Notes on Proofs - Page 1
Name_________________________
Standard Addressed: MA1G3 Students will discover, prove, and apply properties of triangles, quadrilaterals, & other
polygons.
c. Understand and use congruence postulates and theorems for triangles (SSS, SAS, ASA, AAS, HL)
A proof in Geometry seeks to answer the question "Why?" without making leaps
nor assumptions, but rather through a logical step-by-step process.
Throughout the remainder of this unit, you will be required to construct proofs, explaining why
two triangles are congruent. This is slightly different than the work you have been doing.
A
For example, in past lessons, you would be told the following:
B
Consider the diagram to the left. Is there enough information to
prove that ABE is congruent to DCE ?
E
C
D
A student's natural thought process would be as follows:
A
B
"First, I've been given that two pairs of sides are congruent."
"Next, I see that I've got vertical angles that must be congruent. This
angle is the 'included angle' between the two sides mentioned above."
E
vertical
angles
C
D
"So, the two triangles must be congruent by SAS."
In order to construct a proper proof, one simply puts the thought process above into a clear and
concise format. Let's first observe some characteristics about the thought process above.
First, the student observed what was given.
At the beginning of a formal Geometry proof, the given information is always listed first.
Next, notice that the student gathered additional information based on the diagram. He/she
noticed that vertical angles had been formed.
In the middle of a formal Geometry proof, one makes conclusions based on the given
information and the diagram. The student will usually have to recall past Geometry knowledge.
Finally, notice how the student stated that the triangles are congruent by SAS. This is what
he/she was trying to prove in the first place - that the triangles were congruent.
At the end of a formal Geometry proof, one always states what he/she is trying to prove.
One way to construct a proof is as a two-column proof. Here is a two-column proof for the
problem on the front of this page:
A
B
Given: BE  EC , AE  ED
E
Prove: ABE  DCE
C
D
Statements
Reasons
1.
BE  EC , AE  ED
1. Given
2.
BEA  CED
2. Vertical Angles' Theorem
3. ABE  DCE
3. SAS Theorem
Notice that the statements that one is making appear on the left side, and the reasons, or the
"why", the statements are true appear on the right side.
Also, again notice that the given information appears first, and what one is trying to prove (that
the two triangles are congruent) appears last.
--------------------------------------------------------------------------------------------------------------------The diagram below appeared in earlier notes in this unit. Observe how one writes a two-column
proof in order to show that the two triangles are congruent.
L
always first
line of proof
Given: NLQT is a parallelogram.
Q
N
Prove: NLQ  QTN
always LAST
line of proof
T
1.
Statements
NLQT is a parallelogram.
Reasons
1. Given
2.
NL  TQ
2. Opp. sides of a parallelogram are congruent.
3. LQ  NT
3. Opp. sides of a parallelogram are congruent.
4. L  T
4. Opp. angles of a parallelogram are congruent.
5. NLQ  QTN
5. SAS Theorem
L
Note again that the "given" is on the first line, and what one is trying to
prove is on the last line.
In the middle of the proof (lines 2 thru 4), one gathers the information
required to prove the triangles congruent.
Q
N
T
Notes on Proofs - Page 2
Name_________________________
Reflexive Property of Congruence - This property essentially states that a geometric figure is
congruent to itself. It has helped students in the past to think about the idea of a "reflection" in a
mirror.
The Reflexive Property of Congruence can always be used when two triangles share a side.
Suppose one wants to prove that WXY is congruent to ZXY .
One item that he/she could use in the proof is the fact that the two
X
triangles share the same side ( XY ).
W
Z
Y
the triangles
share a side
In the proof, the student could communicate this idea like so:
Statement
Reason
XY  XY
Reflexive Property
In the "statement", observe the same expression appears on both sides of the congruence sign:
XY  XY
same expression
--------------------------------------------------------------------------------------------------------------------Here is the entire two-column proof for the figure above:
X
Given: Point Y is the midpoint of WZ , and WX  XZ .
Prove: WXY  ZXY
W
Statements
1.
Point Y is the midpoint of WZ , and
Y
Z
Reasons
1. Given
WX  XZ .
2.
XY  XY
2. Reflexive Property of Congruence
3.
WY  YZ
3. Since Y is the midpoint of WZ , it divides
WZ into two congruent segments.
4. SSS Theorem
4. WXY  ZXY
Notice on Line 3 how an explanation is given to show the impact of the midpoint in creating the
two congruent segments. Again, the reasons should tell "why" the statements are true.
