Prove Lines are Parallel. p Use angle relationships to prove that lines are parallel. VOCABULARY. Paragraph proof

Size: px

Start display at page:

Download "Prove Lines are Parallel. p Use angle relationships to prove that lines are parallel. VOCABULARY. Paragraph proof"

Delphia Whitehead
6 years ago
Views:

1 Your Notes Prove Lines re Prllel Gol p Use ngle reltionships to prove tht lines re prllel VOCABULARY Prgrph proof POSTULATE 6 CORRESPONDING ANGLES If two lines re cut trnsversl so the corresponding ngles re congruent, then the lines re 6 Emple Find the vlue of tht mes m i n Appl the Corresponding Angles Converse Lines m nd n re prllel if the mred corresponding ngles re congruent 78 ( )8 ( )8 5 Use Postulte 6 to write n eqution 5 Sutrct from ech side 5 Divide ech side The lines m nd n re prllel when 5 m n Checpoint Find the vlue of tht mes i (5 7)8 988 Copright Holt McDougl All rights reserved Lesson Geometr Noteting Guide 69

2 Your Notes Prove Lines re Prllel Gol p Use ngle reltionships to prove tht lines re prllel VOCABULARY Prgrph proof A proof cn e written in prgrph form, clled prgrph proof POSTULATE 6 CORRESPONDING ANGLES If two lines re cut trnsversl so the corresponding ngles re congruent, then the lines re prllel 6 Emple Find the vlue of tht mes m i n Lines m nd n re prllel if the mred corresponding ngles re congruent 78 ( )8 ( ) Use Postulte 6 to write n eqution 5 68 Appl the Corresponding Angles Converse Sutrct from ech side 5 4 Divide ech side The lines m nd n re prllel when 5 4 m n Checpoint Find the vlue of tht mes i (5 7) Copright Holt McDougl All rights reserved Lesson Geometr Noteting Guide 69

3 THEOREM 4 ALTERNATE INTERIOR ANGLES If two lines re cut trnsversl so the lternte interior ngles re congruent, then the lines re 5 4 THEOREM 5 ALTERNATE EXTERIOR ANGLES If two lines re cut trnsversl so the lternte eterior ngles re congruent, then the lines re THEOREM 6 CONSECUTIVE INTERIOR ANGLES If two lines re cut trnsversl so the consecutive interior ngles re supplementr, then the lines re 5 8 Emple Flgs How cn ou tell whether the sides of the flg of Nepl re prllel? Solve rel-world prolem Becuse the re congruent, ou now tht the sides of the flg re Checpoint Cn ou prove tht lines nd re prllel? Eplin wh or wh not m m Lesson Geometr Noteting Guide Copright Holt McDougl All rights reserved

4 THEOREM 4 ALTERNATE INTERIOR ANGLES If two lines re cut trnsversl so the lternte interior ngles re congruent, then the lines re prllel 5 4 THEOREM 5 ALTERNATE EXTERIOR ANGLES If two lines re cut trnsversl so the lternte eterior ngles re congruent, then the lines re prllel THEOREM 6 CONSECUTIVE INTERIOR ANGLES If two lines re cut trnsversl so the consecutive interior ngles re supplementr, then the lines re prllel 5 8 Emple Flgs How cn ou tell whether the sides of the flg of Nepl re prllel? Solve rel-world prolem Becuse the lternte interior ngles re congruent, ou now tht the sides of the flg re prllel Checpoint Cn ou prove tht lines nd re prllel? Eplin wh or wh not m m Yes, ou cn use the Consecutive Interior Angles Converse to prove i 70 Lesson Geometr Noteting Guide Copright Holt McDougl All rights reserved

5 Emple In the figure, i nd is congruent to Prove i Write prgrph proof Loo t the digrm to me pln The digrm suggests tht ou loo t ngles,, nd Also, ou m find it helpful to focus on one pir of lines nd one trnsversl t time Pln for Proof Loo t nd Loo t nd In prgrph proofs, trnsitionl words such s so, then, nd therefore help to me the logic cler ecuse i Pln in Action It is given tht i, so the, > If > then It is lso given tht > Then the Trnsitive Propert of Congruence for ngles Therefore, the, i Checpoint Complete the following eercise In Emple, suppose it is given tht > nd i Complete the following prgrph proof showing tht i It is given tht i B the Eterior Angles Postulte, It is lso given tht > Then the Trnsitive Propert of Congruence for ngles Therefore, the, i Copright Holt McDougl All rights reserved Lesson Geometr Noteting Guide 7

