Geometry Section 3-2 1112
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Transcript of Geometry Section 3-2 1112
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SECTION 3-2ANGLES AND PARALLEL LINES
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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?
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POSTULATES AND THEOREMS
CORRESPONDING ANGLES POSTULATE
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POSTULATES AND THEOREMS
CORRESPONDING ANGLES POSTULATE
IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF
CORRESPONDING ANGLES IS CONGRUENT.
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POSTULATES AND THEOREMS
CORRESPONDING ANGLES POSTULATE
IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF
CORRESPONDING ANGLES IS CONGRUENT.
∠1 ≅ ∠5
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POSTULATES AND THEOREMS
ALTERNATE INTERIOR ANGLES THEOREM
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POSTULATES AND THEOREMS
ALTERNATE INTERIOR ANGLES THEOREM
IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF
ALTERNATE INTERIOR ANGLES IS CONGRUENT.
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POSTULATES AND THEOREMS
ALTERNATE INTERIOR ANGLES THEOREM
IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF
ALTERNATE INTERIOR ANGLES IS CONGRUENT.
∠4 ≅ ∠6
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POSTULATES AND THEOREMS
CONSECUTIVE INTERIOR ANGLES THEOREM
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POSTULATES AND THEOREMS
CONSECUTIVE INTERIOR ANGLES THEOREM
IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF CONSECUTIVE INTERIOR ANGLES IS
SUPPLEMENTARY.
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POSTULATES AND THEOREMS
CONSECUTIVE INTERIOR ANGLES THEOREM
IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF CONSECUTIVE INTERIOR ANGLES IS
SUPPLEMENTARY.
∠4 AND ∠5 ARE SUPPLEMENTARY
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POSTULATES AND THEOREMS
ALTERNATE EXTERIOR ANGLES THEOREM
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POSTULATES AND THEOREMS
ALTERNATE EXTERIOR ANGLES THEOREM
IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF ALTERNATE EXTERIOR ANGLES IS
CONGRUENT.
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POSTULATES AND THEOREMS
ALTERNATE EXTERIOR ANGLES THEOREM
IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF ALTERNATE EXTERIOR ANGLES IS
CONGRUENT.
∠2 ≅ ∠7
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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
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EXAMPLE 1
IN THE FIGURE, m∠4 = 51°. FIND THE MEASURE OF EACH ANGLE. GIVE A JUSTIFICATION TO
YOUR ANSWER.a. m∠2
51°; VERTICAL ANGLESWITH ∠4
b. m∠3
c. m∠6
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EXAMPLE 1
IN THE FIGURE, m∠4 = 51°. FIND THE MEASURE OF EACH ANGLE. GIVE A JUSTIFICATION TO
YOUR ANSWER.a. m∠2
51°; VERTICAL ANGLESWITH ∠4
b. m∠3129°; SUPPLEMENTARY ANGLES WITH ∠4
c. m∠6
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EXAMPLE 1
IN THE FIGURE, m∠4 = 51°. FIND THE MEASURE OF EACH ANGLE. GIVE A JUSTIFICATION TO
YOUR ANSWER.a. m∠2
51°; VERTICAL ANGLESWITH ∠4
b. m∠3129°; SUPPLEMENTARY ANGLES WITH ∠4
c. m∠651°; ALTERNATE INTERIOR ANGLES WITH ∠4
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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.
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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.
m∠1 = 125°; VERTICAL
WITH ∠2
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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.
m∠1 = 125°; VERTICAL
WITH ∠2m∠3 = 55°; CONSECUTIVE
INTERIOR WITH ∠2
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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.
m∠1 = 125°; VERTICAL
WITH ∠2m∠3 = 55°; CONSECUTIVE
INTERIOR WITH ∠2m∠4 = 125°; CONSECUTIVE
INTERIOR WITH ∠3
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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.
m∠1 = 125°; VERTICAL
WITH ∠2m∠3 = 55°; CONSECUTIVE
INTERIOR WITH ∠2m∠4 = 125°; CONSECUTIVE
INTERIOR WITH ∠3
m∠5 = 55°; SUPPLEMENTARY WITH ∠4
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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.
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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.
2x − 10 = x + 15
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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.
2x − 10 = x + 15−x −x
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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.
2x − 10 = x + 15−x −x +10+10
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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.
2x − 10 = x + 15−x −x +10+10
x = 25
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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.
2x − 10 = x + 15−x −x +10+10
x = 25
SINCE THE ANGLES ARE CORRESPONDING, THEY ARE CONGRUENT, MEANING THEY ARE
EQUAL.
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EXAMPLE 3
USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING.
b. IF m∠7 = 4(y − 25) AND
m∠1 = 4y, FIND y.
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EXAMPLE 3
USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING.
b. IF m∠7 = 4(y − 25) AND
m∠1 = 4y, FIND y.
4(y − 25) + 4y = 180
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EXAMPLE 3
USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING.
b. IF m∠7 = 4(y − 25) AND
m∠1 = 4y, FIND y.
4(y − 25) + 4y = 1804y − 100 + 4y = 180
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EXAMPLE 3
USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING.
b. IF m∠7 = 4(y − 25) AND
m∠1 = 4y, FIND y.
4(y − 25) + 4y = 1804y − 100 + 4y = 180
8y − 100 = 180
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EXAMPLE 3
USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING.
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
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EXAMPLE 3
USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING.
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
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CHECK YOUR UNDERSTANDING
CHECK OUT PROBLEMS 1-10 ON PAGE 181.
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PROBLEM SET
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PROBLEM SET
P. 181 #11-35 ODD
“YOU CAN FOOL TOO MANY OF THE PEOPLE TOO MUCH OF THE TIME.” - JAMES THURBER
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