section 4.1 geometry of parallel lines this booklet

Foundations of Math 11 Updated January 2020

1 Adrian Herlaar, School District 61 www.mrherlaar.weebly.com

Section 4.1 – Geometry of Parallel Lines

This booklet belongs to: Block:

First let’s look at some vocabulary

a) Acute – an angle between 0 and 90 degrees

b) Obtuse – an angle between 90 and 180 degrees

c) Straight – angle exactly 180 degrees

d) Right – angle exactly 90 degrees

e) Complementary – two angles that add up to 90 degrees

f) Supplementary – 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 𝑙1 𝑎𝑛𝑑 𝑙2 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 a…

POSTULATE – 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 𝑙1

𝑙2

Transversal



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

∠1 𝑎𝑛𝑑 ∠3 are vertical angles

∠2 𝑎𝑛𝑑 ∠4 are vertical angles

6 5

7 8

4 3

2 1 𝑙1

𝑙2

∠1 = ∠5

∠2 = ∠6

∠3 = ∠7

∠4 = ∠8

This means parallel

𝑙1 ∥ 𝑙2

1 2

3 4



Proof – Vertical Angles are Equal

Given: ∠1 𝑎𝑛𝑑 ∠2 are vertical angles

Prove: ∠1 = ∠2

Proof Statement Reason

1. ∠1 + ∠3 = 180° Angles on a line add to 180° (supplementary) 2. ∠2 + ∠3 = 180° Angles on a line add to 180° (supplementary) 3. ∠1 + ∠3 = ∠2 + ∠3 Both equal to 180° (substitution) 4. ∠1 = ∠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 𝑙1𝑎𝑛𝑑 𝑙2 are intersected by a transversal, the four angles between the lines are

called interior angles

∠3, ∠4, ∠5, 𝑎𝑛𝑑 ∠6 are interior angles

∠3 𝑎𝑛𝑑 ∠6 are alternate interior angles

∠4 𝑎𝑛𝑑 ∠5 are alternate interior angles

Proof – Alternate Interior Angles of Parallel Lines are Equal

Given: 𝑙1 ∥ 𝑙2

Prove: ∠4 = ∠5


1. 𝑙1 ∥ 𝑙2 Given 2. ∠1 = ∠4 Vertical Angles 3. ∠1 = ∠5 Corresponding Angles 4. ∠4 = ∠5 Substitution (both equal to ∠1)

3 2

1

6 5

7 8

4 3 2 1

𝑙1

𝑙2

𝑙1 ∥ 𝑙2

5

4

1 𝑙1

𝑙2

𝑙1 ∥ 𝑙2



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 𝑙1𝑎𝑛𝑑 𝑙2 are intersected by a transversal, then the interior angles on the same

side of the transversal are called co-interior angles

∠3, ∠4, ∠5, 𝑎𝑛𝑑 ∠6 are interior angles

∠3 𝑎𝑛𝑑 ∠5 are co-interior angles

∠4 𝑎𝑛𝑑 ∠6 are co-interior angles

Proof – Co-Interior Angles of Parallel Lines are Supplementary

Given: 𝑙1 ∥ 𝑙2

Prove: ∠3 + ∠5 = 180°


1. 𝑙1 ∥ 𝑙2 Given 2. ∠3 = ∠6 Alternate interior Angles 3. ∠5 + ∠6 = 180° Angles on a line (Supplementary) 4. ∠5 + ∠3 = 180° Substitution (∠3 for ∠6) 5. ∠3 + ∠5 = 180° 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

𝑙1

𝑙2

𝑙1 ∥ 𝑙2

5

6

3

𝑙1

𝑙2

𝑙1 ∥ 𝑙2



The Sum of Angles in a Triangle

We will use our knowledge of parallel lines to prove this most importntat theorem.

Given: ∆𝐴𝐵𝐶

Prove: ∠1 + ∠2 + ∠3 = 180°


1. Draw line DC parallel to AB Construction 2. ∠3 + ∠4 = ∠𝐷𝐶𝐵 Angle Addition 3. ∠𝐷𝐶𝐵 + ∠2 = 180° Co-Interior Angles 4. ∠3 + ∠4 + ∠2 = 180° Substitution (From step 2) 5. ∠1 = ∠4 Alternate Interior Angles 6. ∠1 + ∠2 + ∠3 = 180° Substitution

Angle Sum of a Triangle Theorem The Sum of angles in a triangle is 180°

Summary

Parallel Lines and a Transversal

Vertical Angles

∠1 = ∠4

∠2 = ∠3

∠5 = ∠8

∠6 = ∠7

Corresponding Angles

∠1 = ∠5

∠2 = ∠6

∠3 = ∠7

∠4 = ∠8

Alternate Interior Angles

∠3 = ∠6

∠4 = ∠5

Co-Interior Angles

∠3 + ∠5 = 180°

∠4 + ∠6 = 180°

C

D

D

D

3

D

4

D

2

D

1

D

B

D

A

D

6 5

7 8

4 3 2 1

𝑙1

𝑙2

𝑙1 ∥ 𝑙2



Find all the missing angles and state the reasons for each answer.

