section 4.1 geometry of parallel lines this booklet
TRANSCRIPT
Foundations of Math 11 Updated January 2020
1 Adrian Herlaar, School District 61 www.mrherlaar.weebly.com
Section 4.1 β Geometry of Parallel Lines
This booklet belongs to: Block:
First letβs look at some vocabulary
a) Acute β an angle between 0 and 90 degrees
b) Obtuse β an angle between 90 and 180 degrees
c) Straight β angle exactly 180 degrees
d) Right β angle exactly 90 degrees
e) Complementary β two angles that add up to 90 degrees
f) Supplementary β two angles that add up to 180 degrees
When we look at angle relationships we can tell a lot about ANGLES FORMED BY A TRANSVERSAL
When two lines π1 πππ π2 are intersected by a third line, a transversal, eight angles are formed,
4 around each line.
To study these relationships we start with an assumption, or aβ¦
POSTULATE β accepted assumption without proof
To devise our theorems we will use, postulates, inductive and deductive reasoning
There are a series of rules named after letters of the alphabet, because they create that shape
They all involve two parallel lines being intersected by a transversal
6 5
7 8
4 3
2 1 π1
π2
Transversal
Foundations of Math 11 Updated January 2020
2 Adrian Herlaar, School District 61 www.mrherlaar.weebly.com
Corresponding Angles Postulate (F Rule)
If two parallel lines are cut by a transversal, then the corresponding angles are equal
If two lines are cut by a transversal, and the corresponding angles are equal, then the lines are parallel.
With this Postulate we can now prove many more relationships between parallel lines and transversals
Deductive reasoning will be used repeatedly for these proofs
Vertical Angles
When two lines intersect, they form two pairs of vertical angles
β 1 πππ β 3 are vertical angles
β 2 πππ β 4 are vertical angles
6 5
7 8
4 3
2 1 π1
π2
β 1 = β 5
β 2 = β 6
β 3 = β 7
β 4 = β 8
This means parallel
π1 β₯ π2
1 2
3 4
Foundations of Math 11 Updated January 2020
3 Adrian Herlaar, School District 61 www.mrherlaar.weebly.com
Proof β Vertical Angles are Equal
Given: β 1 πππ β 2 are vertical angles
Prove: β 1 = β 2
Proof Statement Reason
1. β 1 + β 3 = 180Β° Angles on a line add to 180Β° (supplementary) 2. β 2 + β 3 = 180Β° Angles on a line add to 180Β° (supplementary) 3. β 1 + β 3 = β 2 + β 3 Both equal to 180Β° (substitution) 4. β 1 = β 2 Subtraction
Vertical Angle Theorem
If two angles are vertical angles, then the angles are equal.
Proved Statements are called THEOREMS.
Alternate Interior Angles (the Z rule)
When two lines π1πππ π2 are intersected by a transversal, the four angles between the lines are
called interior angles
β 3, β 4, β 5, πππ β 6 are interior angles
β 3 πππ β 6 are alternate interior angles
β 4 πππ β 5 are alternate interior angles
Proof β Alternate Interior Angles of Parallel Lines are Equal
Given: π1 β₯ π2
Prove: β 4 = β 5
Proof Statement Reason
1. π1 β₯ π2 Given 2. β 1 = β 4 Vertical Angles 3. β 1 = β 5 Corresponding Angles 4. β 4 = β 5 Substitution (both equal to β 1)
3 2
1
6 5
7 8
4 3 2 1
π1
π2
π1 β₯ π2
5
4
1 π1
π2
π1 β₯ π2
Foundations of Math 11 Updated January 2020
4 Adrian Herlaar, School District 61 www.mrherlaar.weebly.com
Alternate Interior Angle Theorem (Z Rule) If two parallel lines are cut by a transversal, then the alternate interior angles are equal. If two lines are cut by a transversal, and the alternate interior angles are equal, then the lines are parallel
Co-Interior Angles
When two lines π1πππ π2 are intersected by a transversal, then the interior angles on the same
side of the transversal are called co-interior angles
β 3, β 4, β 5, πππ β 6 are interior angles
β 3 πππ β 5 are co-interior angles
β 4 πππ β 6 are co-interior angles
Proof β Co-Interior Angles of Parallel Lines are Supplementary
Given: π1 β₯ π2
Prove: β 3 + β 5 = 180Β°
Proof Statement Reason
1. π1 β₯ π2 Given 2. β 3 = β 6 Alternate interior Angles 3. β 5 + β 6 = 180Β° Angles on a line (Supplementary) 4. β 5 + β 3 = 180Β° Substitution (β 3 for β 6) 5. β 3 + β 5 = 180Β° Re-write Step 4
Co-Interior Angle Theorem If two parallel lines are cut by a transversal, then the co-interior angles are supplementary. If two lines are cut by a transversal, and the co-interior angles are supplementary, then the lines are parallel.
