1. Angles Formed by Parallel Lines and Transversals
21.1
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Geometry
Holt Geometry

2. Identify each angle pair. 1. 1 and 3 2. 3 and 6 3. 4 and 5
21.1
Warm Up
Identify each angle pair.
1. 1 and 3
2. 3 and 6
3. 4 and 5
4. 6 and 7
corr. s
alt. int. s
alt. ext. s
same-side int s

3. 21.1
Objective
Prove and use theorems about the angles formed by parallel lines and a transversal.

4. 21.1

5. Example 1: Using the Corresponding Angles Postulate
21.1
Example 1: Using the Corresponding Angles Postulate
Find each angle measure.
A. mECF
x = 70
Corr. s Post.
mECF = 70°
B. mDCE
5x = 4x + 22
Corr. s Post.
x = 22
Subtract 4x from both sides.
mDCE = 5x
= 5(22)
Substitute 22 for x.
= 110°

6. 21.1
Example 2
Find mQRS.
x = 118
Corr. s Post.
mQRS + x = 180°
*Def. of Linear Pair*
mQRS = 180° – x
Subtract x from both sides.
= 180° – 118°
Substitute 118° for x.
= 62°

7. 21.1
If a transversal is perpendicular to two parallel lines, all eight angles are congruent.
Helpful Hint

8. 21.1
Remember that postulates are statements that are accepted without proof.
Since the Corresponding Angles Postulate is given as a postulate, it can be used to prove the next three theorems.

9. 21.1

10. 21.1
Example 3
Find each angle measure.
A. mEDG
mEDG = 75°
Alt. Ext. s are
Congruent.
B. mBDG
x – 30° = 75°
Alt. Ext. s are congruent.
x = 105
Add 30 to both sides.
mBDG = 105°

11. 21.1
Example 4
Find x and y in the diagram.
By the Alternate Interior Angles
Theorem, (5x + 4y)° = 55°.
By the Corresponding Angles Postulate, (5x + 5y)° = 60°.
5x + 5y = 60
–(5x + 4y = 55)
y = 5
Subtract the first equation from the second equation.
Substitute 5 for y in 5x + 5y = 60. Simplify and solve for x.
5x + 5(5) = 60
x = 7, y = 5

12. 21.1
Lesson Quiz
State the theorem or postulate that is related to the measures of the angles in each pair. Then find the unknown angle measures.
1. m1 = 120°, m2 = (60x)°
2. m2 = (75x – 30)°,
m3 = (30x + 60)°
Alt. Ext. s Thm.; m2 = 120°
Corr. s Post.; m2 = 120°, m3 = 120°
3. m3 = (50x + 20)°, m4= (100x – 80)°
4. m3 = (45x + 30)°, m5 = (25x + 10)°
Alt. Int. s Thm.; m3 = 120°, m4 =120°
Same-Side Int. s Thm.; m3 = 120°, m5 =60°

13. Proving Lines Parallel
21.2
Proving Lines Parallel
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Geometry
Holt Geometry

14. 21.2
Warm Up
State the converse of each statement.
1. If a = b, then a + c = b + c.
2. If mA + mB = 90°, then A and B are complementary.
3. If AB + BC = AC, then A, B, and C are collinear.
If a + c = b + c, then a = b.
If A and  B are complementary, then mA + mB =90°.
If A, B, and C are collinear, then AB + BC = AC.

15. 21.2
Objective
Use the angles formed by a transversal to prove two lines are parallel.

16. 21.2
Recall that the converse of a theorem is found by exchanging the hypothesis and conclusion. The converse of a theorem is not automatically true. If it is true, it must be stated as a postulate or proved as a separate theorem.

17. 21.2

18. 21.2
The Converse of the Corresponding Angles Postulate is used to construct parallel lines. The Parallel Postulate guarantees that for any line ℓ, you can always construct a parallel line through a point that is not on ℓ.

19. 21.2

20. 21.2
Lesson Quiz: Part I
Name the postulate or theorem
that proves p || r.
1. 4  5
Conv. of Alt. Int. s Thm.
2. 2  7
Conv. of Alt. Ext. s Thm.
3. 3  7
Conv. of Corr. s Post.
4. 3 and 5 are supplementary.
Conv. of Same-Side Int. s Thm.

21. Perpendicular Lines Warm Up Lesson Presentation Lesson Quiz 21.3
Holt McDougal Geometry
Holt Geometry

22. 21.3
Objective
Prove and apply theorems about perpendicular lines.

23. 21.3
Vocabulary
perpendicular bisector
distance from a point to a line

24. 21.3
The perpendicular bisector of a segment is a line perpendicular to a segment at the segment’s midpoint.
The shortest segment from a point to a line is perpendicular to the line. This fact is used to define the distance from a point to a line as the length of the perpendicular segment from the point to the line.

25. Example 5: Distance From a Point to a Line
21.3
Example 5: Distance From a Point to a Line
A. Name the shortest segment from point A to BC. AP
B. Write and solve an inequality for x.
AC > AP
AP is the shortest segment.
x – 8 > 12
Substitute x – 8 for AC and 12 for AP.
+ 8
Add 8 to both sides of the inequality.
x > 20

26. 21.3
HYPOTHESIS
CONCLUSION

27. 21.3
Example 6
Solve to find x and y in the diagram.
x = 9, y = 4.5