Example:

Solution:

∠1 = 50° co-interior angles (2 ∗ 40° + 2𝑥 = 180° → 2𝑥 = 100° → 𝑥 = 50°

∠2 = 90° sum of angles in a triangle 40° + 50° + 𝑦 = 180° → 𝑦 = 90°

Example:

Solution:

∠1 = 70° supplementary angles plus sum of a triangle

∠2 = 70° alternate interior angles

∠3 = 20° supplementary angles plus sum of angles in a triangle

40°

1

90°

00°



Example:

Solution:

∠1 + ∠2 = 180° co-interior angles

𝑥2 − 25𝑥 + 𝑥 = 180

𝑥2 − 24𝑥 − 180 = 0

(𝑥 − 30)(𝑥 + 6) = 0

𝑥 = −6 𝑎𝑛𝑑 30, 𝑟𝑒𝑗𝑒𝑐𝑡 − 6 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 𝑤𝑒 𝑐𝑎𝑛′𝑡ℎ𝑎𝑣𝑒 𝑎 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡

∠1 = 𝑥2 − 25𝑥 → (30)2 − 25(30) → 150°

2

1

∠1 = (𝑥2 − 25𝑥)°

∠2 = 𝑥°

Find the value of ∠1.



Section 4.1 – Practice Problems

For the following questions, solve for the missing angles and give the reason.

1.

2.

3.

∠1 = ________, ________________________________________

∠2 = ________, ________________________________________

∠1 = ________, ________________________________________

∠1 = ________, ________________________________________

∠2 = ________, ________________________________________

2

2



4.

5.

6.

∠1 = ________, ________________________________________

∠2 = ________, ________________________________________

∠1 = ________, ________________________________________

∠2 = ________, ________________________________________

∠3 = ________, ________________________________________

∠1 = ________, ________________________________________

∠2 = ________, ________________________________________

2

3 2

11



7.

8.

9.

10.

∠1 = ________, ________________________________________

∠2 = ________, ________________________________________

∠3 = ________, ________________________________________

∠4 = ________, ________________________________________

∠1 = ________, ________________________________________

∠2 = ________, ________________________________________

∠3 = ________, ________________________________________

∠1 = ________, ________________________________________

∠2 = ________, ________________________________________

∠3 = ________, ________________________________________

∠1 = ________, ________________________________________

∠2 = ________, ________________________________________

3

2



11. X

12. S

13. S

2

5

1

16𝑥 − 5

14𝑥 + 3

∠1 = ________, ________________________________________

∠2 = ________, ________________________________________

∠1 = ________, ________________________________________

∠2 = ________, ________________________________________

∠1 = ________, ________________________________________



14. S

15. S

16. S

6𝑥 + 7

2𝑥 − 3

∠1 = ________, ________________________________________

∠1 = ________, ________________________________________

∠2 = ________, ________________________________________

∠3 = ________, ________________________________________

∠1 = ________, ________________________________________

∠2 = ________, ________________________________________



Answer Key – Section 4.1

Please see Section 4.1 on the Website for Detailed Solutions

1. 𝐴𝑛𝑔𝑙𝑒 1: 80°; 𝐴𝑛𝑔𝑙𝑒 2: 80° 2. 𝐴𝑛𝑔𝑙𝑒 1: 60° 3. 𝐴𝑛𝑔𝑙𝑒 1: 100°; 𝐴𝑛𝑔𝑙𝑒 2: 100° 4. 𝐴𝑛𝑔𝑙𝑒 1: 65°; 𝐴𝑛𝑔𝑙𝑒 2: 115° 5. 𝐴𝑛𝑔𝑙𝑒 1: 20°; 𝐴𝑛𝑔𝑙𝑒 2: 60°; 𝐴𝑛𝑔𝑙𝑒 3: 60° 6. 𝐴𝑛𝑔𝑙𝑒 1: 55°; 𝐴𝑛𝑔𝑙𝑒 2: 15° 7. 𝐴𝑛𝑔𝑙𝑒 1: 120°; 𝐴𝑛𝑔𝑙𝑒 2: 60° 8. 𝐴𝑛𝑔𝑙𝑒 1: 35°; 𝐴𝑛𝑔𝑙𝑒 2: 35°; 𝐴𝑛𝑔𝑙𝑒 3: 55° 9. 𝐴𝑛𝑔𝑙𝑒 1: 57°; 𝐴𝑛𝑔𝑙𝑒 2: 128°; 𝐴𝑛𝑔𝑙𝑒 3: 123° 10. 𝐴𝑛𝑔𝑙𝑒 1: 45°; 𝐴𝑛𝑔𝑙𝑒 2: 70°; 𝐴𝑛𝑔𝑙𝑒 3: 70°; 𝐴𝑛𝑔𝑙𝑒 4: 65° 11. 𝐴𝑛𝑔𝑙𝑒 1: 65°; 𝐴𝑛𝑔𝑙𝑒 2: 115° 12. 𝐴𝑛𝑔𝑙𝑒 1: 20°; 𝐴𝑛𝑔𝑙𝑒 2: 110° 13. 𝐴𝑛𝑔𝑙𝑒 1: 121° 14. 𝐴𝑛𝑔𝑙𝑒 1: 139° 15. 𝐴𝑛𝑔𝑙𝑒 1: 130°; 𝐴𝑛𝑔𝑙𝑒 2: 25°; 𝐴𝑛𝑔𝑙𝑒 3: 65° 16. 𝐴𝑛𝑔𝑙𝑒 1: 100°; 𝐴𝑛𝑔𝑙𝑒 2: 80°



Extra Work Space

section 4.1 geometry of parallel lines this booklet

Documents