Download 3.1: Properties of Parallel Lines

Properties of Equality and Congruence,
and Proving Lines Parallel
Objectives: Students will be able to…
• Use properties of equality and congruence
• Justify each step in solving equations
•Identify angles formed by two lines and a transversal
•Prove and use properties of parallel lines
•Use a transversal to prove two lines parallel
Remember…
 In Geometry, we cannot assume anything is necessarily true
(unless it is a theorem or postulate). We can’t say “That looks
like an acute angle, so it is an acute angle”
 It needs to be told to us, marked on a diagram or we have to
use logic to prove it
 THEOREMS have already been proven true for us
 POSTULATES are Geometric statements that are assumed to
be true
Drawing Conclusions
There are conclusions you can make from a diagram.
You can assume that angles are…
 Adjacent angles
 Adjacent supplementary angles
 Vertical angles
Unless it is marked or you are told, you cannot assume…
 Angles or segments are congruent
 An angle is a right angle
 Lines are parallel or perpendicular
To justify what we do in Geometry, we
can use:
 Properties
 Postulates
 Theorems
 Definitions (ex. Definition of a right angle, definition of an
angle bisector, etc…)
 In summary, we must justify everything we do. This helps us
with logical thinking!!!!
Properties of equality (use with numbers)
If a = b then a + c = b + c
If a = b then a - c = b – c
If a = b, then a ● c = b ● c
If a = b, then
a b, c ≠ 0

c c
a = a
If a = b, then b = a
If a = b and b = c, then a = c
Addition Property of Equality
Subtraction Property of Equality
Multiplication Property of Equality
Division Property of Equality
Reflexive Property of Equality
Symmetric Property of Equality
Transitive Property of Equality
More properties of equality
 Substitution Property:
 If a = b, then b can replace a in any expression
 The Distributive Property:
 a(b + c) = ab + bc
Properties of congruence
 Reflexive Property: AB
 AB
 Symmetric Property: If AB
, A   A
 CD,
then CD  AB
If  A   B, then B
 Transitive Property: If AB  CD and CD  EF

A
, then AB  EF
If  A   B and  B  C, then  A  C
Using Properties of equality and congruence
 Name the property that justifies each statement.
a)
If x = y and y + 4 = 3x, then x + 4 = 3x
b)
If x + 4 = 3x, then 4 = 2x
c)
If P  Q, Q  R and R  S , then P  S
Justify each step of solving the
following problem…
2 (3x + 7) = 26
Two-Column Proof
 Displays steps that prove a statement
 Statements on left; Reasons/justifications on right
 Use theorems, postulates, definition, properties and given statements
for justifications
GIVEN: (what you know)
PROVE: (what you must show)
Statements:
Reasons:
1.
1.
2.
2.
3.
3.
.
.
.
.
.
.
Don’t forget…
Vertical Angles are congruent
Also don’t forget…
 Angles that form a line (straight angle) add up to 180°
Solve for x. Justify each step.
4x
6x-40
Given :
mAOC  4 x
mBOD  6 x  40
Solve for x. Justify each step.
4x+60
2x
Extra examples, if necessary. Find the value of the
variables. Justify each step.
1.
2.
(8t)°
126°
(3x)°
(6x-54)°
(10t)°
TRANSVERSAL:
 Line intersects 2 lines in 2 distinct points
 The intersection of a transversal and the 2 lines form 8 angles
Transversal n
intersects line l
and m
The angles formed when a transversal
intersects 2 lines depends on their position
ALTERNATE INTERIOR ANGLES:
 Non-adjacent
 Lie on opposite sides of the transversal in between the 2
lines it intersects
Alternate Exterior Angles
 Lie outside the 2 lines on opposite sides of the transversal
Same-Side Interior Angles (Co-interior)
 Lie on the same side of the transversal between the two lines
Same-Side Exterior Angles
 Lie outside the 2 lines on same side of transversal
Corresponding Angles
 Lie on the same side of the transversal
 In corresponding positions
Using a protractor, measure the
following:
a.) 1 pair of corresponding angles
b.) 1 pair of alternate interior
angles
c.) 1 pair of same side interior
angles
If a transversal intersects 2 parallel
lines:
 Corresponding angles are congruent
 Alternate interior angles are congruent
 Same side interior angles are supplementary
 Alternate exterior angles are congruent
 Same side exterior angles are supplementary
Finding Measures of Angles
8
7
6
a
2
50°
5
4
1
3
b
c
d
Find the measure of each angle. Justify your answer.
a.)3
c.)5
b.)4
d .)6
e.)7
f .)8
Find the values of x and y. Then find the
measures of the angles.
(2x)°
y°
(y-50)°
What is the converse of this statement? (The
converse of a statement switches the hypothesis and the conclusion)
If a transversal intersects 2 parallel lines, then
corresponding angles are congruent.
Postulate
Converse of the Corresponding Angles Postulate
 If two lines and a transversal form corresponding angles
that are congruent, then the two lines are parallel.
Write the converse:
If a transversal intersects 2 parallel lines, then
alternate interior angles are congruent.
Theorem
Converse of the Alternate Interior Angle Theorem
 If two lines and a transversal form alternate interior angles
that are congruent, then the two lines are parallel.
If C  B, then line 1 ll line 2
Write the converse:
If a transversal intersects 2 parallel lines, then same
side interior angles are supplementary.
Theorem
Converse of the Same-Side Interior Angles Theorem
 If two lines and a transversal form same-side interior angles
that are supplementary, then the two lines are parallel.
If 4 and 6 are supplementary, then m ll n
Which lines or segments are parallel? How do you know??
1.
2.
C
H
e
45°
g
b
c
M
45°
A
R
Which segments are parallel?
J
O
K
N
L
M
mJ  mL  180
Theorem 3-5
 If two lines are parallel to the same line, then they are
parallel to each other.
a ll b
* Lines can be coplanar or noncoplanar
Theorem
 In a plane, if two lines are perpendicular to the same
line, then they are parallel to each other.
k ll l
In City Hall, Corridor 1 and Corridor 2 are both
perpendicular to Corridor 3. What can you say about
corridor 1 and corridor 2?
Solve for x and then solve for each angle such that n ll m
n
14 + 3x
5x - 66
m
Find the value of x so that m||n.
62
m
7x - 8
n
Proof: Given m3  m5  180
Prove: n ll m
3
5
Statements
1. m3  m5  180
2. m5  m7  180
3. m3  m5  m5  m7
4. m3  m7
5. 3  7
6. n ll m
Reasons
1.
2.
3.
4.
5.
6.
7
n
m