triangles
•
equilateral triangles
•
right triangles
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Assignment 17
Once again the formal method of construction will be followed using a compass and
straight edge.
Example 1
Construct an isosceles triangle using AB and CD .
Solution:
Step 1
Draw ray RP .
Step 2
Open the compass the length of segment AB . Put the point of the compass
on R and draw an arc with this same length through the ray. Label the point
where the arc intersects the ray S.
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Assignment 17
Step 3
Open the compass the length of segment CD . Put the point of the compass
on R and draw an arc above the ray with this same length. Keeping the same
length, put the point of the compass on S and draw an arc above the ray
intersecting the previous arc. Label the point where the two arcs intersect T.
RST
is an isosceles triangle.
What would have happened if segment CD would have been used for the base of the
isosceles triangle?
Activity 17.4
This activity will be handed in with your assignment.
•
•
•
•
•
Choose any length of segment.
Construct an equilateral triangle with each side of the triangle being congruent to
this segment.
Use the same steps as were used with constructing the isosceles triangle.
State in each step what you are doing in the construction.
What is the measure of each angle?
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Assignment 17
Constructing Right Triangles
In order to construct a right triangle you must first be able to construct a perpendicular
line. The steps for this are as follows:
Construct a perpendicular bisector of a line segment AB using a straightedge and
compass.






Begin by placing the point of your compass on A.
Draw arcs 1 and 2
Now place your compass on point B and draw arcs 3 and 4.
Label the intersecting arcs C and D.
Construct CD
CD  AB
C
The figure below shows the steps for constructing a perpendicular line through a point on
the line.
• Begin by placing the point of your compass on P and make arcs 1 and 2.
• Put the point of the compass on A and make arcs 3 and 4.
• Put the point of the compass on B and make arcs 5 and 6.
P
A
B
Example 2
Mathematics 20
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Assignment 17
Construct a right triangle XYZ where the length of the hypotenuse is
congruent to segment AB and the length of a leg is congruent to
segment CD .
Solution:
Step 1
Draw a line. Label a point Y on this line.
Step 2
Open the compass to the length of segment CD . With this measure and the
point of the compass at Y, draw an arc crossing the line. Label this point Z.
Step 3
Construct a perpendicular line at point Y using the steps as outlined in this
section.
Mathematics 20
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Assignment 17
Step 4
Open the compass to the length of segment AB . With this measure and the
point of the compass at Z, draw an arc crossing the constructed perpendicular
line. Label point X.
X
XYZ is a right triangle with the length of the hypotenuse congruent to segment AB and
the length of a leg congruent to segment CD .
Exercise 17.4
1.
Construct an isosceles triangle with base b and altitude h to the base.
(Hint: the altitude bisects the base.)
2.
Construct a triangle with given altitude h, base b, and one side c.
Mathematics 20
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Assignment 17
3.
Construct a triangle with given angle C and sides a and b include the angle.
4.
Given the segments with lengths of XY and the altitude h to XY and given the
measure of X . Construct XYZ .
5.
Construct an equilateral triangle with a 3.8 cm altitude.
(Hint: Use the triangle in Activity 17.4 to construct a 60 angle.
Mathematics 20
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Assignment 17
Conclusion
In the introduction you were given an Irrational Number Math problem. The solution to
this problem is:
Let x =
2+
2+
2+
2+
..... .
Then x =
2 + x , and
x2 = 2 + x
x2  x  2 = 0
( x  2)( x + 1) = 0
x = 2,  1
Reject  1 since the root is the principal root.
Summary
Some of the concepts that you have learned in this lesson are:
•
Two triangles can be proven congruent using the congruence postulates.
•
The information in the given can be used to prove that corresponding parts are
congruent simply by stating that the information is given.
•
Congruence is an equivalence relation on the set of segments and angles as stated
in the reflexive, symmetric and transitive properties.
•
Any two right angles are congruent.
•
Perpendicular lines meet to form right angles.
•
The midpoint of a segment or the bisector of a segment divides the segment into two
congruent segments.
•
The bisector of an angle bisects the angle into two congruent angles.
