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What’s man’s first duty?
The answer’s brief: to be himself
Henrik Ibsen, 1828-1906
Norwegian writer, dramatist, poet.
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GEOMETRY
Friday, April 28, 2017
Proving Corresponding Parts Equal
Suppose you are told that
ABC ≅
RST. Then you
know that six things must be true. You should be able
to complete the statements below
C
A
<A≅ <R
<B≅ <S
<C≅ <T
T
B
R
S
AB = RS
BC = ST
AC = RT
1
The six statements are true because of the definition of
congruent triangles. Corresponding parts of congruent
triangles are equal.
A strategy for proving that two segments or two angles
are equal:
1 Find two triangles in which the two sides or the two
angles are corresponding parts.
2. Prove that the two triangles are congruent.
3. State that the two parts are equal, using as reason,
CPCTC (Corresponding parts of congruent triangles
are congruent.)
This strategy will be used in the examples that follow:
1.
P
J
K
Q
2
Given: JP = JQ
PK = QK
Prove: < P ≅ < Q
STATEMENTS
1. JP = JQ
2. PK = QK
3. JK = JK
4. PJK ≅
QJK
5. < P ≅ < Q
REASONS
Given
Given
Reflexive Property of
Equality or Same Segment
SSS Postulate
CPCTC
W
2.
X
M
Z
Y
Given: M is the midpoint of ̅̅̅̅
𝑿𝒀
M is the midpoint of ̅̅̅̅̅
𝒁𝑾
Prove: XZ = YZ
3
STATEMENTS
1. XM = YM
2. ZM = WM
REASONS
Given
Given
3. < XMZ = < YMW
4. XMZ ≅ YMW
5. XZ = YW
Vertical Angles are ≅
SAS Postulate
CPCTC
TRY:
3. In the following exercise, possible reasons are provided.
Select the reason that support each statement and
complete the proof.
C
I)
11(p132)
1 2
A
B
D
Given: AC = BC
⃗⃗⃗⃗⃗⃗
𝑪𝑫 bisects <ACB
Prove: AD = BD
4
STATEMENTS
1. AC = BC
2. <1= <2
3. CD = CD
4. ACD ≅
5. AD = BD
REASONS
BCD
REASONS
a)
b)
c)
d)
e)
f)
g)
h)
i)
Reflexive Property of Equality or Same segment.
SSS Postulate
ASA Postulate
SAS Postulate
Given
Substitution Postulate
AAS Theorem
Definition of Rhombus
CPCTC (Corresponding Parts of Congruent Triangles are
Congruent).
5
More examples:
4.
E
S
22S
R
O
Given: <O and <E are right angles.
RO = ES
Prove: ̅̅̅̅̅
𝑹𝑶 // ̅̅̅̅
𝑬𝑺
STATEMENTS
1. <O and <E are right
angles.
2. RO = ES
3. RS = RS
4.
ROS ≅ SER
5. < SRO ≅ < ESR
6. ̅̅̅̅̅
𝑹𝑶 // ̅̅̅̅
𝑬𝑺
REASONS
Given
Given
Reflexive Property of Equality
or Same Segment
HL Theorem: If the hypotenuse
and a leg of one right triangle
are equal to the corresponding
parts of another right triangle,
the triangles are congruent.
CPCTC
Theorem: If two lines and a
transversal form equal
alternate interior angles, then
the lines are parallel.
6
5. Given: <A = <B
<AYX = <BXY
Prove: AX = BY
A
B
X
Y
STATEMENTS
1. < A = < B; <AYX = <BXY
2. XY = XY
3. AXY ≅
4. AX = BY
BYX
REASONS
Given
Reflexive Property of Equality
or Same segment
AAS Theorem
CPCTC
7
6. Given: ̅̅̅̅
𝐸𝐵
̅̅̅̅
𝐴𝐶
<E=<C
EB = BC
Prove: AX = BY
E
D
A
STATEMENTS
1. < EBA and <DBC are right
angles.
2. <EBA = <DBC
3. <E = <C; EB = BC
4. EBA ≅ CBD
5. AB = BD
B
C
REASONS
̅̅̅̅
Given: ̅̅̅̅
𝐸𝐵
𝐴𝐶
Substitution Postulate: If a = b then a
can be substituted for b in any
equation or inequality.
Given
ASA Postulate
CPCTC
Remark:
Reasons used in a proof:
1.
2.
3.
4.
Given information
Definitions
Postulates (These are statements accepted without proof.)
Theorems (These are statements which have been proved.)
8
9