Module 4: Lines and Angles Date: 4.2 Transversals and Parallel Lines __________________: a line that intersects 2 coplanar lines at 2 distinct points

Vertical Angles definition vs. Vertical Angles Theorem

Proof of Vertical Angles Theorem

Recall: Linear Pair definition vs. Linear Pair Theorem

Ex.

Name all pairs of Corresponding Angles:

Name all pairs of Same-Side Interior Angles:

Name all pairs of Alternate Interior Angles:

Name all pairs of Alternate Exterior Angles:

Name all pairs of Same-Side Exterior Angles:

Name all the linear pairs:

Name all pairs of vertical angles:

1) ∠3 𝑎𝑛𝑑 ∠6 are in the interior of 𝑚 and 𝑛. Name 2 more pairs.

2) Name two pairs of exterior angles.

3) ∠4 𝑎𝑛𝑑 ∠6 are on different sides of the transversal. Name 2 more pairs. These angles are alternate.

4) Name 2 pairs of angles on the same side of 𝑡. These angles are consecutive.

𝑚 1 2

3 4

7

5 6

8

𝑡

𝑛

dhdkeytolb. AH

10/14/16

Transversal

2L 's whosesidesfam If Verticals,

Given

:CHlaare

apamsofoprrays thenanqesautpwerwgtpgs,HELIE.ie#?Y0r0pp.L'Sformedby3ztsYEPpg;3totheoremintersecting.us#yr*aear*I5EHE4HiBfdsuMsostYin64=416 .

subtractionHDObefore

proof ! Adjacent Lisformedbyopprays ftp.thenanglesare supplementary

detailedidentifies theoremtulsyoumwe

inside the -

( \ - linesL4andL5 ,L4andL3

is outside the -7 Handled ,L2andL8linesI3 and < 5

,L2andL4

* comeback * doafterpnofssamecokr⇒± -

different color 'D svppbmentany ( ymwygeethemassames ;dgL4and<5)<2MdBL1dL5 ,L2

.lt/L4&L7,L3K8L4tL5,L3rL6L44L6/L3aL5L2aL7,L1&L8L1&L7,L2&L8L1&L2/L2&L3/L3&LY/L4&Ll,L5+L6/L6&L8/L8tL7

,LFKS

LIDB ,L2&L4 , L5.lk/L6&L7

Work Together:

x Using the two parallel lines, draw a transversal that intersects them.

x Label the lines 𝑝 and 𝑞 and the transversal 𝑡.

x Label the angles 1 through 8 x Use a protractor to measure each of the

eight angles formed. Record the measures on the diagram.

x Make conjectures about the measures of corresponding angles, alternate interior and exterior angles and same-side interior and exterior angles. Compare your results with those of your classmates.

Conjectures:

Constructing parallel lines:

Which conjecture does this construction illustrate/ prove?

Same-side Interior Angles Postulate

If two parallel lines are cut by a transversal, then the pairs of same-side interior angles are supplementary.

1. Explain how you can find 𝑚∠3 in the postulate diagram if 𝑝 ∥ 𝑞 and 𝑚∠6 = 61°.

2. In the postulate diagram, suppose 𝑝 ∥ 𝑞 and line 𝑡 is perpendicular to line 𝑝. Can you conclude that line 𝑡 is perpendicular to line 𝑞? Explain.

𝑡 𝑝

𝑞

*

Protract

corresponding L 's → ±

8 Akin . L 's → ±

����������� ������������������ !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~��������������������������������� ¡¢£¤¥¦§¨©ª«¬­®¯°±²³´µ¶·¸¹º»¼½¾¿ÀÁÂÃÄÅÆÇÈÉÊËÌÍÎÏÐÑÒÓÔÕÖ×ØÙÚÛÜÝÞßàáâãäåæçèéêëìíîïðñòóôõö÷øùúûüýþÿ3����������� ������������������ !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~��������������������������������� ¡¢£¤¥¦§¨©ª«¬­®¯°±²³´µ¶·¸¹º»¼½¾¿ÀÁÂÃÄÅÆÇÈÉÊËÌÍÎÏÐÑÒÓÔÕÖ×ØÙÚÛÜÝÞßàáâãäåæçèéêëìíîïðñòóôõö÷øùúûüýþÿ Same side interior L 's → 1800

Method 1 : Replicate ants Alternate method : Rhombus method ssidebngths

←* ¥¥eresponding L 's theorem (Method 1)← why postulate ?

If p 11 of ,then

L3 and Lb are

supplementary

L 3 and Lb supp (by postulate) So LZ +61=180 L3=÷If ttp ,

then Ll,

< 2,13 ,M are 900

.

