Reasoning and Proof
Algebra Properties of Equality
If a b and x y, then a x b y.
If a b and x y, then ax by.
. ,segment someGiven ABABAB
. , ngle someGiven AAAa
If a b and b c, then a c.
Addition
(APE)
Multiplication
(MPE)
Reflexive
Substitution
l m
x y
z
Given: Intersecting lines l and m with two
vertical angles having measures x and y.
Prove: x = y
Statements Reasons
1. x + z = 1800
2. y + z = 1800
3. x + z = y + z
4. x = y
1. Definition of a straight
angle
2.
3.
4.
Definition of a straight
angle
Substitution (steps 1 & 2)
APE
Statements Reasons
1.
2.
3.
4.
5.
6.
7.
8.
9.
Prove the following conditional:
If two angles are congruent and supplementary, then they
are right angles.
anglesright are B andA :Prove
.arysupplement are B ndA and :Given
aBA A B
BmAm oBmAm 180oAmAm 180oAm 1802
oAm 90oBm 90
anglesright are B andA
Def. of supplementary angles
Substitution (steps 1 & 3)
Simplification
MPE
Substitution (steps 1 & 6)
Definition of a right angle
Definition of congruent angles
m A m B
A and B are supplementary Given
Given
Congruent Triangles
Two figures are CONGRUENT if they have
exactly the same size and shape.
When two figures are congruent, we always list
CORRESPONDING LABELS in the same order.
Triangles have 6 parts (3 angles and 3 sides). When
triangles are congruent, their corresponding parts are also
congruent. List the 6 pairs of corresponding parts from the
congruent triangles above. DEAB
DFAC
EFBC
EDFBAC
FEDCBA
EFDBCD
Congruent Triangles
When all six parts of one triangle are
congruent to the corresponding 6 parts of
another, then the two triangles are congruent.
For example, would knowing that two pairs of sides are
congruent be enough?
However, what is the minimum amount of information
required to conclude that two triangles are congruent?
Key Concept:
If the information you provide is sufficient, then you should
be able to create only one unique triangle with the given
conditions.
Use the straws provided to represent sides of triangles and
determine what the minimum conditions are to conclude that
two triangles are congruent. Make a list of these conditions.
Congruent Triangles
Why doesn’t Angle, Side, Side work?
Reason 1:
ASS is a BAD word!!
A
S
S S
Also, there is a counterexample.
The triangle below is not the only one possible with
the given conditions.
Congruent Triangles
Why does AAS work?
A
B
C D
E
F
o25
o25
o75o75
8
8
What do you know about angles C and F?
o80 o80
So what other triangle congruence theorem
does AAS come from?
Congruent Triangles
However, ASS does work when the
Hypotenuse and one Leg of a Right
Triangle are congruent. Why?
C
A
C
A
B B
The Pythagorean Theorem forces the third pair of sides
to be congruent, giving us SAS. This Triangle
Congruence Postulate Is called Hypotenuse-Leg (HL).
Congruent Triangles
What if the legs of two right triangles
are congruent? Are the triangles
congruent?
A A
B B
This Triangle Congruence Postulate Is called Leg-Leg (LL).
It is derived from SAS.
Congruent Triangles
So, the Triangle Congruency Postulates are:
Side-Angle-Side (SAS)
Two congruent sides and their included angle.
Side-Side-Side (SSS)
All three sides are congruent.
Angle-Side-Angle (ASA)
Two angles and their included Side are congruent.
Angle-Angle-Side (AAS)
Two angles and a side that is NOT included by them.
These postulates represent the minimum amount of information
considered sufficient for two triangles to be congruent.
Congruent Triangles
The Right Triangle Congruency Postulates are:
Hypotenuse-Leg (HL)
If the hypoteni and one leg of two right triangles are congruent,
then the triangles are congruent.
In order to use these postulates as reasons for triangle congruency
in proofs, you would have to first demonstrate that the triangles are
right triangles.
Leg-Leg (LL)
If the legs of two right triangles are congruent, then the
triangles are congruent.