•
An altitude of a triangle is the perpendicular segment from a vertex to the line
containing the opposite side.
Mathematics 20
190
Assignment 17
•
A median of a triangle is a segment from a vertex to the midpoint of the opposite
side.
•
Supplementary angles have a sum of 1800.
•
If two angles form a linear pair, then they are supplementary.
•
Supplements of congruent angles are congruent.
•
If two angles are congruent and supplementary, then each is a right angle.
•
Vertical angles are congruent.
•
If two lines are cut by a transversal, then:
•
the alternate interior angles are congruent.
•
the corresponding angles are congruent.
•
the same side interior angles are supplementary.
•
Use the problem solving steps when proving that two triangles are congruent.
•
Once you have proven that two triangles are congruent, you can also prove that the
corresponding parts of these triangles are also congruent.
•
When constructing right triangles, it is necessary to be able to construct a
perpendicular line from one of the legs.
Mathematics 20
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Assignment 17
Mathematics 20
192
Assignment 17
Solutions to Exercises
Exercise 17.1
1.
2.
Mathematics 20
a.
Angles A and B form a linear pair.
b.
Angles A and B are vertically opposite angles.
c.
Equilateral ABC, BD is a median to AC .
d.
ABC, ABD, AD  AC , AB  AB , DAB  CAB
e.
Each leg of one right triangle is congruent to a leg of a
second right triangle.
a.
HL Theorem
b.
AAS Postulate
c.
Definition of congruent segments
Definition of congruent angles
d.
Transitive property of congruence of angles
In the same triangle angles opposite  sides a r e  .
In the same triangle sides opposite  s are  .
e.
DC is the perpendicular bisector of AB or Definition of
bisector.
Perpendicular lines meet to form right angles and any
two right angles are congruent.
f.
Definition of altitude
Definition of perpendicular lines
Any two right angles are congruent.
g.
Vertically opposite angles are congruent.
If two parallel lines are cut by a transversal, then
corresponding angles are congruent.
Transitive property of congruence of angles
193
Assignment 17
Exercise 17.2
1.
Statement
Reason
1.
Given
2.
1   2
3  4
BD  BD
2.
3.
ABD  CBD
3.
Reflexive Property
of congruence of
segments
ASA Postulate
Reason
1.
Given
2.
Statement
RW  SX
WS  XR
RS  SR
3.
RSW  SRX
3.
1.
2.
3.
Statement
CF  DE
FCD  EDC
CD  DC
4.
CFD  DEC
Reason
1.
Given
2.
Given
3.
Reflexive Property
of congruence of
segments
4.
SAS Postulate
1.
Statement
1  2
2.
1  5
3.
2  5
4.
m 2  m 4  180 
5.
m 5  m 4  180 
1.
2.
1.
3.
4.
Mathematics 20
194
2.
Reflexive Property
of congruence of
segments
SSS Postulate
Reason
1.
Base angles of an
isosceles  are  .
2.
Definition of
vertical angles
3.
Transitive
property.
4.
Definition of
supplementary
' s .
5.
Substitution
Assignment 17
5.
Exercise 17.3
1.
1.
Statement
DBC  ECB
2.
BC  CB
3.
m EBC = m DCB
4.
EBC  DCB
5.
DBC  ECB
Given:
Prove:
Reason
1.
Base angles of an
isosceles triangle
are congruent.
2.
Reflexive Property
of congruence of
segments
3.
Halves of equals
are equal.
4.
Angles of equal
measure are
congruent.
5.
ASA Postulate
ABC with AB  AC
B  C
Plan 1
1.
Statement
A  A
2.
3.
4.
5.
AB  AC
AC  AB
ABC  ACB
B  C
Reason
1.
Reflexive property
of congruence of
angles
2.
Given
3.
Given
4.
SAS Postulate
5.
CPCTC
Plan 2
Mathematics 20
1.
2.
Statement
AB  AC
BAD  CAD
3.
4.
5.
AD  AD
BAD  CAD
B  C
195
Reason
1.
Given
2.
Definition of
angle bisector
3.
Reflexive Property
4.
SAS Postulate
5.
CPCTC
Assignment 17
2.