If one is going to use LL or HL to prove that two triangles are congruent, then he/she must state
that the two triangles are right triangles in the proof.
K
Given: K and P are right angles, LN  NS , and point N is the
L
midpoint of KP .
N
Prove: KLN  PSN
P
S
Statements
Reasons
1. K and P are right angles, LN  NS ,
1. Given
and point N is the midpoint of KP .
2. KLN and PSN are right triangles.
2. They both contain a right interior angle.
3.
KN  NP
4. KLN  PSN
3. Since N is the midpoint of KP , it divides
KP into two congruent segments.
4. HL Theorem
Look at Line 2. One must state that the two triangles are right triangles in order to use
HL Theorem (used at the end of Line 4).
--------------------------------------------------------------------------------------------------------------------Here are some questions which can guide every triangle proof in this unit:
1.
What is given? This is always the first line of the proof.
2.
Do the triangles share a side? If so, you can use the Reflexive Property of Congruence.
3.
Is there a special quadrilateral like a parallelogram, kite, etc.? If so, what properties does
it have that could help?
4.
Are there any special angles (vertical angles, alternate interior angles, etc.)?
5.
Are there any other terms in the "given" or in the problem (perpendicular, midpoint,
bisect, altitudes, circumcenter) that could contribute to the proof?
6.
What are you trying to prove? This is always the last statement of the proof.
---------------------------------------------------------------------------------------------------------------------
Homework on Proofs
Name_________________________
C
A
E
Given: AB  CD , BC  DE , and
1.
C is the midpoint of AE .
Prove: ABC  CDE
B
Statements
Reasons
1.
1. Given
2.
2.
3.
D
3.
ABC  CDE
--------------------------------------------------------------------------------------------------------------------J
Given: IK is perpendicular to JL , and IJ  IL .
2.
K
Prove : IKJ  IKL by HL Theorem
L
I
Statements
Reasons
1.
1.
2.
JKI and LKI are right angles.
2.
3.
IKJ and IKL are right triangles.
3.
4.
4. Reflexive Property of Congruence
5.
5.
--------------------------------------------------------------------------------------------------------------------R
S
3.
Given: RSTU is an isosceles trapezoid, and RU  ST .
Prove: RTU  SUT
T
U
Statements
1.
Reasons
1.
2.
UT  UT
3.
US  RT
4.
RTU  SUT
2.
3.
4.
V
T
4.
Given: TV is parallel to XY , TV  XY , and W is the midpoint of XV .
Prove: TVW  YXW
W
Y
X
Statements
1.
2.
Reasons
1. Given
V  X
2.
3.
3.
4. SAS Theorem
4.
--------------------------------------------------------------------------------------------------------------------5. In the figure below, if FHMI
6. In the figure below,
7. If ED & RT are parallel,
is a kite, why is FM
why are the two marked
why are the two marked
perpendicular to IH ?
angles congruent?
angles congruent?
R
F
E
H
I
D
M
T
Question 1-----------------------------------------------------------------------------------------------------------------------------------------------------------------1. AB  CD , BC  DE , and C is the midpoint of AE .
1. Given
2. AC  CE
2. Since C is the midpoint of AE , it divides AE into 2 congruent segments.
3. ABC  CDE
3. SSS Theorem
Question 2-----------------------------------------------------------------------------------------------------------------------------------------------------------------1. IK is perpendicular to JL , and IJ  IL .
1. Given
2. JKI and LKI are right angles.
2. Right angles are formed by perpendicular segments.
3. IKJ and IKL are right triangles.
3. They both contain right interior angles.
4. IK  IK
4. Reflexive Property of Congruence
5. IKJ  IKL
5. HL Theorem
Question 3-----------------------------------------------------------------------------------------------------------------------------------------------------------------1. RSTU is an isosceles trapezoid, and RU  ST .
1. Given
2. UT  UT
2. Reflexive Property of Congruence
3. US  RT
3. The diagonals of an isosceles trapezoid are congruent.
4. RTU  SUT
4. SSS Theorem
Question 4-----------------------------------------------------------------------------------------------------------------------------------------------------------------1. TV is parallel to XY , TV  XY , and W is the midpoint of XV 1. Given
2. V  X
2. Alternate Interior Angles' Theorem
3. VW  WX
3. Since W is the midpoint of XW , it divides XW into 2 congruent
segments.
4. TVW  YXW
4. SAS Theorem
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Question 5 - Because the diagonals of a kite are perpendicular
Question 6 - Because the angles are vertical angles, and vertical angles are congruent.
Question 7 - Because the angles are alternate interior angles, and they are congruent to each other.