6 5
7 8
4 3 2 1
π1
π2
π1 β₯ π2
5
6
3
π1
π2
π1 β₯ π2
Foundations of Math 11 Updated January 2020
5 Adrian Herlaar, School District 61 www.mrherlaar.weebly.com
The Sum of Angles in a Triangle
We will use our knowledge of parallel lines to prove this most importntat theorem.
Given: βπ΄π΅πΆ
Prove: β 1 + β 2 + β 3 = 180Β°
Proof Statement Reason
1. Draw line DC parallel to AB Construction 2. β 3 + β 4 = β π·πΆπ΅ Angle Addition 3. β π·πΆπ΅ + β 2 = 180Β° Co-Interior Angles 4. β 3 + β 4 + β 2 = 180Β° Substitution (From step 2) 5. β 1 = β 4 Alternate Interior Angles 6. β 1 + β 2 + β 3 = 180Β° Substitution
Angle Sum of a Triangle Theorem The Sum of angles in a triangle is 180Β°
Summary
Parallel Lines and a Transversal
Vertical Angles
β 1 = β 4
β 2 = β 3
β 5 = β 8
β 6 = β 7
Corresponding Angles
β 1 = β 5
β 2 = β 6
β 3 = β 7
β 4 = β 8
Alternate Interior Angles
β 3 = β 6
β 4 = β 5
Co-Interior Angles
β 3 + β 5 = 180Β°
β 4 + β 6 = 180Β°
C
D
D
D
3
D
4
D
2
D
1
D
B
D
A
D
6 5
7 8
4 3 2 1
π1
π2
π1 β₯ π2
Foundations of Math 11 Updated January 2020
6 Adrian Herlaar, School District 61 www.mrherlaar.weebly.com
Find all the missing angles and state the reasons for each answer.
Example:
Solution:
β 1 = 50Β° co-interior angles (2 β 40Β° + 2π₯ = 180Β° β 2π₯ = 100Β° β π₯ = 50Β°
β 2 = 90Β° sum of angles in a triangle 40Β° + 50Β° + π¦ = 180Β° β π¦ = 90Β°
Example:
Solution:
β 1 = 70Β° supplementary angles plus sum of a triangle
β 2 = 70Β° alternate interior angles
β 3 = 20Β° supplementary angles plus sum of angles in a triangle
40Β°
1
90Β°
00Β°
Foundations of Math 11 Updated January 2020
7 Adrian Herlaar, School District 61 www.mrherlaar.weebly.com
Example:
Solution:
β 1 + β 2 = 180Β° co-interior angles
π₯2 β 25π₯ + π₯ = 180
π₯2 β 24π₯ β 180 = 0
(π₯ β 30)(π₯ + 6) = 0
π₯ = β6 πππ 30, ππππππ‘ β 6 πππππ’π π π€π πππβ²π‘βππ£π π πππππ‘ππ£π ππππ π’ππππππ‘
β 1 = π₯2 β 25π₯ β (30)2 β 25(30) β 150Β°
2
1
β 1 = (π₯2 β 25π₯)Β°
β 2 = π₯Β°
Find the value of β 1.
Foundations of Math 11 Updated January 2020
8 Adrian Herlaar, School District 61 www.mrherlaar.weebly.com
Section 4.1 β Practice Problems
For the following questions, solve for the missing angles and give the reason.
