Direct and

1
Chapter 6 Reasoning
6.6
Direct and Indirect Proof
6.6.1
MATHPOWERTM 11, WESTERN EDITION
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Direct and Indirect Proof
Direct Proof begins with given information and
deductively reaches a
conclusion.
Indirect Proof is known as proof by
contradiction. The desired conclusion is
assumed to be false. If this assumption
leads to a contradiction, then it can be
concluded that the assumption was
incorrect and the desired conclusion is
true.
6.6.2
3
Theorems Used In Proofs

• Parallel Line Theorem
• If a transversal intersects two lines, making
  the alternate angles
• equal, then the lines are parallel.
• If a transversal intersects two lines, making
  the corresponding
• angles equal, then the lines are parallel.
• If a transversal intersects two lines, making
  the co-interior angles
• on the same side of the transversal
  supplementary, then the lines
• are parallel.

Triangle Sum Theorem The sum of the interior
angles of a triangle is 1800.
Third Angle Theorem If two angles of one triangle
are congruent to two angles of a second
triangle, then the third angles of the two
triangles are congruent.
6.6.3
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Theorems Used In Proofs contd
SSS Congruence Theorem If three sides of one
triangle are equal to three sides of
another triangle, then the triangles are
congruent.
SAS Congruence Theorem It two sides and the
contained angle of one triangle are equal to two
sides and the contained angle of another
triangle, then the triangles are congruent.
ASA Congruence Theorem If two angles and the
contained side of one triangle are equal to two
angles and the contained side of another
triangle, then the two triangles are congruent.
Perpendicular Bisector Theorem Any point on the
perpendicular bisector of a line segment
is equidistant from the endpoints of the segment.
6.6.4
5
Theorems Used In Proofs contd
Exterior Angle Theorem The exterior angle of a
triangle is equal to the sum of the interior
opposite angles.
Isosceles Triangle Theorem In any isosceles
triangle, the angles opposite the equal sides are
equal.
Hypotenuse-Side Congruence If the hypotenuse and
one side of one right triangle are congruent to
the hypotenuse and one side of another right
triangle, the triangles are congruent.
6.6.5
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Geometric Proofs
In the diagram, P is any point on the
perpendicular bisector of line segment AB.
b) Prove that PA PB.
Given
In ?PAM and ?PBM
AM BM
SAT
PM PM
Reflexive
Since the triangles are congruent, PA PB.
6.6.6
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Indirect Proof - the Process
Step 1 State the result to be proved as either
true or false. Step 2 Assume
that the result to be proved is false. Step 3
Show that a conclusion can be reached that
contradicts the known facts. Step 4
Since there is a contradiction, the assumption
in Step 2 is incorrect. Therefore, we
can conclude that the assumption
made in Step 2 is true.
6.6.7
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Using Indirect Proof
Step 1 PQ ? PR is a true statement.
Step 2 Assume that PQ ? PR is a false
statement. Hence PQ PR.
Step 3 Reach a conclusion given the known
facts.
In a scalene triangle, no two sides are the
same. Therefore, PQ ? PR.
Step 4 Since the assumption made in Step 2 is
false, then the statement in
Step 1 is true.
6.6.8
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Assignment
Suggested Questions
Page 372 1-5, 8, 10, 15-19
6.6.9