D
Definitions, Postulates, and Theorems
A B C
Segment Addition Postulate (SAP)
ACBCAB
B
A C
m BAD m DAC m BAC
Angle Addition Postulate (AAP) A B C
D
180om ABC
180om ABD m DBC
arysupplement are and DBCABD
Definition of a Straight Angle
D
Definitions, Postulates, and Theorems
A B C
B
A C
Definition of a Segment Bisector
Definition of an Angle Bisector
Altitude
A
B C D
Median
Statements Reasons
1
2
3
4
5
6
7
8
Given:
Prove:
DCAB and,AB ofmidpoint theis C
BDAD
AB ofmidpoint theis C
CBAC
anglesright are BCD and ACD
BDAD
Given
CDAB Given
Midpoint a of Def
A C B
D
SAS
Theorem Congruency AngleRight
of Definition
Property ReflexiveCDCD
BCD ACD
BCD ACD
CP
Equally Wet
Statements Reasons
1
2
3
4
5
6
7
8
Given:
Prove:
DCAB and,AB ofmidpoint theis C
BDAD
AB ofmidpoint theis C
CBAC
anglesright are BCD and ACD
BDAD
Given
CDAB Given
Midpoint a of Def
A C B
D
LL
of Definition
Property ReflexiveCDCD
BCD ACD
CP
Equally Wet
nglesright tria are BCD and ACD ngleright tria a of Definition
Given:
Prove:
D ofbisector angle the and with DCBDADADB
ABDCCBAC and
BDCADC
BDAD
DCDC
BDCADC
CBAC
BCDACD
anglesright are B and CDACD
AB DC
Given
DDC bisects Given
bisector anglean ofn Defintitio
Property ReflexiveA C B
D
SAS
CP
CP
Thm. sSupplementCongruent
of Definition
anglestraight a is ACB Given
arysupplement are nd BCDaACD anglestraight a of Definition
Theorem 4.9
Statements Reasons
A C
B
The Base Angles Theorem
CBDABD
Isosceles triangles are symmetric. If we draw
in the angle bisector of B, the triangle is
symmetric about this axis of symmetry.
Since the triangle is symmetric:
CA CP
What is the Converse of
this Theorem?
Is it true?
The Base Angles Theorem
How could we use this theorem to explain why
all equilateral triangles are also equiangular?
A C
B What is the
Converse of
this Theorem?
Is It True?
The Base Angles Theorem
How could we use this theorem to explain why
all equilateral triangles are also equiangular?
A C
B What is the
Converse of
this Theorem?
Is It True?
Exterior Angle Theorem
a d c
b
Which groups of angles equal 180?
Can you use substitution to show that angle d is equal to the
sum of angles a and b?
ocba 180 odc 180
dccba
dba
The measure of an exterior angle of a triangle is equal to the
sum of the measures of the non adjacent interior angles (and
consequently greater than the measures of either).
1
2
3
4
5
6
7
8
Given:
Prove:
AB to DCsuch that drawn is DC , DBAD
AB ofbisector theis DC
DBAD
DCDC
BDCADC
CBAC
Given
Property Reflexive
A C B
D
HL
CP
The Perpendicular Bisector Theorem Converse
ABDC
ABDC ofBisector theis Bisector a of Definition
Given
s'right are and BCDACD of Definition
sright are and BCDACD right of Definition
1
2
3
4
5
6
7
8
9
Given:
Prove:
AB to DCsuch that drawn is DC , DBAD
AB ofbisector theis DC
DBAD
CBAC
Given
A C B
D
CP
The Perpendicular Bisector Theorem Converse
ABDC
ABDC ofBisector theis Bisector a of Definition
Given
s'right are and BCDACD of Definition
BCDACD Theorem Right
Isosceles is ABD Triangles Isosceles of Def
CBDCAD Theorem Angles Base
CBDCAD AAS
1
2
3
4
5
6
7
8
Given:
Prove: AB ofmidpoint theis Cuch that drawn is DC
s
DBAD
AB ofbisector theis DC
DBAD
DCDC
BDCADC
Isosceles is ADB
BDCADC
Given
midpoint a of Definition
Property Reflexive
A C B
D
SSS
Triangles Isosceles of Def
CP
The Perpendicular Bisector Theorem Converse
CBAC
4.9 Theorem
ADB bisects DC bisector of Def
ABDC ofBisector theis
1
2
3
4
5
6
7
8
9
10
Given:
Prove: AB ofmidpoint theis Cuch that drawn is DC
s
DBAD
AB ofbisector theis DC
DBAD
DCDC
BDCADC
supp. are and BCDACD
BCDACD
Given
midpoint a of Definition
Property Reflexive
A C B
D
SSS
AngleStraight a of Def
CP
The Perpendicular Bisector Theorem Converse
CBAC
sSupplementCongruent
straight a is ACB Given
sBCDACD right are and
ABDC Definition
ABDC ofBisector theis Bisector a of Definition