If two angles of a triangle are congruent, then the sides opposite
those angles are congruent.
Given:
Prove:
ABC with B  C
AC  AB
Plan 1
1.
Statement
BC  CB
B  C
C  B
ABC  ACB
AC  AB
2.
3.
4.
Reason
1.
Reflexive property
of congruence of
segments
2.
Given
3.
4.
ASA Postulate
CPCTC
Plan 2
Mathematics 20
1.
2.
Statement
B  C
BAD  CAD
3.
4.
5.
AD  AD
BAD  CAD
AB  AC
196
Reason
1.
Given
2.
Definition of angle
bisector
3.
Reflexive property
4.
AAS
5.
CPCTC
Assignment 17
3.
Reason
1.
Given
2.
Given
3.
Statement
AS  XS
S is the midpoint
of WB
BS  WS
4.
ASB  XSW
4.
5.
6.
ASB  XSW
AB  XW
5.
6.
1.
2.
Statement
AC  BC
CAD  CBE
3.
4.
5.
6.
AD  BE
ACD  BCE
CD  CE
DCE is isosceles.
Reason
1.
Given
2.
If two sides of a
triangle are  ,
then the angles
opposite those
sides are  .
3.
Given
4.
SAS Postulate
5.
CPCTC
6.
Definition of
1.
2.
3.
Statement
CF  DE
FCD  EDC
CD  DC
4.
5.
6.
FCD  EDC
CFD  DEC
FGC  EGD
7.
8.
9.
CFG  DEG
CG  DG
EGD is isosceles
1.
2.
4.
5.
Mathematics 20
197
3.
Definition of
midpoint
Definition of
vertical angles.
SAS Postulate
CPCTC
Reason
1.
Given
2.
Given
3.
Reflexive property
of congruence of
segments
4.
SAS Postulate
5.
CPCTC
6.
Vertically opposite
angles are
congruent.
7.
AAS Postulate
8.
CPCTC
9.
Definition of
isosceles  .
Assignment 17
Here is another way to prove question 5.
6.
Given:
Prove:
1.
2.
3.
4.
5.
6.
Statement
CF  DE
FCD  EDC
CD  DC
FCD  EDC
FDC  ECD
GC  GD
7.
CDG is isosceles
JEN  ANE
JE NA
1.
2.
3.
Mathematics 20
Reason
1.
Given
2.
Given
3.
Reflexive property
4.
SAS Postulate
5.
CPCTC
6.
If two angles of a
triangle are  ,
then the sides
opposite those
angles are  .
7.
Definition of
isosceles  .
Statement
JEN  ANE
JEN  ANE
JE NA
198
Reason
1.
Given
2.
CPCTC
3.
If alternate
interior angles are
 , the lines are
parallel
Assignment 17
7.
For this proof diagonal BD was constructed.
Given:
Prove:
Exercise 17.4
Mathematics 20
1.
Rhombus ABCD with diagonal BD .
BD bisects ADC
BD bisects ABC
1.
Statement
AD  CD
2.
AB  CB
3.
BD  BD
4.
5.
6.
7.
ABD  CBD
ADB  CDB
ABD  CBD
BD bisects ADC
8.
BD bisects ABC
Reason
1.
Definition of
rhombus
2.
Definition of
rhombus
3.
Reflexive property
of congruence of
segments
4.
SSS Postulate
5.
CPCTC
6.
CPCTC
7.
Definition of
angle bisector
8.
Definition of
angle bisector
Construct the base of length b and construct the perpendicular
bisector of b. On the perpendicular bisector mark the altitude h.
Join the end point of the altitude to each end point of the base.
199
Assignment 17
2.
Construct the base of length b (arc 1) and construct a
perpendicular, l2, to b at any point on b (point T, arcs 2-4). On
the perpendicular (step 5), mark the altitude h (arc 6) and then
construct another perpendicular, l3, to the first perpendicular, l2,
at the end point of the altitude h (arcs 7-9). With the centre of
the compass at one endpoint of b (diagram used point 1) and
radius c mark a point X on the second perpendicular, l3. Join X
to the end points of the base.
l2
7
l3
8
X
8
6
9
9
7
4
3
c
h
h
6
T
2
4
2
1
l1
3
5
Mathematics 20
200
Assignment 17
Mathematics 20
3.
