Transcript Proving Lines Parallel

Showing Lines are Parallel
3.5
Objectives
• Show that two lines are parallel.
Key Vocabulary
• Conditional Statement
• Converse
• Hypothesis
• Conclusion
Postulates
• 9 Corresponding Angles Converse
Theorems
• 3.8 Alternate Interior Angles Converse
• 3.9 Alternate Exterior Angles Converse
• 3.10 Same-Side Interior Angles Converse
Conditional Statement
A conditional statement is a statement that can be written in if-then form.
Example: If an animal has hair, then it is a mammal.
Conditional statements are always written “if p then q.”
The conditional “if p then q” is made up of two parts;
1.
2.
Hypothesis – statement p or the “if part” in a conditional.
Conclusion – statement q or the “then part” a conditional.
Conditional Statement
1. Given the conditional, “If John is not at work then John is sick.”
Identify the hypothesis and the conclusion.
−
−
Hypothesis: John is not at work.
Conclusion: John is sick.
2. Given the conditional, “If a number is divisible by four then it is
even.” Identify the hypothesis and the conclusion.
–
–
Hypothesis: A number is divisible by four.
Conclusion: A number is even.
Example 1a:
Identify the hypothesis and conclusion of the following statement.
If a polygon has 6 sides, then it is a hexagon.
If a polygon has 6 sides, then it is a hexagon.
hypothesis
Answer: Hypothesis: a polygon has 6 sides
Conclusion: it is a hexagon
conclusion
Example 1b:
Identify the hypothesis and conclusion of the following statement.
If Tamika completes the maze in her computer game, then she will advance to
the next level of play .
Answer: Hypothesis: Tamika completes the maze in her computer game
Conclusion: she will advance to the next level of play
Your Turn:
Identify the hypothesis and conclusion of each statement.
a. If you are a baby, then you will cry.
Answer: Hypothesis: you are a baby
Conclusion: you will cry
b. If you want to find the distance between two points, then you can use the
Distance Formula.
Answer: Hypothesis: you want to find the distance between two points
Conclusion: you can use the Distance Formula
Converse
• From a conditional we can also create additional statements
referred to as related conditionals. These include the converse.
• Given the conditional “if p then q”;
– Converse is “if q then p” (reverse the hypothesis and the conclusion).
Conditional and Converse
Conditional and Converse
• Conditional: If a quadrilateral is a rectangle then a
quadrilateral is a parallelogram.
• Find the converse.
• Converse; If a quadrilateral is a parallelogram then a
quadrilateral is a rectangle.
13
Example 2:
Write the converse.
Conditional: If a shape is a square, then it is a rectangle.
The conditional statement is true.
Write the converse by switching the hypothesis and conclusion of the
conditional.
Converse: If a shape is a rectangle, then it is a square.
The converse is false.
The converse of a true conditional statement may
or may not be true.
Your Turn:
Write the converse of the conditional; “The sum of the measures of two
complementary angles is 90.” Determine whether each statement is true or false.
If a statement is false, give a counterexample.
Answer: Conditional: If two angles are complementary, then the sum of their
measures is 90; true.
Converse: If the sum of the measures of two angles is 90, then they
are complementary; true.
Example 3:
Statement: If two segments are congruent, then the
two segments have the same length.
a.
Write the converse of the true statement above.
b.
Determine whether the converse is true.
SOLUTION
a.
Converse: If two segments have the same length,
then the two segments are congruent.
b.
The converse is a true statement.
Your Turn:
Write the converse of the true statement. Then
determine whether the converse is true.
1.
If two angles have the same measure, then the
two angles are congruent.
ANSWER
2.
If two angles are congruent, then the two
angles have the same measure; true
If 3 and 4 are complementary, then m3 + m4 = 90°.
ANSWER
If m3 + m4 = 90°, then 3 and 4 are
complementary; true
Your Turn:
Write the converse of the true statement. Then
determine whether the converse is true.
3.
If 1 and 2 are right angles, then 1  2.
ANSWER
If 1  2, then 1 and 2 are right
angles; false
Review
• Recall that the converse of a theorem is found by exchanging the
hypothesis and conclusion. The converse of a theorem is not
automatically true. If it is true, it must be stated as a postulate or
proved as a separate theorem.
• Conditional: “if p then q” ⇒ Converse: “if q then p”
Practice:
State the converse of each statement.
1. If a = b, then a + c = b + c.
If a + c = b + c, then a = b.
2. If mA + mB = 90°, then A and B are
complementary.
If A and  B are complementary, then mA + mB =90°.
3. If AB + BC = AC, then A, B, and C are collinear.
If A, B, and C are collinear, then AB + BC = AC.
Postulate 9
Corresponding Angles Converse
If two lines in a plane are cut by a transversal so that corresponding
angles are congruent, then the lines are parallel.
Abbreviation: If corr. s are , then lines are ║.
Example:
1
j
2
k
If ∠1≅∠2,then j ll k
Example 4: Using the Corresponding
Angles Converse Postulate
Use the Corresponding Angles Converse Postulate and the given information to
show that ℓ || m.
Given: 4  8
4  8
ℓ || m
4 and 8 are corresponding angles.
Corr. s Conv. Post.
Example 5: Using the Corresponding
Angles Converse Postulate
Use the Corresponding Angles Converse Postulate and the given information to
show that ℓ || m.
m3 = (4x – 80)°,
m7 = (3x – 50)°, x = 30
m3 = 4(30) – 80 = 40
m8 = 3(30) – 50 = 40
m3 = m8
3  8
ℓ || m
Substitute 30 for x.
