3.3 Prove Lines are Parallel

Transcription

1 Warm-up! Turn in your proof to me and pick up a different one, grade it on our 5 point scale! If it is not a 5 write on the paper what they need to do to improve it. Return to the proof writer! 1

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3 3.3 Prove Lines are Parallel 3

4 So, far we have discussed that if we have a pair of parallel lines, then certain pairs of angles created by a transversal are congruent or supplementary. Now we will consider the converse of these statements. REMEMBER: the converse of our conditional statement does NOT always have to be true, so each converse of a theorem must be proved true 4

5 Postulate 15: Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent

6 Postulate 16: Corresponding Angles Converse If two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel

7 Example What value of x makes m n? (15x+15) 75 7

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9 Theorem 3.1: Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent

10 Theorem 3.4: Alternate Interior Angles Converse If two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel

11 Prove: Alternate Interior Angles Converse Given: 4 5 Prove: g h g h Statements and 4 are vert. angles g h Reasons 1. Given 2. Defn Vert. Angles 3. Vertical Angle 4. Transitive Prop. of 1,3 5. Corresponding Angles Converse 11

12 Theorem 3.2: Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent

13 Theorem 3.5: Alternate Exterior Angles Converse If two lines are cut by a transversal so the alternate exterior angles are congruent, then the lines are parallel

14 Prove: Alternate Exterior Angles Converse Homework!

15 Theorem 3.3: Consecutive Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. m 4+m 6=

16 Theorem 3.6: Consecutive Interior Angles Converse If two lines are cut by a transversal so the consecutive interior angles are supplementary, then the lines are parallel. 6 4 m 4+m 6=180 16

17 Prove: Consecutive Interior Angles Converse Homework!! 6 4 m 4+m 6=180 17

18 Explain why 1 and 2 and are not corresponding angles with respect to any pair of lines and transversal. There is NO transitive property of corresponding angles. Only congruence and equality 18

19 Paragraph Proof! The statements and reasons in a paragraph proof are written in sentences, using words to explain the logical flow of the arguments. 19

20 Prove: Alternate Interior Angles Converse Given: 4 5 Prove: g h g h Statements and 4 are vert. angles g h Reasons 1. Given 2. Defn Vert. Angles 3. Vertical Angle 4. Transitive Prop. of 1,3 5. Corresponding Angles Converse We are given that angle 4 is congruent to angle 5. By the definition of vertical angles angle 1 and angle 4 are vertical angles. By the vertical angle theorem angle 1 is congruent to angle 4. By the transitive property of congruency angle 1 is congruent to angle 5. Therefore, by the corresponding angles converse we can conclude g is parallel to h. 20

21 Prove: Using a paragraph proof Given: 3 6 Prove: l m 3 6 We are given that angle 3 and angle 6 are congruent. 7 By definition of vertical angles angle 6 is vertical to angle l 7. So, angle 6 is congruent to angle 7 by vertical angle m theorem. Angle 3 is congruent to angle 7 by substitution. By definition angle 3 and 7 are corresponding. By the corresponding angles CONVERSE l is parallel to m It is given that 3 6. Then 6 7 by the Vertical Angle Congruence Theorem. So, 3 7 by the Transitive Prop. of Congruence for angles. Thus, l m by the corresponding Angles Converse. 21

22 Theorem 3.7: Transitive Property of If two lines are parallel to the same line, then they are parallel to each other. Parallel Lines If a b and b c, then a c. a b c 22

23 Types of angles Properties of Parallel lines 23

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25 D 60 H A E Find Angle ABC B F 65 I C G 25

26 D H A E Find Angle ABC J F B I C K G DHB HBK m DHB = m HBK BIF IBK m BIF = m IBK m ABC = m HBK + m IBK m ABC = substitution =

27 Homework Pg 165: 4 19, 22 24, do on a separate piece of paper! 27