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Geometry, Chapter 4
Section 4.1: Deductive reasoning; conditional statements; the converse, inverse and contrapositive of conditional
statements
1.
Statement – A group of words and/or symbols that can be classified collectively as true or false.
Examples: Classify the each statement as true, false, or neither.
a. 8 – 6 = 2
b. An isosceles triangle has at least two congruent sides.
2.
c. 10 < 3
d. Hello!
Conditional Statement (or implication) – a statement of the form “If P, then Q.”
A conditional statement is classified as true or false as a whole. A conditional statement can be written in
equivalent forms.
Vertical Angles: Two non-adjacent angles formed by two intersecting lines.
In the figure at right 1 and 2 are vertical angles as are 3 and 4.
1
4
Ex: Vertical Angle Theorem:
“If two lines intersect, then the vertical angles formed are congruent.”
3.
2
3
Deductive Reasoning – The type of reasoning in which a conclusion is demonstrated by a sequence of
logically valid statements based on a set of accepted assumptions.
4.
Laws of Detachment and Syllogism
Two Forms of Valid Argument used in Deductive Reasoning
Premise – A statement that will be presumed true for the purposes of that argument.
Law of Detachment:
1. P  Q
2. P
Premise 1
Premise 2
true.
 Q
Conclusion
In other words, if P  Q is a true
conditional statement and P is true, then Q
is
(Note:  is a symbol which means ‘therefore’.)
Law of Syllogism:
5.
1. P  Q
Premise 1
2. Q  R
Premise 2
Using words rather than symbols,
1. if P, then Q
2. if Q, then R
 P R
Conclusion
 if P, then R
Three Variations of the Conditional, “If P, then Q” or p  q
1. Converse – interchanges its hypothesis and conclusion. “If Q, then P.” or q  p.
The converse of a given statement is not necessarily true.
2. Inverse – the negation of the hypothesis and conclusion. “If not P, then not Q.” or ~p  ~q.
The inverse of a given statement is not necessarily true.
3. Contrapositive – the negation of the hypothesis and conclusion are interchanged. “If not Q, then not P.”
or ~q  ~p.
If the given statement is true, then its contrapositive is true.
Geometry, Chapter 4
Example. Write the converse, inverse and contrapositive of the given statement and state whether they are
true or false.
Statement: “If a person lives in San Luis Obispo, then that person lives in California.”
Converse:
Inverse:
Contrapositive:
6.
Biconditional Statement – combination of a conditional and its converse when they are both true.
“If P, then Q” combined with “If Q, then P” becomes “P if and only if Q”. p  q & q  p gives p  q.
Example: “A triangle is equilateral if and only if it is equiangular.” is a biconditional statement.
Section 4.2: Congruent Triangles; Triangle Congruency Postulates
1.
Congruent Geometric Figures – figures with the same shape and same size.
2.
Corresponding Parts of Congruent Triangles - the parts of the triangles that coincide when one is placed
on top of the other.
3.
Congruent Triangles – Two triangles, ABC and DEF are congruent, written ABC  DEF, whenever
A  D, B  E, and C  F, AB  DE , BC  EF , and AC  DF .
E
B

A
4.
C
ABC  DEF
F
D
Triangle Congruence Postulates
E
B
a. SAS Congruence
(side-angle-side)
C
A
b. ASA Congruence
(angle-side-angle)
c. SSS Congruence
(side-side-side)

ABC  DEF

ABC  DEF
E
C
F
D
B
A
ABC  DEF
F
D
B
A

E
C
D
F
Geometry, Chapter 4
5.
Right Triangle Congruence Theorems - these follow from the SAS Congruence.
a. LL Congruence
A
b. HL Congruence
D
A

C
B F
D

ABC  DEF
E
C
B
F
ABC  DEF
E
Section 4.3: Important theorems we can prove with congruent triangles.
1.
Once you have proved that two triangles are congruent using one of the triangle congruences, it necessarily
follows that all remaining corresponding parts are congruent. Reason: Corresponding Parts
2.
Median of a Triangle – a line segment drawn from a vertex to the midpoint of the side opposite the vertex.
3.
Theorem – If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
B

Note: We are talking about an isosceles triangle here. This theorem can be
restated, as “the base angles of an isosceles triangle are congruent.”
4.
B
C
A
A
Theorem – If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
B
B
C

5.
C
A
A
Theorem - If a ray bisects the vertex angle in an isosceles triangle, then the ray bisects the base and is
perpendicular to it.
B
B

C
A
6.
C
A
Perpendicular Bisector Theorem – A point is on the perpendicular bisector of a line segment if and only if
it is equidistant from the endpoint of the segment.
P
P

A
7.
B
A
B
The Exterior Angle Theorem (Version 1) – The measure of an exterior angle of a triangle is greater than the measure
of either of the nonadjacent interior angles.
2
1
3

3  1 and 3 > 2
C