1. Parallel lines and transversals
TC2MA212

2. PARALLEL LINES l Def: line that do not intersect. Illustration:
Notation: l || m AB || CD A B C D l m

3. PERPENDICULAR LINES m n
Def: Lines that intersect to form a right angle.
Illustration:
Notation: m  n
Key Fact: 4 right angles are formed. m n

4. Transversal Def: a line that intersects two lines at different points
Illustration: t

5. 1   4 2   3 5   8 6   7 Vertical Angles
Two angles that are opposite angles. 1 2 3 4 5 6 7 8 t
1   4
2   3
5   8
6   7

6. Theorem:
Vertical angles are congruent. 1 2 3 4 5 6 7 8 t

7. Vertical Angles Find the measures of the missing angles t 125  ?
55  ?
55 

8. Worksheet

9. Supplementary Angles/ Linear Pair
Two angles that form a line (sum=180) 1 2 3 4 5 6 7 8 t
1+2=180
2+4=180
4+3=180
3+1=180
5+6=180
6+8=180
8+7=180
7+5=180

10. Supplementary Angles/ Linear Pair
Find the measures of the missing angles t ?
108 
72  ?
108 

11. 1   5 2   6 3   7 4   8 Corresponding Angles
Two angles that occupy corresponding positions. t
1   5
2   6
3   7
4   8
Top Left
Top Right 1 2 3 4 5 6 7 8
Bottom Left
Bottom Right
Top Left
Top Right
Bottom Left
Bottom Right

12. Corresponding Angles Postulate
If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 1 2
1 ≅ 2

13. Corresponding Angles Find the measures of the missing angles t 145 
35  ?
145 

14. Alternate Interior Angles
Two angles that lie between parallel lines on opposite sides of the transversal t
3   6
4   5 1 2 3 4 5 6 7 8

15. Theorem 1: Alternate Interior Angles
If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. 3 4
3 ≅ 4

16. Proving the Alternate Interior Angles Theorem
Given: p ║ q
Prove: 1 ≅ 2 1 2 3

17. Proof Statements: p ║ q 1 ≅ 3 3 ≅ 2 1 ≅ 2 Reasons: Given
Corresponding Angles Postulate
Vertical Angles Theorem
Transitive Property of Congruence

18. Alternate Interior Angles
Find the measures of the missing angles t
82 
98  ?
82 

19. Alternate Exterior Angles
Two angles that lie outside parallel lines on opposite sides of the transversal t
2   7
1   8 1 2 3 4 5 6 7 8

20. Alternate Exterior Angles
Find the measures of the missing angles t
120  ?
60 
120 

21. Theorem 3: Alternate Exterior Angles
If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. 7 8
7 ≅ 8

22. Proof

23. Consecutive Interior Angles
Two angles that lie between parallel lines on the same sides of the transversal t
3 +5 = 180
4 +6 = 180 1 2 3 4 5 6 7 8

24. Consecutive Interior Angles
Find the measures of the missing angles t
135  ?
45 

25. Theorem2: Consecutive Interior Angles
If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. 5 6
5 + 6 = 180°

26. Proof

27. Using properties of parallel lines
Given that m 5 = 65°, find each measure. Tell which postulate or theorem you use.
A. m 6 B. m 7
C. m 8 D. m 9 9 6 8 5 7

28. Solutions: m 6 = m 5 = 65° m 7 = 180° - m 5 =115°
Vertical Angles Theorem
m 7 = 180° - m 5 =115°
Linear Pair postulate
m 8 = m 5 = 65°
Corresponding Angles Postulate
m 9 = m 7 = 115°
Alternate Exterior Angles Theorem