Download Properties of Parallel Lines

Warm-up 3.1
3.1 Properties of Parallel
Lines
Mrs. Gonzales
Standards/Objectives:
Standard 3: Students will learn and apply
geometric concepts.
Objectives:
• Prove and use results about parallel lines
and transversals.
• Use properties of parallel lines to solve
real-life problems, such as estimating the
Earth’s circumference
Definitions
• A transversal is a line that intersects two
other coplanar lines at two distinct points.
Postulate 15 Corresponding Angles
Postulate
• If two parallel lines are cut by a
transversal, then the pairs of
corresponding angles are congruent.
1
2
1 ≅ 2
Theorem 3.4 Alternate Interior
Angles
• If two parallel lines are cut by a
transversal, then the pairs of alternate
interior angles are congruent.
3
4
3 ≅ 4
Theorem 3.5 Consecutive Interior
Angles (Same-side)
• If two parallel lines are cut by a
transversal, then the pairs of consecutive
interior angles are supplementary.
5
6
5 + 6 = 180°
Theorem 3.6 Alternate Exterior
Angles
• If two parallel lines are cut by a
transversal, then the pairs of alternate
exterior angles are congruent.
7
8
7 ≅ 8
Theorem 3.7 Perpendicular
Transversal
• If a transversal is perpendicular to one of
the two parallel lines, then it is
perpendicular to the other.
j
h
k
jk
Two – column proof
• Given: a||b
• Prove:
• Statements
Diagram
Reasons
Example 1: Proving the Alternate
Interior Angles Theorem
• Given: p ║ q
• Prove: 1 ≅ 2
1
2
3
Proof
Statements:
1. p ║ q
2. 1 ≅ 3
3. 3 ≅ 2
4. 1 ≅ 2
Reasons:
1. Given
2. Corresponding
Angles Postulate
3. Vertical Angles
Theorem
4. Transitive Property
of Congruence
Example 2: Using properties of
parallel lines
• Given that m 5 = 65°, find each measure.
Tell which postulate or theorem you use.
• A. m 6
B. m 7
• C. m 8
D. m 9
9
6
5
7
8
Solutions:
a. m 6 = m 5 = 65°
•
Vertical Angles Theorem
b. m 7 = 180° - m 5 =115°
•
Linear Pair postulate
c. m 8 = m 5 = 65°
•
Corresponding Angles Postulate
d. m 9 = m 7 = 115°
•
Alternate Exterior Angles Theorem
Ex. 3—Classifying Leaves
BOTANY—Some plants are classified by the
arrangement of the veins in their leaves.
In the diagram below, j ║ k. What is m 1?
j
k
120° 1
Solution
1. m 1 + 120° = 180°
2. m 1 = 60°
1. Consecutive Interior
angles Theorem
2. Subtraction POE
Ex. 4: Using properties of parallel
lines
• Use the properties of parallel lines to find
the value of x.
125°
4
(x + 15)°
Proof
Statements:
1. m4 = 125°
2. m4 +(x+15)°=180°
3. 125°+(x+15)°= 180°
4. x = 40°
Reasons:
1. Corresponding
Angles Postulate
2. Linear Pair Postulate
3. Substitution POE
4. Subtraction POE
Algebra to Find Angle Measures
• Find the values of x and y.
50°
x°
y°
70°
Find the Values
• Find the values and then find the angle
measures.
2x°
y°
Y-50°
Properties of Parallel Lines
LESSON 3-1
Additional Examples
Use the diagram above. Identify which angle forms a pair of same-side
interior angles with 1. Identify which angle forms a pair of corresponding
angles with 1.
Same-side interior angles are on the same side of transversal t
between lines p and q.
4, 8, and 5 are on the same side of the transversal as
but only 1 and 8 are interior.
So
1 and
8 are same-side interior angles.
1,
Properties of Parallel Lines
LESSON 3-1
Additional Examples
(continued)
Corresponding angles also lie on the same side of the transversal.
One angle must be an interior angle, and the other must be an exterior angle.
The angle corresponding to 1 must lie in the same position relative to
line q as 1 lies relative to line p. Because 1 is an interior angle, 1 and
are corresponding angles.
5
Quick Check
Properties of Parallel Lines
LESSON 3-1
Additional Examples
Compare 2 and the vertical angle of 1. Classify the angles
as alternate interior angles, same-side interior angles, or
corresponding angles.
The vertical angle of
1 is between the parallel runway segments.
2 is between the runway segments and on the opposite side of
the transversal runway.
Because alternate interior angles are not adjacent and lie between
the lines on opposite sides of the transversal, 2 and the vertical angle
of 1 are alternate interior angles.
Quick Check
Properties of Parallel Lines
LESSON 3-1
Additional Examples
Use the given that a b and the diagram to write a two-column proof
that 1 and 4 are supplementary.
Statements
Reasons
1. a  b
2. m 1 = m 3
1. Given
2. Corresponding Angles Postulate
3. m 3 + m 4 = 180
3. Angle Addition Postulate
4. m 1 + m 4 = 180
4. Substitution
5. 1 and 4 are
supplementary
5. Definition of supplementary angles
Quick Check
Properties of Parallel Lines
LESSON 3-1
Additional Examples
In the diagram above,
|| m. Find m 1 and then m
2.
1 and the 42° angle are corresponding angles. Because
m 1 = 42 by the Corresponding Angles Postulate.
|| m,
Because 1 and 2 are adjacent angles that form a straight angle,
m 1 + m 2 = 180 by the Angle Addition Postulate.
If you substitute 42 for m 1, the equation becomes 42 + m
Subtract 42 from each side to find m 2 = 138.
2 = 180.
Quick Check
Properties of Parallel Lines
LESSON 3-1
Additional Examples
In the diagram above,
|| m. Find the values of a, b, and c.
a = 65
Alternate Interior Angles Theorem
c = 40
Alternate Interior Angles Theorem
a + b + c = 180
65 + b + 40 = 180
b = 75
Angle Addition Postulate
Substitution Property of Equality
Subtraction Property of Equality
Quick Check
Assignment
• Pg. 118 5-8, 10-16, 26, 45-50