3.1: Properties of Parallel Lines
Objectives: Students will be able toUse properties of equality
and congruence Justify each step in solving equations Identify
angles formed by two lines and a transversalProve and use
properties of parallel linesUse a transversal to prove two lines
parallelProperties of Equality and Congruence, and Proving Lines
ParallelRememberIn Geometry, we cannot assume anything is
necessarily true (unless it is a theorem or postulate). We cant 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
trueDrawing ConclusionsThere are conclusions you can make from a
diagram.
You can assume that angles areAdjacent anglesAdjacent
supplementary anglesVertical angles
Unless it is marked or you are told, you cannot assumeAngles or
segments are congruentAn angle is a right angleLines are parallel
or perpendicular To justify what we do in Geometry, we can
use:PropertiesPostulatesTheoremsDefinitions (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 Addition Property of EqualityIf a = b then a -
c = b c Subtraction Property of EqualityIf a = b, then a c = b c
Multiplication Property of EqualityIf a = b, then , c 0 Division
Property of Equality
a = a Reflexive Property of EqualityIf a = b, then b = a
Symmetric Property of EqualityIf a = b and b = c, then a = c
Transitive Property of Equality
More properties of equalitySubstitution Property:If a = b, then
b can replace a in any expression
The Distributive Property:a(b + c) = ab + bc Properties of
congruenceReflexive Property: , A A
Symmetric Property: If , then If A B, then B A
Transitive Property: If and , then If A B and B C, then A C
Using Properties of equality and congruenceName the property
that justifies each statement.
If x = y and y + 4 = 3x, then x + 4 = 3x
If x + 4 = 3x, then 4 = 2x
If
Justify each step of solving the following problem2 (3x + 7) =
26Two-Column ProofDisplays steps that prove a statementStatements
on left; Reasons/justifications on rightUse 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.. .. .. .Dont forgetVertical
Angles are congruent
Also dont forgetAngles that form a line (straight angle) add up
to 180
Solve for x. Justify each step.
6x-40
4x
Solve for x. Justify each step.
4x+602xExtra examples, if necessary. Find the value of the
variables. Justify each step. 1. 2.
126(3x)(6x-54)(8t)(10t)TRANSVERSAL:Line intersects 2 lines in 2
distinct pointsThe intersection of a transversal and the 2 lines
form 8 angles
Transversal n intersects line l and mThe angles formed when a
transversal intersects 2 lines depends on their positionALTERNATE
INTERIOR ANGLES:Non-adjacentLie on opposite sides of the
transversal in between the 2 lines it intersects
Alternate Exterior AnglesLie 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 AnglesLie outside the 2 lines on same side of
transversal
Corresponding AnglesLie on the same side of the transversalIn
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 anglesIf a transversal
intersects 2 parallel lines:Corresponding angles are
congruentAlternate interior angles are congruentSame side interior
angles are supplementaryAlternate exterior angles are congruentSame
side exterior angles are supplementary
Finding Measures of Angles a
b
c dFind the measure of each angle. Justify your answer.
5073514628Find 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. PostulateConverse 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.
TheoremConverse 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 , then line 1 ll line 2
Write the converse:If a transversal intersects 2 parallel lines,
then same side interior angles are supplementary. TheoremConverse
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 are supplementary, then m ll n
Which lines or segments are parallel? How do you know??1. 2.
bgecHCRAM4545Which segments are parallel?
JKLONMTheorem 3-5If two lines are parallel to the same line,
then they are parallel to each other.
a ll b
* Lines can be coplanar or noncoplanar
TheoremIn 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 Prove: n ll m StatementsReasons
1. 1. 2. 2.3. 3.4. 4. 5. 5.6. n ll m 6.
nm753
