Download Mth 97 Fall 2012 Sections 5.1 and 5.2 Section 5.1 – Indirect

Mth 97
Fall 2012
Sections 5.1 and 5.2
Section 5.1 – Indirect Reasoning and the Parallel Postulates
Indirect reasoning or indirect proof assumes that the hypothesis of the result we are trying to prove
and the negation (or opposite) of the conclusion. Then we show that this situation leads to a
contradiction. Because the situation is not possible, we conclude that our assumption must be incorrect
and the conclusion as given in the original statement must follow from the hypothesis. (See page 241.)
Parallel Lines (vocabulary)
______________ lines are lines that do not lie in the same plane and do not intersect.
___________________ lines are lines that lie in the same plane and do not intersect. (same slope)
A ________________________ is a line that intersects two other lines.
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____________________ __________________ angles are interior angles on opposite sides of a transversal.
____________________ __________________ angles are exterior angles on opposite sides of a transversal.
_____________________________ angles are non-adjacent angles on the same side of the transversal, one
in the interior and one in the exterior.
Theorems that can be used to show that two lines are parallel
Theorem 5.1 – If two lines cut by a transversal form a pair of congruent alternate interior angles, then the
lines are parallel.
(See sketch above)
If 3  5 or 4  6 then
Corollary 5.2 – If tw o lines are both perpendicular to a transversal, then the lines are parallel.
4 6
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The converse is also true.
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Mth 97
Fall 2012
Sections 5.1 and 5.2
Corollary 5.3 – If two lines cut by a transversal form a pair of congruent corresponding angles with the
transversal, then the lines are parallel.
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Corollary 5.4 – If two lines cut by a transversal form a pair of supplementary interior angles on the same side
of the transversal, then the lines are parallel.
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Postulate 5.1 – The Parallel Postulate (for Euclidean Geometry)
Given a line l and a point P not on l, there is only one line m containing P such that l m .
P
Theorem 5.5 and Corollaries 5.7 and 5.8 are the converses of Corollaries 5.1, 5.3 and 5.4 and are summarized
below.
If two lines are parallel, then…

x y
Alternate interior angles are congruent.
1 2
4 3
x

Corresponding angles are congruent.

If x
Consecutive interior angles are supplementary.
y
5 6
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y and the m2  115 , find the measures of the rest of the angles.
m1 
m3 
m4 
m6 
m7 
m8 
m5 
Do ICA 8
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Mth 97
Fall 2012
Sections 5.1 and 5.2
Section 5.2 – Important Theorems based on the Parallel Postulate
Theorem 5.9 – Angle Sum in a Triangle Theorem – The sum of the angle measures in a triangle is 180°.
Corollary 5.10 – The Exterior Angle Theorem – an exterior triangle of a triangle is equal to the sum of the
measures of the two non-adjacent interior angles.
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Given:
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Prove:
Statement
Reason
m3  m4  180
m1  m2  m3  180
m3  m4  m1  m2  m3
m4  m1  m2
1.
2.
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4.
1.
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More Triangular Congruence
Corollary 5.11 – AAS – If two angles and a side of one triangle are congruent to the corresponding angles and
side in another triangle, then the triangles are congruent.
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If
Y
then
Theorem 5.12 – HA Congruence – If the hypotenuse and an acute angle of one right triangle are congruent to
the hypotenuse and an acute angle of another right triangle, then the two triangles are congruent.
R
A
D
T
If
O
G
then
Theorem 5.13 – Angle Bisector Theorem – A point is on the bisector of an angle if and only if it is equidistant
from the sides of the angle.
P is on the bisector of  A if and only if PB = PC.
Proof is on pages 253-254.
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