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.Hypothesis – statement p or the “if part” in a conditional. 2.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.

Identify the hypothesis and conclusion of the following statement. If a polygon has 6 sides, then it is a hexagon. Answer: Hypothesis: a polygon has 6 sides Conclusion: it is a hexagon hypothesis conclusion If a polygon has 6 sides, then it is a hexagon. Example 1a:

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 Identify the hypothesis and conclusion of the following statement. Example 1b:

Identify the hypothesis and conclusion of each statement. a. If you are a baby, then you will cry. b. If you want to find the distance between two points, then you can use the Distance Formula. Answer: Hypothesis: you are a baby Conclusion: you will cry Answer: Hypothesis: you want to find the distance between two points Conclusion: you can use the Distance Formula Your Turn:

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: 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

Write the converse. Conditional: If a shape is a square, then it is a rectangle. Write the converse by switching the hypothesis and conclusion of the conditional. Converse: If a shape is a rectangle, then it is a square. Example 2: The conditional statement is true. The converse is false. The converse of a true conditional statement may or may not be true.

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. Your Turn:

Write the converse of the true statement above. a. Determine whether the converse is true. b. Statement: If two segments are congruent, then the two segments have the same length. SOLUTION a. Converse: If two segments have the same length, then the two segments are congruent. b. The converse is a true statement. Example 3:

ANSWER 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°. 2. If two angles have the same measure, then the two angles are congruent. 1. Write the converse of the true statement. Then determine whether the converse is true. ANSWER If m  3 + m  4 = 90°, then  3 and  4 are complementary; true Your Turn:

ANSWER If  1   2, then  1 and  2 are right angles; false If  1 and  2 are right angles, then  1   Write the converse of the true statement. Then determine whether the converse is true. Your Turn:

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”

State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m  A + m  B = 90°, then  A and  B are complementary. 3. If AB + BC = AC, then A, B, and C are collinear. If a + c = b + c, then a = b. If  A and  B are complementary, then m  A + m  B =90°. If A, B, and C are collinear, then AB + BC = AC. Practice:

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 ║. j k If ∠ 1 ≅∠ 2,then j ll k Example: 1 2

Use the Corresponding Angles Converse Postulate and the given information to show that ℓ || m. Example 4: Using the Corresponding Angles Converse Postulate Given:  4   8  4   8  4 and  8 are corresponding angles. ℓ || m Corr.  s Conv. Post.

Use the Corresponding Angles Converse Postulate and the given information to show that ℓ || m. Example 5: Using the Corresponding Angles Converse Postulate m  3 = (4x – 80)°, m  7 = (3x – 50)°, x = 30 m  3 = 4(30) – 80 = 40Substitute 30 for x. m  8 = 3(30) – 50 = 40Substitute 30 for x. ℓ || m Conv. of Corr.  s Post.  3   8Def. of   s. m  3 = m  8Trans. Prop. of Equality

Your Turn Use the Corresponding Angles Converse Postulate and the given information to show that ℓ || m. Given; m  1 = m  3  1   3  1 and  3 are corresponding angles. ℓ || m 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 = 77Substitute 13 for x. m  5 = 5(13) + 12 = 77Substitute 13 for x. ℓ || m Corr.  s Conv. Post.  7   5 Def. of   s. m  7 = m  5Trans. Prop. of Equality

a. Is enough information given to conclude that BD  EG ? Explain. 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. Example 6:

b. No. You are not given any information about the angles formed where EG intersects CG. 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. b. SOLUTION c. SOLUTION Is enough information given to conclude that BD  EG ? Explain.

ANSWER Yes. Two angles are corresponding and congruent. By the Corresponding Angles Converse, the lines are parallel. Is enough information given to conclude that RT  XZ ? Explain. 1. Your Turn:

2. Is enough information given to conclude that RT  XZ ? Explain. ANSWER No. There is no information given about the angles formed where SY intersects XZ. Your Turn:

3. Is enough information given to conclude that RT  XZ ? Explain. 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. Your Turn:

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 ║. j k Example: 1 3 If ∠ 1 ≅∠ 3,then j ll 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 ║. j k 4 5 Example: 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 Same- Side interior angles is supplementary, then the lines are parallel. Abbreviation: If same-side int.  s are supp., then lines are ║. If m ∠ 1 + m ∠ 2 = 180 ˚, then j ll k j k Example: 1 2

SOLUTION a. Yes. The angle congruence marks on the diagram allow you to conclude that m  n by the Alternate Interior Angles Converse. Does the diagram give enough information to conclude that m  n ? a.b. No. Not enough information is given to conclude that m  n. Example 7:

ANSWER Yes, by the Alternate Exterior Angles Converse 1. Does the diagram give enough information to conclude that c  d ? Explain. Your Turn:

5x° + 115° = 180° Supplementary angles 5x = 65 Subtract 115 from each side. x = 13 Divide each side by 5. ANSWER So, if x = 13, then j  k. SOLUTION Lines j and k are parallel if the marked angles are supplementary. Find the value of x so that j  k. j k n 115  5x5x Example 8:

Find the value of x so that v  w. 1. ANSWER ANSWER 30 ANSWER 68 Your Turn:

Determine which lines, if any, are parallel. consecutive interior angles are supplementary. So, consecutive interior angles are not supplementary. So, c is not parallel to a or b. Answer: Example 9:

Determine which lines, if any, are parallel. Answer: Your Turn:

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:

Alternate exterior angles Subtract 7x from each side. Substitution Add 25 to each side. Divide each side by 4. Solve Example 10:

Answer: Original equation Simplify. Examine Verify the angle measure by using the value of x to find Since Example 10:

ALGEBRA Find x and m  GBA so that Answer: Your Turn:

Ways to Prove 2 Lines Parallel 1.Show that a pair of corresponding angles are congruent. 2.Show that a pair of alternate interior angles are congruent. 3.Show that a pair of alternate exterior angles are congruent. 4.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 : #1-43 odd