Construct an angle congruent to angle C. With centre of the
compass at the vertex, mark a length a along one arm and a
length b along the other arm of the angle. Join the two marks to
form the third side.
4.
Draw given base XY . At X construct given X . At O, any
point in XY , draw OM  XY . With centre O and radius h draw
an arc cutting OM at T. At T draw TS  OM cutting arm of
X at Z. Join Z Y . ZXY is required  .
201
Assignment 17
Construct two lines parallel to each other (l1 and l2) and 3.8 cm
apart (arcs 1-7). Construct an angle congruent to the angle of
the equilateral triangle in Activity 17.4 so that one arm
coincides with one of the lines (arc 8, 9). Extend the other arm
so that it intersects the second line. With a compass measure
the length of the segment between the two parallel lines. From
the vertex of the angle mark the same length along the other
arm (arc 10). Join the marks to form a triangle.
5.
5
7
3
2
7
4
6
l2
6
8
9
5
1
1
3
8
10
l1
2
Can you think of another way to construct this proof?
Mathematics 20
202
Assignment 17
Mathematics 20
Module 3
Assignment 17
Mathematics 20
203
Assignment 17
Mathematics 20
204
Assignment 17
Optional insert: Assignment #17 frontal sheet here.
Mathematics 20
205
Assignment 17
Mathematics 20
206
Assignment 17
Assignment 17
Values
(30)
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.
2.
The reason for the statement A  B is ***.
____
a.
____
b.
____
____
c.
d.
If ABC  FED , then ***.
____
____
____
____
3.
a.
b.
c.
d.
AB  DE
BC  EF
AC  FE
AC  FD
If ABC  DEF and DEF  XYZ , then ***.
____
____
____
Mathematics 20
congruent angles are angles which have the same
measure
congruent segments are segments which have the same
measure
in an isosceles triangle two angles have the same measure
ASA
a.
b.
c.
AB  XY
BC  XY
AB  XZ
207
Assignment 17
____
Mathematics 20
d.
BC  XZ
208
Assignment 17
4.
If ABC  DFE and DEF  XYZ , then A  X by ***.
____
____
____
____
5.
Mathematics 20
a.
b.
c.
d.
congruent
vertically opposite
supplementary and form a linear pair
complementary and form a linear pair
The angles A and B are ***.
____
____
____
____
7.
reflexive property
symmetric property
transitive property
triangle property
The angles A and B are ***.
____
____
____
____
6.
a.
b.
c.
d.
a.
b.
c.
d.
a linear pair
supplementary
supplementary and form a linear pair
complementary
The converse of the statement "If two angles form a linear pair, they
are supplementary" is ***.
____
____
____
a.
b.
c.
____
d.
Supplementary angles form a linear pair
Linear pairs are supplementary
If the measures of two angles add up to 90 , they are
complementary
Given a linear pair, the angles are supplementary.
209
Assignment 17
8.
To have ABC  DEF by ASA the additional condition required is
***.
____
____
____
____
9.
a.
b.
c.
d.
A= B
A = B
m A  m B
m A = m B
For parallel lines a and b the reason 1  2 is that ***.
____
____
____
____
Mathematics 20
A  D
AB  DE
AC  DF
C  F
The statement which is equivalent to the statement A  B is ***.
____
____
____
____
10.
a.
b.
c.
d.
a.
b.
c.
d.
corresponding angles are congruent
same side interior angles are congruent
vertically opposite angles are congruent
alternate interior angles are congruent
210
Assignment 17
11.
1.
2.
Given
__________
AB = CD
AB  CD
The reason for the second step is ***.
____
____
____
____
12.
a.
b.
c.
d.
Definition of congruence of segments
Symmetric property of congruence of segments
Definition of median of a triangle
Definition of midpoint of a segment
The reason that ABD  ACD is ***.
____
____
____
____
Mathematics 20
Definition of congruence of segments
SSS Postulate
Given
Definition of equality of segments
If BD is a median, the reason for the statement AD  CD is ***.
