Transcript 3.3 Parallel Lines & Transversals

3.3 Parallel Lines
& Transversals
Ms. Reser
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
Homework
• Pgs. 146-148 #1-30
• REMINDER: There is a quiz after 3.3. If
you want a glance at what it kind of looks
like, check out pg. 149. You will be doing
this for homework next class meeting.
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
• 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
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
NOTE:
• You must show all your work. Check your
syllabus . . . it tells you everything I expect.
We are moving into the next quarter
shortly, and I expect that your work will be
even more professional, neat, organized,
and will show even at a casual glance that
you did your homework. IF IT EVEN
LOOKS COPIED . . . NO CREDIT!!!!