1. Chapter 3: Parallel and Perpendicular Lines
3.1 Lines & Angles
Chapter 3: Parallel and Perpendicular Lines

2. 3.1 Properties of Parallel Lines
Transversal: line that intersects two coplanar lines at two distinct points
Transversal

3. 3.1 Properties of Parallel Lines
The pairs of angles formed have special names… t
Transversal 5 6 1 3 l 4 2 m 7 8

4. Alternate Interior Angles
<1 and <2 <3 and <4 t 5 6 1 3 l 4 2 m 7 8

5. Same-side Interior Angles
<1 and <4 <2 and <3 t 5 6 1 3 l 4 2 m 7 8

6. Corresponding Angles
<1 and <7 <2 and <6 <3 and <8 <4 and <5 t 5 6 1 3 l 4 2 m 7 8

7. Properties of Parallel Lines
Postulate 3-1 Corresponding Angles Postulate If a transversal intersects two parallel lines, then corresponding angles are congruent t
line l || line m 1 l 2 m
<1 = <2

8. Properties of Parallel Lines
Theorem 3-1 Alternate Interior Angles Theorem If a transversal intersects two parallel lines, then alternate interior angles are congruent. t
line l || line m l 3 1 2 m
<2 = <3

9. Properties of Parallel Lines
Theorem 3-2 Same-Side Interior Angles Theorem If a transversal intersects two parallel lines, then same-side interior angles are supplementary.
line l || line m t l 3 1 2 m
m<1 + m<2 = 180

10. Properties of Parallel Lines
Theorem 3-3 Alternate Exterior Angles Theorem If a transversal intersects two parallel lines, then alternate exterior angles are congruent.
line l || line m t 1 l 3 m 2
m<1 = m<2

11. Properties of Parallel Lines
Theorem 3-4 Same Side Exterior Angles Theorem If a transversal intersects two parallel lines, then same-side exterior angles are supplementary.
line l || line m t 1 l 3 m 2
m<1 + m<2 = 180

12. Two-Column Proof Given: a || b Prove: <1 = <3 Statements Reasons4 3 1 b 1. 1. 2. 2. 3. 3. 4. 4.
* This proves why alternate interior angles are congruent *

13. Two-Column Proof a b t 2 3 1
Given: a || b Prove: <1 and <2 are supplementary Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5.

14. Finding Angle Measures
a || b
c || d c d
<1 <2 <3 <4 <5 <6 <7 <8 8 7 6 a
50° 2 5 4 b 1 3
(1, 2, 4, 3, 8, 7, 5, 6)

15. Using Algebra to Find Angle Measures
Find the value of x and y. x = y = ▲ ▲
50° y
70° x ▲ 2x y
(y – 50) ▲

16. Properties of Parallel Lines
Theorem 3-3 Alternate Exterior Angles Theorem If a transversal intersects two parallel lines, then alternate exterior angles are congruent.
line l || line m t 1 l 3 m 2
m<1 = m<2

17. Properties of Parallel Lines
Theorem 3-4 Same Side Exterior Angles Theorem If a transversal intersects two parallel lines, then same-side exterior angles are supplementary.
line l || line m t 1 l 3 m 2
m<1 + m<2 = 180

18. 3.2 Proving Lines Parallel
Chapter 3: Parallel and Perpendicular Lines

19. What are we learning? Students will…
To use a transversal and prove lines are parallel.
Evidence Outcome: Prove geometric theorems (lines, angles, triangles, parallelograms).
Purpose (Relevancy): To show how to paint parallel lines on a parking lot.

20. Proving Lines Parallel1 2 m
Postulate 3-2: Converse (Opposite) of Corresponding Angles Postulate: If two lines and a transversal form corresponding angles that are congruent, then the two lines are parallel. Limon Language: If corresponding angles are congruent, then the lines are parallel.

21. Proving Lines Parallel
72°
We know these two lines are parallel!!
72°
108°
72°
If Alternate Interior Angles are congruent we can assume lines are parallel too!

22. Proving Lines Parallel1 4 2 m
Theorem 3-5: Converse of Alternate Interior Angles Theorem
If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel.
Limon Language: If alternate interior angles are congruent, then the lines are parallel.

23. Proof of Theorem 3-5 (C of AIAT)l 3 1 m 2
Statements
Reasons 1. 2. 3. 4.

24. Proving Lines Parallel
72°
We know these two lines are parallel!!
72°
108°
72°
If Same-Side Interior Angles are supplementary, we can assume lines are parallel too!!

25. Proving Lines Parallel1 4 2 m
Theorem 3-6: Converse of Same-Side Interior Angles Theorem
If two lines and a transversal form same-side interior angles that are supplementary, then the two lines are parallel.
Limon Language: If same side interior angles add up to 180 degrees, then the lines are parallel.

26. Proving Lines Parallel
108°
72°
We know these two lines are parallel!!
72°
108°
72°
108°
If Alternate Exterior Angles are congruent, we can assume lines are parallel!!!

27. Proving Lines Parallel1 b 3 2
Theorem 3-7: Converse of Alternate Exterior Angles Theorem
If two lines and a transversal intersects form alternate exterior angles that are congruent, then the two lines are parallel. Limon Language: If alternate exterior angles are congruent, then the lines are parallel.

28. Proof of Theorem 3-7 (C of AEAT)1 4 b 2
Statements
Reasons 1. 2. 3. 4.

29. Proving Lines Parallel
108°
72°
We know these two lines are parallel!!
72°
108°
72°
108°
If Same-Side Exterior Angles are supplementary, we can assume lines are parallel!!!

30. Proving Lines Parallel1 b 3 2
Theorem 3-8: Converse of Same-Side Exterior Angles Theorem
If two lines and a transversal intersects form same-side exterior angles that are supplementary, then the two lines are parallel. Limon Language: If same-side exterior angles add up to 180 degrees, then the lines are parallel.

31. Let’s Apply What We Have Learned, K?
Find the value of x for which l || m l
40° m
(2x + 6)°

32. You Try One!
Find the value of x for which a || b a
(7x - 8)° b
62°

33. Assignments: SHOW ALL WORK FOR CREDIT Workbook pages 59-60 evens only
page all
page evens only
page all
SHOW ALL WORK FOR CREDIT
Pearson Realize: Continue Chapter 3 Assignments
Try to complete through 3-4 by tomorrow.
Offline Alternate: Student Companion pages 58-73