Geometry Section 3-2 1112

SECTION 3-2ANGLES AND PARALLEL LINES

Wednesday, December 14, 2011

ESSENTIAL QUESTIONS

HOW DO YOU USE THEOREMS TO DETERMINE THE RELATIONSHIPS BETWEEN SPECIFIC PAIRS OF ANGLES?

HOW DO YOU USE ALGEBRA TO FIND ANGLE MEASUREMENTS?


POSTULATES AND THEOREMS

CORRESPONDING ANGLES POSTULATE




IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF

CORRESPONDING ANGLES IS CONGRUENT.





CORRESPONDING ANGLES IS CONGRUENT.

∠1 ≅ ∠5



ALTERNATE INTERIOR ANGLES THEOREM





ALTERNATE INTERIOR ANGLES IS CONGRUENT.





ALTERNATE INTERIOR ANGLES IS CONGRUENT.

∠4 ≅ ∠6



CONSECUTIVE INTERIOR ANGLES THEOREM




IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF CONSECUTIVE INTERIOR ANGLES IS

SUPPLEMENTARY.




IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF CONSECUTIVE INTERIOR ANGLES IS

SUPPLEMENTARY.

∠4 AND ∠5 ARE SUPPLEMENTARY



ALTERNATE EXTERIOR ANGLES THEOREM




IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF ALTERNATE EXTERIOR ANGLES IS

CONGRUENT.




IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF ALTERNATE EXTERIOR ANGLES IS

CONGRUENT.

∠2 ≅ ∠7


EXAMPLE 1

IN THE FIGURE, m∠4 = 51°. FIND THE MEASURE OF EACH ANGLE. GIVE A JUSTIFICATION TO

YOUR ANSWER.a. m∠2

b. m∠3

c. m∠6


EXAMPLE 1



51°; VERTICAL ANGLESWITH ∠4

b. m∠3

c. m∠6


EXAMPLE 1




b. m∠3129°; SUPPLEMENTARY ANGLES WITH ∠4

c. m∠6


EXAMPLE 1




b. m∠3129°; SUPPLEMENTARY ANGLES WITH ∠4

c. m∠651°; ALTERNATE INTERIOR ANGLES WITH ∠4


EXAMPLE 2

USE THE FIGURE, IN WHICH a||b, m∠2 = 125°, AND c||d||e, TO FIND THE MEASURE OF EACH NUMBERED ANGLE. PROVIDE A REASON FOR

THE ANSWER FOR EACH MEASURE.


EXAMPLE 2



m∠1 = 125°; VERTICAL

WITH ∠2


EXAMPLE 2




WITH ∠2m∠3 = 55°; CONSECUTIVE

INTERIOR WITH ∠2


EXAMPLE 2





INTERIOR WITH ∠2m∠4 = 125°; CONSECUTIVE

INTERIOR WITH ∠3


EXAMPLE 2





INTERIOR WITH ∠2m∠4 = 125°; CONSECUTIVE

INTERIOR WITH ∠3

m∠5 = 55°; SUPPLEMENTARY WITH ∠4


EXAMPLE 3

USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING.

a. IF m∠2 = 2x − 10 AND

m∠6 = x + 15, FIND x.


EXAMPLE 3


a. IF m∠2 = 2x − 10 AND

m∠6 = x + 15, FIND x.

2x − 10 = x + 15


EXAMPLE 3


a. IF m∠2 = 2x − 10 AND

m∠6 = x + 15, FIND x.

2x − 10 = x + 15−x −x


EXAMPLE 3


a. IF m∠2 = 2x − 10 AND

m∠6 = x + 15, FIND x.

2x − 10 = x + 15−x −x +10+10


EXAMPLE 3


a. IF m∠2 = 2x − 10 AND

m∠6 = x + 15, FIND x.

2x − 10 = x + 15−x −x +10+10

x = 25


EXAMPLE 3


a. IF m∠2 = 2x − 10 AND

m∠6 = x + 15, FIND x.

2x − 10 = x + 15−x −x +10+10

x = 25

SINCE THE ANGLES ARE CORRESPONDING, THEY ARE CONGRUENT, MEANING THEY ARE

EQUAL.


EXAMPLE 3


b. IF m∠7 = 4(y − 25) AND

m∠1 = 4y, FIND y.


EXAMPLE 3


b. IF m∠7 = 4(y − 25) AND

m∠1 = 4y, FIND y.

4(y − 25) + 4y = 180


EXAMPLE 3


b. IF m∠7 = 4(y − 25) AND

m∠1 = 4y, FIND y.

4(y − 25) + 4y = 1804y − 100 + 4y = 180


EXAMPLE 3


b. IF m∠7 = 4(y − 25) AND

m∠1 = 4y, FIND y.

4(y − 25) + 4y = 1804y − 100 + 4y = 180

8y − 100 = 180


EXAMPLE 3


b. IF m∠7 = 4(y − 25) AND

m∠1 = 4y, FIND y.

4(y − 25) + 4y = 1804y − 100 + 4y = 180

8y − 100 = 1808y = 280


EXAMPLE 3


b. IF m∠7 = 4(y − 25) AND

m∠1 = 4y, FIND y.

4(y − 25) + 4y = 1804y − 100 + 4y = 180

8y − 100 = 1808y = 280

y = 35


CHECK YOUR UNDERSTANDING

CHECK OUT PROBLEMS 1-10 ON PAGE 181.


PROBLEM SET


PROBLEM SET

P. 181 #11-35 ODD

“YOU CAN FOOL TOO MANY OF THE PEOPLE TOO MUCH OF THE TIME.” - JAMES THURBER


Geometry Section 3-2 1112

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