Substitute 30 for x.
Trans. Prop. of Equality
Def. of  s.
Conv. of Corr. s Post.
Your Turn
Use the Corresponding Angles Converse Postulate and the given information to
show that ℓ || m.
Given; m1 = m3
1  3
ℓ || m
1 and 3 are
corresponding angles.
Conv. of Corr. s Post.
Your Turn
Use the Corresponding Angles Converse Postulate and the given information to
show that ℓ || m.
m7 = (4x + 25)°,
m5 = (5x + 12)°, x = 13
m7 = 4(13) + 25 = 77
m5 = 5(13) + 12 = 77
m7 = m5
7  5
ℓ || m
Substitute 13 for x.
Substitute 13 for x.
Trans. Prop. of Equality
Def. of  s.
Corr. s Conv. Post.
Example 6:
Is enough information given to conclude that BD  EG?
Explain.
a.
SOLUTION
a.
Yes. The two marked angles are corresponding
and congruent. There is enough information to use
the Corresponding Angles Converse to conclude
that BD  EG.
Is enough information given to conclude that BD  EG?
Explain.
b.
c.
SOLUTION
b.
No. You are not given any information about the
angles formed where EG intersects CG.
SOLUTION
c.
Yes. You can conclude that mEFC = 100º. So,
there is enough information to use the
Corresponding Angles Converse to conclude
that BD  EG.
Your Turn:
Is enough information given to conclude that RT  XZ?
Explain.
1.
ANSWER
Yes. Two angles are corresponding and
congruent. By the Corresponding Angles
Converse, the lines are parallel.
Your Turn:
Is enough information given to conclude that RT  XZ?
Explain.
2.
ANSWER
No. There is no information given about
the angles formed where SY intersects
XZ.
Your Turn:
Is enough information given to conclude that RT  XZ?
Explain.
3.
ANSWER
Yes. Sample answer: Since RT  SY, all four
angles with vertex S are right angles.
Corresponding angles are both right angles,
and all right angles are congruent. By the
Corresponding Angles Converse, the lines
are parallel.
Theorem 3.8
Alternate Interior Angles Converse
If two lines in a plane are cut by a transversal so that a pair of alternate
interior angles is congruent, then the lines are parallel.
Abbreviation: If alt. int. s are , then lines are ║.
Example:
j
3
1
If ∠1≅∠3,then j ll k
k
Theorem 3.9
Alternate Exterior Angles Converse
If two lines in a plane are cut by a transversal so that a pair of alternate
exterior angles is congruent, then the two lines are parallel.
Abbreviation: If alt ext. s are , then lines are ║.
Example:
4
j
k
5
If ∠4≅∠5, then j ll k
Theorem 3.10
Same-Side Interior Angles Converse
If two lines in a plane are cut by a transversal so that a pair of SameSide interior angles is supplementary, then the lines are parallel.
Abbreviation: If same-side int. s are supp., then lines are ║.
Example:
j
2
1
If m∠1 + m∠2 = 180˚, then j ll k
k
Example 7:
Does the diagram give enough information to conclude
that m  n?
a.
b.
SOLUTION
a.
Yes. The angle congruence marks on the diagram
allow you to conclude that m  n by the Alternate
Interior Angles Converse.
b.
No. Not enough information is given to conclude
that m  n.
Your Turn:
1.
Does the diagram give enough information to
conclude that c  d? Explain.
ANSWER
Yes, by the Alternate Exterior Angles
Converse
Example 8:
Find the value of x so that j  k.
115
5x
j
SOLUTION
k
Lines j and k are parallel if the
marked angles are supplementary.
5x° + 115° = 180°
5x = 65
x = 13
ANSWER
Supplementary angles
Subtract 115 from each side.
Divide each side by 5.
So, if x = 13, then j  k.
n
Your Turn:
Find the value of x so that v  w.
1.
ANSWER
55
ANSWER
30
ANSWER
68
2.
3.
Example 9:
Determine which lines,
if any, are parallel.
consecutive interior angles are supplementary.
So,
consecutive
is not parallel to a or b.
Answer:
interior angles are not supplementary. So, c
Your Turn:
Determine which lines, if any, are parallel.
Answer:
Example 10:
ALGEBRA Find x and mZYN so that
Explore From the figure, you know that
and
You also know that
are alternate exterior angles.
Example 10:
Solve
Alternate exterior angles
Substitution
Subtract 7x from each side.
Add 25 to each side.
Divide each side by 4.
Example 10:
Original equation
Simplify.
Examine Verify the angle measure by using the value of x to
Since
Answer:
find
Your Turn:
ALGEBRA Find x and mGBA so that
Answer:
Ways to Prove 2 Lines Parallel
1.
2.
3.
4.
Show that a pair of corresponding angles are congruent.
Show that a pair of alternate interior angles are congruent.
Show that a pair of alternate exterior angles are congruent.
Show that a pair of same-side interior angles are
supplementary.
Joke Time
• What’s a cow’s favorite painting?
• The Moona Lisa
• What does the tooth fairy give for half a tooth?
• Nothing. She wants the tooth, the whole tooth, and nothing but the tooth!
• What do you get if you take a native Alaskan and divide its circumference by
its diameter?
• Eskimo pi
Assignment
• Section 3.5, pg. 139-142: #1-43 odd