6 Emple In the figure, i nd is congruent to Prove i Write prgrph proof Loo t the digrm to me pln The digrm suggests tht ou loo t ngles,, nd Also, ou m find it helpful to focus on one pir of lines nd one trnsversl t time Pln for Proof Loo t nd Loo t nd In prgrph proofs, trnsitionl words such s so, then, nd therefore help to me the logic cler > ecuse i If > then i Pln in Action It is given tht i, so the Corresponding Angles Postulte, > It is lso given tht > Then > the Trnsitive Propert of Congruence for ngles Therefore, the Alternte Eterior Angles Converse, i Checpoint Complete the following eercise In Emple, suppose it is given tht > nd i Complete the following prgrph proof showing tht i It is given tht i B the Eterior Angles Postulte, > It is lso given tht > Then > the Trnsitive Propert of Congruence for ngles Therefore, the Corresponding Angles Converse, i Copright Holt McDougl All rights reserved Lesson Geometr Noteting Guide 7

7 THEOREM 7 TRANSITIVE PROPERTY OF PARALLEL LINES If two lines re prllel to the sme line, then the re to ech other p q r Emple 4 Utilit poles Ech utilit pole shown is prllel to the pole immeditel to its right Eplin wh the leftmost pole is prllel to the rightmost pole Use the Trnsitive Propert of Prllel Lines t t t t 4 t 5 t 6 When ou nme severl similr items, ou cn use one vrile with suscripts to eep trc of the items The poles from left to right cn e nmed t, t, t,, t 6 Ech pole is prllel to the one to its right, so t i, t i, nd so on Then t i t the Similrl, ecuse t i t 4, it follows tht t i B continuing this resoning, t i So, the leftmost pole is prllel to the rightmost pole Checpoint Complete the following eercise 4 Ech horizontl piece of the s window linds shown is clled slt Ech slt is prllel to the slt immeditel elow it Eplin wh the top slt is prllel to the ottom slt s 6 Homewor 7 Lesson Geometr Noteting Guide Copright Holt McDougl All rights reserved

8 THEOREM 7 TRANSITIVE PROPERTY OF PARALLEL LINES If two lines re prllel to the sme line, then the re prllel to ech other p q r Emple 4 Utilit poles Ech utilit pole shown is prllel to the pole immeditel to its right Eplin wh the leftmost pole is prllel to the rightmost pole Use the Trnsitive Propert of Prllel Lines t t t t 4 t 5 t 6 When ou nme severl similr items, ou cn use one vrile with suscripts to eep trc of the items The poles from left to right cn e nmed t, t, t,, t 6 Ech pole is prllel to the one to its right, so t i t, t i t, nd so on Then t i t the Trnsitive Propert of Prllel Lines Similrl, ecuse t i t 4, it follows tht t i t 4 B continuing this resoning, t i t 6 So, the leftmost pole is prllel to the rightmost pole Checpoint Complete the following eercise Homewor 4 Ech horizontl piece of the s window linds shown is clled slt Ech slt is prllel to the slt immeditel elow it Eplin wh the top slt is prllel to the ottom slt The slts from top to ottom cn e nmed s, s, s,, s 6 Ech slt is prllel to the one elow it, so s i s, s i s, nd so on Then s i s the Trnsitive Propert of Prllel Lines Similrl, ecuse s i s 4, it follows tht s i s 4 B continuing this resoning, s i s 6 So, the top slt is prllel to the ottom slt s 6 7 Lesson Geometr Noteting Guide Copright Holt McDougl All rights reserved

Name Class Date. Line AB is parallel to line CD. skew. ABDC } plane EFHG. In Exercises 4 7, use the diagram to name each of the following.

Name Class Date. Line AB is parallel to line CD. skew. ABDC } plane EFHG. In Exercises 4 7, use the diagram to name each of the following. Reteching Lines nd Angles Not ll lines nd plnes intersect. prllel plnes. prllel. } shows tht lines or plnes re prllel: < > < > A } ens Line A is prllel to line. skew. A } plne EFHG A plne FH } plne AEG