1. Angles and Parallel Lines
Geometry D – Section 3.2

2. Angles and Parallel Lines
We are going to investigate the relationship of various angles created by two parallel lines and a transversal.
Obtain a ½ sheet of graph paper and a protractor. Construct two || lines and a transversal similar to the image on the next slide.

3. Angles and Parallel Lines
Extend your lines the full height and width of the paper.
Pause for time to work!

4. Angles and Parallel Lines
Label the angles as shown below. 1 2 3 4 5 6 7 8
Pause for time to work!

5. Angles and Parallel Lines
Measure all angles using a protractor to the nearest degree. 1 2 3 4 5 6 7 8
Pause for time to work!

6. Angles and Parallel Lines
Measure all angles using a protractor to the nearest degree.
127o
53o 1 2
Note: Your measurements may be different values but should be in the same pattern. 3 4
53o
127o
127o 5 6
53o 7 8
53o
127o

7. Angles and Parallel Lines
Identify the relationship between the following angles?
127o
53o 1 2 3 4
53o
127o
127o 5 6
53o 7 8
53o
From Chapter 2, the angles are linear pairs.
127o
What can be said about the measures of the linear pairs?
Linear pairs are supplementary (sum to 180o).

8. Angles and Parallel Lines
Identify the relationship between the following angles?
127o
53o 1 2 3 4
53o
127o
127o 5 6
53o 7 8
53o
From Chapter 2, the angles are vertical angles.
127o
What can be said about the measures of the vertical angles?
Vertical angles are congruent angles.

9. Angles and Parallel Lines
Identify the relationship between the following angles?
127o
53o 1 2 3 4
53o
127o
127o 5 6
53o 7 8
53o
The angles are corresponding angles.
127o
What can be said about the measures of the corresponding angles?
The measures are equal and the angles are congruent.

10. Angles and Parallel Lines
Corresponding Angles Postulate –
If two parallel lines are cut by a transversal, then each pair of corresponding angles are congruent.

11. Angles and Parallel Lines
Identify the relationship between the following angles?
127o
53o 1 2 3 4
53o
127o
127o 5 6
53o 7 8
53o
The angles are alternate interior angles.
127o
What can be said about the measures of the alternate interior angles?
The measures are equal and the angles are congruent.

12. Angles and Parallel Lines
Alternate Interior Angles Theorem –
If two parallel lines are cut by a transversal, then each pair of alternate interior angles are congruent.
You will prove this theorem as a homework problem!

13. Angles and Parallel Lines
Identify the relationship between the following angles?
127o
53o 1 2 3 4
53o
127o
127o 5 6
53o 7 8
53o
The angles are alternate interior angles.
127o
What can be said about the measures of the alternate interior angles?
The measures add to 180o.

14. Angles and Parallel Lines
Consecutive Interior Angles Theorem –
If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary (sum to 180o).
You will prove this theorem as a homework problem!

15. Angles and Parallel Lines
Identify the relationship between the following angles?
127o
53o 1 2 3 4
53o
127o
127o 5 6
53o 7 8
53o
The angles are alternate exterior angles.
127o
What can be said about the measures of the alternate interior angles?
The measures are equal and the angles are congruent.

16. Angles and Parallel Lines
Alternate Exterior Angles Theorem –
If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent.
Prove:
Statement Reason ?
p || q, t is a transversal of p & q
Given t
Corresponding ‘s are 1 2 ? p 3 4 ? 5 6
Vertical ‘s are q 7 8 ?
Transitive Property

17. Angles and Parallel Lines
Perpendicular Transversal Theorem –
In a plane, if a line is perpendicular to one of two perpendicular lines, then it is perpendicular to the other. t p q
If t is perpendicular ( ) to p, then it is also perpendicular to q.
You will prove this theorem as a homework problem!

18. Angles and Parallel Lines
Applications –
Gather into groups of not more than 3.
Work the following problems in your group. Compare your answers to those provided.

19. Angles and Parallel Lines
Given j || k,
Applications –
Make a sketch of the problem in your notes.
Find the measure of 3 4 5 2 1.
43o 1 7 8 6 9
Corresponds with 1. 11 10 2.
24o 12
Alternate exterior with 13 3.
156o
Linear pair with o – 24o = 156o 14

20. Angles and Parallel Lines
Given j || k,
Find the measure of
Applications – 3 4 5 2 4.
137o 1 7 8 6 9
Linear pair with 3. 11 10 5.
156o 12
Vertical angle with 13 6.
43o
Vertical with Alternate Interior of 3. 14

21. Angles and Parallel Lines
Applications –
Find the values of x and y in each figure. Find the measure of each given angle. Note: Figures are not drawn to scale.
Given:
Pause for time to work!

22. Angles and Parallel Lines
Applications – Solution
Given:
Linear pairs are supplementary.
(5x + 2) + (9x + 10) = 180o
14x + 12 = 180
14x = 168
x = 12
Linear Pair
By corresponding angles,
62o
118o
3y – 1 = 62
62o
3y = 63
y = 21 and

23. Angles and Parallel Lines
Applications –
Find the values of x, y and z in each figure.
(2z)o
(3x–3)o
(4y+2)o
66o
Pause for time to work!

24. Angles and Parallel Lines
is a corresponding angle with the angle of 66o.
Applications –
(3x – 3)o and 66o are linear pairs and sum to 180o.
(3x – 3)o + 66o = 180o 3x + 63 = 180 3x = 117, x = 39
(2z)o
(3x–3)o
66o
(4y + 2)o and 66o are congruent alternate interior angles.
(4y + 2)o = 66o 4y = 64, y = 16
(4y+2)o
66o
(3x–3)o and (2z)o are congruent alternate interior angles.
(3x–3)o = 3(39) – 3 = 114o
(2z)o = 114o, z = 57

25. Angles and Parallel Lines
Applications –
Find the values of x, y and z in each figure.
There are other ways of doing this problem correctly.
If you worked it a different way, would you be willing to share how you did it?
(2z)o
(3x–3)o
(4y+2)o
66o

26. Angles and Parallel Lines
Applications –
Find the measures of all the angles on the object if the measure of angle 1 is 30o.
Pause for time to work!

27. Angles and Parallel Lines
Applications –
Find the measures of all the angles on the object if the measure of angle 1 is 30o.
Perpendicular transversal theorem.
Perpendicular lines intersect in 4 right (90o) angles.
90o
90o
90o
90o
90o

28. Angles and Parallel Lines
Applications –
Find the measures of all the angles on the object if the measure of angle 1 is 30o.
Vertical Angle
Vertical Angle
Alternate interior angle with angle 1.
30o
150o
150o
30o
90o
30o
Linear pairs are supplementary.
90o
30o
Given
90o
90o
90o
Vertical Angles

29. Angles and Parallel Lines
Applications –
Find the measures of all the angles on the object if the measure of angle 1 is 30o.
30o
Vertical angles.
150o
150o
30o
30o
90o
60o
All angles have been found!
60o
90o
30o
90o
90o
Since the transversal is , these two angles must add to 90o using angle addition.
90o

30. Angles and Parallel Lines
Assignment –
3.2 / 17-20, 24, 26, 30, 32, , 38, 40, 43, 48, 55, , 60, 62
Please return your protractor!!!!
Thank you Mr. Matzke!!!!