This document discusses angles and parallel lines. It presents three postulates and theorems about corresponding angles, alternate interior angles, and consecutive interior angles when two parallel lines are cut by a transversal. It then provides three multi-part examples that apply these postulates and theorems to find angle measurements or variables in diagrams. The examples demonstrate using vertical angles, supplementary angles, congruent corresponding and alternate interior angles, and algebra to solve for variables.

1. SECTION 3-2
ANGLES AND PARALLEL LINES
Wednesday, December 14, 2011

2. 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?
Wednesday, December 14, 2011

3. POSTULATES AND THEOREMS
CORRESPONDING ANGLES POSTULATE
Wednesday, December 14, 2011

4. POSTULATES AND THEOREMS
CORRESPONDING ANGLES POSTULATE
IF TWO PARALLEL LINES ARE CUT BY A
TRANSVERSAL, THEN EACH PAIR OF
CORRESPONDING ANGLES IS CONGRUENT.
Wednesday, December 14, 2011

5. 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
Wednesday, December 14, 2011

6. POSTULATES AND THEOREMS
ALTERNATE INTERIOR ANGLES THEOREM
Wednesday, December 14, 2011

7. 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.
Wednesday, December 14, 2011

8. 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
Wednesday, December 14, 2011

9. POSTULATES AND THEOREMS
CONSECUTIVE INTERIOR ANGLES THEOREM
Wednesday, December 14, 2011

10. 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.
Wednesday, December 14, 2011

11. 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
Wednesday, December 14, 2011

12. POSTULATES AND THEOREMS
ALTERNATE EXTERIOR ANGLES THEOREM
Wednesday, December 14, 2011

13. 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.
Wednesday, December 14, 2011

14. 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
Wednesday, December 14, 2011

15. 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
Wednesday, December 14, 2011

16. 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 ANGLES
WITH ∠4
b. m∠3
c. m∠6
Wednesday, December 14, 2011

17. 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 ANGLES
WITH ∠4
b. m∠3
129°; SUPPLEMENTARY ANGLES WITH ∠4
c. m∠6
Wednesday, December 14, 2011

18. 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 ANGLES
WITH ∠4
b. m∠3
129°; SUPPLEMENTARY ANGLES WITH ∠4
c. m∠6
51°; ALTERNATE INTERIOR ANGLES WITH ∠4
Wednesday, December 14, 2011

19. 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.
Wednesday, December 14, 2011

20. 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
Wednesday, December 14, 2011

21. 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
m∠3 = 55°; CONSECUTIVE
INTERIOR WITH ∠2
Wednesday, December 14, 2011

22. 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
m∠3 = 55°; CONSECUTIVE
INTERIOR WITH ∠2
m∠4 = 125°; CONSECUTIVE
INTERIOR WITH ∠3
Wednesday, December 14, 2011

23. 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
m∠3 = 55°; CONSECUTIVE
INTERIOR WITH ∠2
m∠4 = 125°; CONSECUTIVE
INTERIOR WITH ∠3
m∠5 = 55°; SUPPLEMENTARY WITH ∠4
Wednesday, December 14, 2011

24. 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.
Wednesday, December 14, 2011

25. 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
Wednesday, December 14, 2011

26. 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
Wednesday, December 14, 2011

27. 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 +10 −x +10
Wednesday, December 14, 2011

28. 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 +10 −x +10
x = 25
Wednesday, December 14, 2011

29. 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 +10 −x +10
x = 25
SINCE THE ANGLES ARE CORRESPONDING,
THEY ARE CONGRUENT, MEANING THEY ARE
EQUAL.
Wednesday, December 14, 2011

30. 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.
Wednesday, December 14, 2011

31. 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
Wednesday, December 14, 2011

32. 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
4y − 100 + 4y = 180
Wednesday, December 14, 2011

33. 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
4y − 100 + 4y = 180
8y − 100 = 180
Wednesday, December 14, 2011

34. 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
4y − 100 + 4y = 180
8y − 100 = 180
8y = 280
Wednesday, December 14, 2011

35. 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
4y − 100 + 4y = 180
8y − 100 = 180
8y = 280
y = 35
Wednesday, December 14, 2011

36. CHECK YOUR UNDERSTANDING
CHECK OUT PROBLEMS 1-10 ON PAGE 181.
Wednesday, December 14, 2011

37. PROBLEM SET
Wednesday, December 14, 2011

38. PROBLEM SET
P. 181 #11-35 ODD
“YOU CAN FOOL TOO MANY OF THE PEOPLE TOO
MUCH OF THE TIME.” - JAMES THURBER
Wednesday, December 14, 2011