1.
2.
3.
β 1 = ________, ________________________________________
β 2 = ________, ________________________________________
β 1 = ________, ________________________________________
β 1 = ________, ________________________________________
β 2 = ________, ________________________________________
2
2
Foundations of Math 11 Updated January 2020
9 Adrian Herlaar, School District 61 www.mrherlaar.weebly.com
4.
5.
6.
β 1 = ________, ________________________________________
β 2 = ________, ________________________________________
β 1 = ________, ________________________________________
β 2 = ________, ________________________________________
β 3 = ________, ________________________________________
β 1 = ________, ________________________________________
β 2 = ________, ________________________________________
2
3 2
11
Foundations of Math 11 Updated January 2020
10 Adrian Herlaar, School District 61 www.mrherlaar.weebly.com
7.
8.
9.
10.
β 1 = ________, ________________________________________
β 2 = ________, ________________________________________
β 3 = ________, ________________________________________
β 4 = ________, ________________________________________
β 1 = ________, ________________________________________
β 2 = ________, ________________________________________
β 3 = ________, ________________________________________
β 1 = ________, ________________________________________
β 2 = ________, ________________________________________
β 3 = ________, ________________________________________
β 1 = ________, ________________________________________
β 2 = ________, ________________________________________
3
2
Foundations of Math 11 Updated January 2020
11 Adrian Herlaar, School District 61 www.mrherlaar.weebly.com
11. X
12. S
13. S
2
5
1
16π₯ β 5
14π₯ + 3
β 1 = ________, ________________________________________
β 2 = ________, ________________________________________
β 1 = ________, ________________________________________
β 2 = ________, ________________________________________
β 1 = ________, ________________________________________
Foundations of Math 11 Updated January 2020
12 Adrian Herlaar, School District 61 www.mrherlaar.weebly.com
14. S
15. S
16. S
6π₯ + 7
2π₯ β 3
β 1 = ________, ________________________________________
β 1 = ________, ________________________________________
β 2 = ________, ________________________________________
β 3 = ________, ________________________________________
β 1 = ________, ________________________________________
β 2 = ________, ________________________________________
Foundations of Math 11 Updated January 2020
13 Adrian Herlaar, School District 61 www.mrherlaar.weebly.com
Answer Key β Section 4.1
Please see Section 4.1 on the Website for Detailed Solutions
1. π΄ππππ 1: 80Β°; π΄ππππ 2: 80Β° 2. π΄ππππ 1: 60Β° 3. π΄ππππ 1: 100Β°; π΄ππππ 2: 100Β° 4. π΄ππππ 1: 65Β°; π΄ππππ 2: 115Β° 5. π΄ππππ 1: 20Β°; π΄ππππ 2: 60Β°; π΄ππππ 3: 60Β° 6. π΄ππππ 1: 55Β°; π΄ππππ 2: 15Β° 7. π΄ππππ 1: 120Β°; π΄ππππ 2: 60Β° 8. π΄ππππ 1: 35Β°; π΄ππππ 2: 35Β°; π΄ππππ 3: 55Β° 9. π΄ππππ 1: 57Β°; π΄ππππ 2: 128Β°; π΄ππππ 3: 123Β° 10. π΄ππππ 1: 45Β°; π΄ππππ 2: 70Β°; π΄ππππ 3: 70Β°; π΄ππππ 4: 65Β° 11. π΄ππππ 1: 65Β°; π΄ππππ 2: 115Β° 12. π΄ππππ 1: 20Β°; π΄ππππ 2: 110Β° 13. π΄ππππ 1: 121Β° 14. π΄ππππ 1: 139Β° 15. π΄ππππ 1: 130Β°; π΄ππππ 2: 25Β°; π΄ππππ 3: 65Β° 16. π΄ππππ 1: 100Β°; π΄ππππ 2: 80Β°
Foundations of Math 11 Updated January 2020
14 Adrian Herlaar, School District 61 www.mrherlaar.weebly.com
Extra Work Space