1. Pearson Unit 1 Topic 3: Parallel and Perpendicular Lines 3-2: Properties of Parallel Lines Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007

2. TEKS Focus:
(6)(A) Verify theorems about angles formed by the intersection of lines and line segments, including vertical angles, and angle formed by parallel lines cut by a transversal and prove equidistance between the endpoints of a segment and points on its perpendicular bisector and apply these relationships to solve problems.
(1)(G) Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
(1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas.
(5)(A) Investigate patterns to make conjectures about geometric relationships, including angles formed by parallel lines cut by a transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons, and special segments and angles of circles choosing from a variety of tools.

3. Theorem

5. Postulate

7. Example: 1 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°

8. Example: 2 Find each angle measure. A. mECF mCBG = 70 Given
Corr. s Post.
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°

9. Example: 3 Find each angle measure. A. mFBC 2x – 135 = x - 30
Corr. s Thm.
x – 135 = - 30
Subtraction Property of =
x = 105
Addition Property of =
mFBC = mEDG = 105 – 30 = 75 °
Alt. Ext. s Thm.
B. mBDG
x = 105
mBDG = 105°

10. Example: 4 Find mABD. 2x + 10° = 3x – 15° Alt. Int. s Thm.
Subtract 2x and add 15 to both sides.
x = 25
mABD = 2(25) + 10 = 60°
Substitute 25 for x.

11. Example: 5 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
Create a system of two equations.
Subtract the first equation from the second equation.
5x + 5(5) = 60
5x = 35
x = 7
Substitute 5 for y in 5x + 5y = 60. Simplify and solve for x.
x = 7, y = 5

12. Example 6 1. ℓ || m 1. Given 2. 4  8 2. Corr. Angles Postulate
Use the Corresponding Angles Postulate
and the given information to prove
the Alternate Interior Angles Theorem.
Given: ℓ || m
Prove: 4  5
Statements
Reasons
1. ℓ || m
1. Given
2. 4  8
2. Corr. Angles Postulate
3. 8  5
3. Vertical Angles Thm.
4. 4  5
4. Transitive Prop. of 

13. Example: 7 Given: a || b and ℓ || m Prove: 2  15 1. a || b 1. Given
9 10
11 12
13 14
15 16
a b
Use the Corresponding Angles Postulate and the given to complete the proof.
Given: a || b and ℓ || m
Prove: 2  15
Statements
Reasons
1. a || b
1. Given
2. 2  10
2. Corr. Angles Post.
3. 10  11
3. Vertical Angles Thm.
4. ℓ || m
4. Given
5. 11  15
5. Corr. Angles Post.
6. Transitive Prop. of 
6. 2  15

14. EXTRA EXAMPLES NOT USED IN COMPOSITION BOOK FOLLOW.
ALSO REMEMBER TO LOG-ON TO YOUR PEARSON ACCOUNT TO LOOK AT OTHER EXAMPLES BEFORE BEGINNING THE ON-LINE HW AND THE WRITTEN HW.

15. Example: 8
The treble strings of a grand piano are parallel. Viewed from above, the bass strings form transversals to the treble strings. Find the measures of the acute angles in the diagram.
By the Alternate Exterior Angles
Theorem, (25x + 5y)° = 125°.
By the Corresponding Angles Postulate, (25x + 4y)° = 120°.
Create a system and subtract the second
equation from the first. y = 5
Substitute y = 5 in either equation and
solve for x.
25x + 5(5) = 125
25 x = 100
x = 4
An acute angle will be 180° – 125°, or 55°.
The other acute angle will be 180° – 120°, or 60°.

16. Example: 8 (continued)
Did you really have to use algebra to find the acute angle measures in the diagram?
NO! 180 – 125 = 55 and 180 – 120 – 60.