Mth 97
Fall 2013
Chapter 5
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.
t
1
2
4
l
3
5
6
8
7
m
____________________ __________________ 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
r
s
t
The converse is also true.
1
Mth 97
Fall 2013
Chapter 5
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.
m
n
t
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.
m
n
t
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
8 7
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
2
Mth 97
Fall 2013
Chapter 5
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.
2
1
E
3
K
Given:
4
G
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.
3.
4.
1.
2.
3.
4.
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.
N
E
D
W
A
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.
Next: City Designer Project Directions
3
Mth 97
Fall 2013
Chapter 5
Section 5.3 – Parallelograms and Rhombuses
Parallelograms
Theorem 5.14 – A diagonal of a parallelogram forms two congruent triangles.
A
B
If
then
C
D
Corollary 5.15 – In a parallelogram, the opposite sides are congruent and the opposite angles are
congruent.
A
B
If
then
C
D
Corollary 5.16 – Parallel lines are everywhere equidistant.
If
m
then
n
m
A
B
D
C
n
If
, then
Theorem 5.17 – If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is
a parallelogram.
Is the converse of 5.15?
A
B
If
then
D
C
Theorem 5.18 – If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral
is a parallelogram.
A
B
If
then
D
C
4
Mth 97
Fall 2013
Chapter 5
Theorem 5.19 – If a quadrilateral has two sides that are parallel and congruent, then it is a parallelogram.
A
B
If
then
C
D
Theorem 5.20 – The diagonals of a parallelogram bisect each other.
A
B
A
B
M
C
D
C
If
then
D
Theorem 5.21 – If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
This theorem is the _______________________ of 5.20. Write the biconditional statement.
Rhombuses
E
Theorem 5.22 – Every rhombus is a parallelogram.
Previous theorems about parallelograms tell us…
1.
H
M
F
The diagonals of a rhombus bisect each other.
2. The opposite angles of a rhombus are congruent.
G
3. The diagonals of a rhombus form two congruent triangles.
Theorem 5.23 – The diagonals of a rhombus are perpendicular to each other.
A
If
D
B
then
C
Theorem 5.24 – If the diagonals of a parallelogram are perpendicular to each other, then the
parallelogram is a rhombus. Since this is the converse of theorem 5.23 write the biconditional statement.
5
Mth 97
Fall 2013
Chapter 5
Theorem 5.25 – A parallelogram is a rhombus if and only if the diagonals bisect the opposite angles.
F
Write both statements that this biconditional statement says are true.
E
G
1)
H
2)
Do ICA 10
Section 5.4 – Rectangles, Squares and Trapezoids
Rectangles
Theorem 5.26 – Every rectangle is a parallelogram.
Previous theorems about parallelograms tell us…
1. The opposite sides are parallel.
A
B
M
D
C
2. The opposite sides are congruent.
3. The diagonals bisect each other.
Theorem 5.27 – A parallelogram with one right angle is a rectangle.
E
F
If EFGH is a parallelogram and ‫ﮮ‬F is a right angle,
then EFGH is a rectangle.
Given:
Prove: EFGH is a rectangle
H
G
Subgoals: Prove ‫ﮮ‬G is a right angle by
Prove ‫ﮮ‬E is a right angle by
Prove ‫ﮮ‬H is a right angle by
Write the inverse of Theorem 5.27 and tell whether it is true or false.
Write the contrapositive of Theorem 5.27 and tell whether it is true or false.
Theorem 5.28 – The diagonals of a rectangle are congruent.
S
T
If
then
V
U
Theorem 5.29 – If a parallelogram has congruent diagonals, then it is a rectangle.
Is this the converse of Theorem 5.28?
6
Mth 97
Fall 2013
Chapter 5
Squares
Earlier we defined a square to be a quadrilateral with all sides congruent and four right angles.
Other possible definitions might be…
1. A square is a rhombus with one right angle.
A
B
2. A square is a rectangle with two adjacent sides congruent.
D
C
3. A square is a rhombus with congruent diagonals.
4. A square is a rectangle with perpendicular diagonals.
Isosceles Trapezoids
An isosceles trapezoid has congruent legs.
E
H
F
G
Theorem 5.30 – The base angles of an isosceles triangle are congruent.
If EFGH is a trapezoid with EF GH and _________________,then E  F and _______________
See page 273 for proof.
Theorem 5.31 is the converse of Theorem 5.30. Write it below.
Write the inverse of Theorem 5.30 and tell whether it is true or false.
Write the contrapositive of Theorem 5.30 and tell whether it is true or false.
7
Mth 97
Fall 2013
Chapter 5
Section 5.5 – Geometric Constructions Involving Parallel Lines
To construct a line, through a point, parallel to a given line
a) Draw a line through the given point that passes through your given line.
b) Copy the angle created by your original line and the transversal you drew in the first step to
create either alternate interior angles or corresponding angles with your vertex at the given
point.
To construct a rectangle with given side lengths a and b
a) Construct two lines m and n perpendicular at a point.
b) Copy the length a to one of the lines you constructed in the first step line, and b length to the
other line.
c) From the new endpoint of the a length segment you constructed in the second step, swing an
arc with b length.
d) From the new endpoint of the b length segment you constructed in the second step, swing an
arc with c length.
e) Using the intersection point of the arcs you drew as the fourth vertex, draw the remaining two
sides of your rectangle.
a
Use for both a and b
b
This will construct a square.
8
Mth 97
Fall 2013
Chapter 5
To construct a rhombus given lengths a and b of its two diagonals
a)
b)
c)
d)
Copy the length of one of your two line segments.
Construct the perpendicular bisector of the line segment you constructed in the first step.
Construct the perpendicular bisector of the segment you did not copy.
Copy the “half” length of the segment you just bisected to the perpendicular bisector of you
created in the second step using the midpoint of the bisected segment as a center.
e) You now have the diagonals of your rhombus. Draw segments connecting the endpoints of the
segment constructed in the first step and those found in the step just preceding this one.
a
b
Use for both a and b
What will this construct?
To divide a segment into a specified number of congruent segments
a) Draw any ray using one endpoint of the segment to be divided as the endpoint of the new ray.
b) Mark off the specified number of congruent segments of any length on the ray.
c) Draw a segment connecting the second endpoint of the segment to be divided with the last
point marked on the ray in the second step. This creates a triangle.
d) Copy the angle created by your ray and the segment created in the last step to each equally
marked point on your ray. This creates parallel lines. Where these lines cross the original
segment, congruent segments are formed.
Divide the segment into three congruent segments.
9