Download Mth 97 Winter 2013 Sections 4.1 and 4.2 4.1 Reasoning and Proof

Mth 97
Winter 2013
Sections 4.1 and 4.2
4.1 Reasoning and Proof in Geometry
Direct reasoning or ________________________reasoning is used to draw a conclusion from a series of
statements. Conditional statements, “if p, then q,” play a central role in deductive reasoning.
“If p, the q” is written ______________, and can also be read “p ___________ q” or” p only if q.”
p is the _____________________ and q is the___________________
What can you deduce from the following statements?
If ∆ABC is an equilateral triangle, then it is also an equiangular triangle.
∆ABC has 3 congruent sides.
Therefore (symbol
, a 3 dot triangle), ∆ABC is an _______________________________.
Not all statements are written in conditional form (if…then), but we can write most statements in this
form. Rewrite the following statements as conditional statements.
The only polyhedron with four vertices is a triangular pyramid.
_____________________________________________________________________________________
A cylinder is not a polyhedron.
_____________________________________________________________________________________
Law of Detachment
Example:
If p → q is a true conditional statement and p is true, then q is true.
1. If an angle is obtuse, its measure is greater than 90° and less than 180°.
2. ‫ﮮ‬B is obtuse.
3. __________________________________________
Law of Syllogism
Example:
If p → q and q → r are true conditional statements, then p → r is also true.
1. If M is the midpoint of AB , then AM = MB.
2. If the measures of two segments are equal, then they are congruent.
3. _____________________________________
Other forms of the conditional “If p, then q.”
Converse of p → q
q→p
Inverse of p → q
not p →not q
Contrapositive of p → q
not q →not p
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Mth 97
Winter 2013
Sections 4.1 and 4.2
Write the converse, inverse and contrapositive of “If ABCD is a square, it has four right angles.”
Converse ____________________________________________________________________________
Inverse ______________________________________________________________________________
Contrapositive _________________________________________________________________________
A biconditional statement exists only when the conditional statement and its converse are both _________
If p → q and q →p are true, then_____________, read “p if and only if q.” (Shorthand for this is p iff q.)
Example: Write the converse of the statement below. If both the statement and its converse are true,
write the biconditional statement.
If three sides of a triangle are congruent, then the three angles of the triangle are congruent.
Converse: _____________________________________________________________________________
Biconditional: ___________________________________________________________________________
Do ICA 6
Vertical Angles are formed by a pair of intersecting lines and are opposite each other. In the sketch below,
the pair ‫ﮮ‬1 and ‫ﮮ‬3 and the pair ‫ﮮ‬2 and ‫ﮮ‬4 are vertical angles.
Theorem 4.1 – If two angles are a pair of vertical angles,
2
1
3
4
.
their measures are equal. m‫ﮮ‬1 = m‫ﮮ‬3 and m‫ﮮ‬2 = m‫ﮮ‬4
See proof on top of page 191
Proof – A proof is a convincing mathematical argument. The two kinds of proof used extensively in
Geometry are paragraph proof and statement-reason proof (2 column proof).
If two angles are congruent and supplementary, then they are right angles.
Given: 1  2 and 1 and 2 are supplementary
1 2
Prove: 1 and 2 are right angles
Statement
1.
2.
3.
4.
5.
6.
1  2 ; m 1 + m 2 = 180
m 1 = m 2
m 2 + m 2 =180; 2(m 2 ) = 180
m 2 = 90
m 1 = 90
1 and 2 are right angles
Reason (Other samples of this proof are on page 191.)
1.
2.
3.
4.
5.
6.
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Mth 97
Winter 2013
Sections 4.1 and 4.2
4.2 Triangular Congruence Conditions
Definition of Congruent Triangles: ∆ABC  ∆DEF, if and only if
1. All 3 pairs of corresponding angles are congruent.
2. All 3 pairs of corresponding sides are congruent.
Reflexive Property – Something is congruent or equal to ________________
Examples:
∆DEF  ∆DEF or 15 = 15
Symmetric Property – It doesn’t matter which side of the = or  you are on.
Examples:
If ∆ABC  ∆DEF, then ∆DEF  ∆ABC. If 5 + 2 = 7, then 7 = ______________
Transitive Property – If two thing are  or = to the same thing, then they are  or = to each other.
Examples: If ∆ABC  ∆DEF and ∆DEF  ∆GHI, then If ∆ABC  ∆GHI.
or If 4 + 4 = 8 and 8 = (2)(4), then 4 + 4 = ______________
SAS Postulate
LL Theorem
Postulates and Theorems used to prove Two Triangles are Congruent
If two sides and the included angle of one
triangle are congruent respectively to two
sides and the included angle of another
triangle, then the two triangles are
congruent.
If two legs of one right triangle are
congruent respectively to two legs of
another right triangle, then the two
triangles are congruent.
HL Theorem
If the hypotenuse and a leg of one right
triangle are congruent respectively to the
hypotenuse and the leg of another right
triangle, then the two triangles are
congruent.
ASA Postulate If two angles and the included side of one
triangle are congruent respectively to two
angles and the included side of another
triangle, then the two triangles are
congruent.
SSS Postulate If three sides of one triangle are congruent
respectively to three sides of another
triangle, then the two triangles are
congruent.
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Mth 97
Winter 2013
Sections 4.1 and 4.2
Theorem 4.4 – Converse of the Pythagorean Theorem
If the sum of the squares of the lengths of two sides of a triangle equals
the square of the third side, then the triangle is a right triangle.
For each pair of triangles decide whether they are necessarily congruent. If so, write and appropriate
congruence statement and specify which congruence principle applies.
F
a)
A
B
b)
D
c)
C
I
H
d)
T
H
G
U
O
M
P
A
Y
X
W
V
N
Proof using triangular congruence
Given: RA  TA, PA  CA, RP  CT
R
A
C
Prove : R  T
P
Subgoal:
T
Statement
RA  TA, PA  CA, RP  CT
2. ∆RAP  ________________
3. R  T
1.
Reason
1.
2.
3.
4