L3 a Lb supp . ( psoFt.ws#edDSo9ootL6=18oo

BY vertical L theorem,

L 6%8,

so Lego . 26=900By LP theorem

, L5+L6 supp , So 4+90=180 , 25=90°

By Vert. L 's theorem

, L5=L7,

so a = go ,So tlq also

.QED ! Bam!

Proving that Alternate interior angles are congruent

Alternate Interior Angles Theorem

Given: 𝒑 ∥ 𝒒

Prove: 𝒎∠𝟑 = 𝒎∠𝟓

Statements Reasons

1. 𝑝 ∥ 𝑞 1.

2. ∠3 and ∠6 are same side interior angles

2.

3. ∠3 and ∠6 are supplementary angles

3.

4. ∠3 + ∠6 = 180 4.

5. ∠5 and ∠6 are a linear pair

5.

6. ∠5 + ∠6 are supplementary

6.

7. ∠5 + ∠6 = 180 7.

8. ∠3 + ∠6 = ∠5 + ∠6 8.

9. ∠3 = ∠5 9.

Proving that Corresponding Angles are congruent

Corresponding Angles Theorem Given: Prove:

Practice: Use the figure to find angle measures given 𝑝||𝑞. Justify your answers with a theorem, postulate, definition, etc.

(

Wehavedefsandsamesideintt

's postulate)

givendefsamesideintl 'sIf 2 lines are parallel andcutby

same . side int .L postulateatransvesal, then alternate

interior angles are ] def . supplementary

deferPthlrem

def. supplementarysubstitution

subtraction( canuseakintl 's theorem)

p.1=25

statements Reasons

l.pl/q 1. givenH2 lines are parallel and cut 2

.L1andL3hAH2

. def . Vert.L 'sbyatransversal

,then corresponding 3.4=23 3.vert.is theorem

Angles out. 4 .L3aL5aualt.4.defalt.int .L 's

int .c 's

5.23=25 5.AH.int .li stheorem

6.4=6 6.

transitive

pri )

mL5=98° Samesideint.

Mlb '75° samesideirt.

L 's postulate L 's postulate

MLS :l22° dttinttsthm.

Mlb '76°At

.in#LstheoremmLl=l09oCarespmdirgliSthm.mL2=740correspmdingListhm

.

Use the figure to find angle measures given 𝑚||𝑛 and 𝑥||𝑦. Justify your answers with a theorem, postulate, definition, etc.

)m4O÷69° Altintlsthm mL6=H5° Altihttthm .

1×119,

transversal m xlly

transversal-

Ny ,

mL7=118o AH .int.Listhm

.

L4&L7 Vertical so L7=72°( VEAL 's third transversal

xlfhtansvesaln Htlklsolsamesideinttspostjsouklogo

4+46=1800 Csamesideext.ci#thEansversdL5=40CAltint.isthm)44=46 W I T H 0 V T G E 0 M e T R y KfhitransksalmCoral 's thm < 10=42 CON .L'S

tmlki L If E I s P 0 I N T L E S s theoremtransversal (

mllntransvrsaA

So M42=s6°44=660

ygo , yo ¥1550601-

145 1400

- ) Yoo 550 ,

75,

85° 1400

55° 40750

1250450 85+40

600

)54*4-7Nl / 160

126Cgy

6×-20+2×-40=1800

) soo 63 sfamesideintlypostj7600 63 8×-60=180

4 X=3@30

54 #l3 . 16

160pg 13×+2=15×-6

TO 63Homlisthm )

8=2×180#160+20

All Rights Reserved © MathBits.com “Ah-Bach” Series

For questions 8-12:

a || b _R_ 8. m < 1 _U__10. m < 3 85 140 _M_ 9. m < 2 _H__11. m < 4 55 40

_S__12. m < 1 + m < 4 125 For questions 13-16:

_E___13. the value of x. 4 _L___14. value of 13x + 2 54 _T___15. m < 1 126 _O___16. m < 2 63

Parallel Lines Name___ANSWERS______________ Solve each problem and find the LETTER of the MATCHING answer in the Answer Bank. Decode the secret message by placing the associated letter above the number of the question. W I T H O U T G E O M E T R Y, 5 2 15 11 16 10 15 4 13 16 9 13 15 8 7 L I F E I S P O I N T L E S S! 14 2 6 13 2 12 1 16 2 3 15 14 13 12 12

For questions 1-3: a || b with transversals t and s.

__P_ 1. m < 1 75

__I__ 2. m < 2 45

_N__ 3. m < 3 60 For questions 4-7: a || b with transversal t.

_G_ 4. the value of x 30

_W_ 5. value of 6x – 20 160 _F_ 6. m < y 20 _Y__7. m < z + m < y 180

ANSWER BANK

A – 2 B – 3 E – 4 F – 20 G – 30 H – 40 I – 45 K – 50 L – 54 M – 55 N – 60 O – 63 P – 75 R – 85 S – 125 T – 126 U – 140 W – 160 Y – 180