____
____
____
____
13.
a.
b.
c.
d.
a.
b.
c.
d.
HL
LL
SSS
SAS
211
Assignment 17
14.
The angles shown in the diagram are ***.
____
____
____
____
15.
A only
ABC, ADC, ABD
30 , 40 
BAC, CAD, BAD
The reason that ABC  DEC is ***.
____
____
____
____
Mathematics 20
a.
b.
c.
d.
a.
b.
c.
d.
Vertically opposite angles are congruent
SAS
HL
SSS
212
Assignment 17
Mathematics 20
213
Assignment 17
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 separate paper.
Evaluation of your solution to each problem will be based on the following:
(40)
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 missing statements labelled a, b, c, and d.
1.
Prove that if ABCD is a rhombus, the diagonals bisect each other.
(Use Exercise 17.3, Question 6 as a reason for one of your statements.)
D
A
E
C
Given:
Rhombus ABCD
Diagonals AC and BD
Prove:
AC bisects BD
BD bisects AC
B
BE  DE 
AE  CE 
Proof:
Statement
Mathematics 20
Reason
1.
AD // BC
1. a.
_____________________
2.
ADE  CBD
2. b.
_____________________
3.
AED  CEB
3. c.
_____________________
4.
BC  AD
4. d.
_____________________
5.
ADE  CBE
5.
AAS Theorem
6.
BE  DE , AE  CE
6.
CPCTC
214
Assignment 17
2.
Mathematics 20
A pair of tongs is made as shown in the diagram. Prove that the
distance AB is always the same as the distance XY.
215
Assignment 17
3.
Prove that if a point is on the perpendicular bisector of a segment, it is
equidistant from the end points of the segment.
P
A
Given:
PD is  bisector of AB at D.
Prove:
a. _________________________
B
D
Proof:
Statement
4.
Reason
1.
AD  BD
1.
Definition of bisector
of a segment.
2.
PD  PD
2. b.
_____________________
3.
ADP  BDP
3. c.
_____________________
4.
APD  BPD
4.
SAS
5.
AP  BP
5. d.
_____________________
Prove that an altitude to the base of an isosceles triangle is the
bisector of the base.
A
B
D
Given:
Isosceles ABC , AB  AC .
a.
_______________________
Prove:
AD bisects BC at D.
C
Proof:
Statement
Mathematics 20
Reason
1. b.
____________________
1.
Reflexive property of
congruence of segments.
2.
AB  AC
2. c.
_____________________
3.
ABD  ACD
3.
Base angles of an isosceles
triangle are congruent.
4.
ADB  ADC
4.
Right s are  .
5.
ABD  ACD
5. d.
_____________________
6.
BD  CD
6. e.
_____________________
216
Assignment 17
5.
Complete the proof:
Given:
Prove:
ABC with CD  AB and 3  4 .
ACD  BCD
C
3 4
A
1 2
D
B
Answer Part C on separate lined paper. Please include any tables or graphs that
you are required to do with the assignment.
(30)
C.
1.
Submit Activity 17.4
2.
Construct two different triangles with the given altitude h and the two
given angles A and B .
Mathematics 20
217
Assignment 17
3.
Angle Addition Postulate If P is a point in the interior of ABC , then
m ABC = m ABP + m PBC .
Complete the proof that if BD is a median and AD  BD  CD , then
ABC is a right triangle. Use the Angle Addition Postulate as one of
the Reasons.
Mathematics 20
Given:
a.
________________________________________________
Prove:
m ABC  90 
218
Assignment 17
Proof:
Statement
Reason
1.
AD  BD
1. b.
______________________
2.
m 5  m 4
2. c.
______________________
3.
BD  CD
3. d.
______________________
4.
m 1  m 3
4. e.
______________________
5.
m 5  m 4  m 1  m 3  180 
5. f.
______________________
6.
m 4  m 4  m 1  m 1  180 
6. g.
______________________
7.
2m 4  m 1  180 
7. h. ______________________
8.
m 4  m 1  90 
8.
Division by 2.
9.
m ABC  90 
9. i.
______________________
10. j. ______________________
10. ABC is a right  .
100
Mathematics 20
219
Assignment 17