Foundations of Math 11 Updated January 2020
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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
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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
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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
Proof Statement Reason
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
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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°
Proof Statement Reason
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
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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°
Proof Statement Reason
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
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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°
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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.
Foundations of Math 11 Updated January 2020
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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
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4.
5.
6.
∠1 = ________, ________________________________________
∠2 = ________, ________________________________________
∠1 = ________, ________________________________________
∠2 = ________, ________________________________________
∠3 = ________, ________________________________________
∠1 = ________, ________________________________________
∠2 = ________, ________________________________________
2
3 2
11
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7.
8.
9.
10.
∠1 = ________, ________________________________________
∠2 = ________, ________________________________________
∠3 = ________, ________________________________________
∠4 = ________, ________________________________________
∠1 = ________, ________________________________________
∠2 = ________, ________________________________________
∠3 = ________, ________________________________________
∠1 = ________, ________________________________________
∠2 = ________, ________________________________________
∠3 = ________, ________________________________________
∠1 = ________, ________________________________________
∠2 = ________, ________________________________________
3
2
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11. X
12. S
13. S
2
5
1
16𝑥 − 5
14𝑥 + 3
∠1 = ________, ________________________________________
∠2 = ________, ________________________________________
∠1 = ________, ________________________________________
∠2 = ________, ________________________________________
∠1 = ________, ________________________________________
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14. S
15. S
16. S
6𝑥 + 7
2𝑥 − 3
∠1 = ________, ________________________________________
∠1 = ________, ________________________________________
∠2 = ________, ________________________________________
∠3 = ________, ________________________________________
∠1 = ________, ________________________________________
∠2 = ________, ________________________________________
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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°
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Extra Work Space