Math 10-3

Unit 5 Angles and Parallel Lines

Lesson 53 Non-Parallel Lines and Transversals

Formula page

I expect that you will be able to use a formula page for this class,  but that will be determined by your teacher.

Your teacher may have a formula page they expect you to use.

Below is a link to the formula page I will use (at least for now!) when I’m doing questions for this class.

This may change, depending on feedback from your teacher.

I encourage you to download this page and print it so you have it handy when you are doing math.

LINK
https://drive.google.com/open?id=1EMauePx-P6WrtTIcd7-AtNnmJt8zqXpI

Explore the Math Pages 198 - 201

Read/Work through this section of your textbook.
These ideas will be developed in the examples

Your teacher will let you know which pages you need to do for this lesson.

This lesson has TONS of vocabulary.  TONS....

In the sections that follow,  I have provided video clips that go through all the words and ideas of the lesson, as well as providing links to interactive demonstrations of the ideas.

There is a LOT to cover here, but, on the other hand, it’s very practical and visual.

In the next lesson, we discuss almost EXACTLY THE SAME IDEAS as THIS lesson.

In this lesson we discuss crossing two lines with a transversal, and in the next lesson, we look at the SPECIAL CASE where we have two PARALLEL lines and a transversal.

It doesn’t really make sense to me to discuss this separately, so....  in the video tutorials that follow,  I will talk about the ideas of THIS lesson AND the NEXT lesson.

So this lesson will take a bit longer, but then you will have covered the ideas of the next lesson as well.

Explore the Math  (video and online interactive demonstrations)

Vertically Opposite Angles

vertically opposite angles: 
angles created by intersecting lines that share only a vertex
(note!  bad choice of words for this,  the angles certainly DON’T need to be vertical!!)

Demonstration  (Open Math Reference)

LINK  (Geogebra)

Video Tutorial A

Lines and Transversals

transversal: 
a line that intersects two or more lines

LINK  (Geogebra)

Demonstration  (Open Math Reference)

Video Tutorial B

Corresponding angles

corresponding angles: 
angles that occupy the same relative position in two different intersections

Demonstration

LINK  (Geogebra)

Video Tutorial C

Interior angles of a transversal

Interior angles of a transversal
Created where a transversal crosses two (usually parallel) lines. 
Each pair of interior angles are inside the parallel lines, and on the same side of the transversal.

Demonstration

Video Tutorial D

Exterior angles of a transversal

Exterior Angles of a transversal. 
Exterior Angles are created where a transversal crosses two (usually parallel) lines. 
Each pair of these angles are outside the parallel lines, and on the same side of the transversal.

Demonstration

Video Tutorial E

Alternate Interior Angles

alternate interior angles: 
angles in opposite positions between two lines intersected by a transversal

Open Math Reference

Geogebra

Video Tutorial F

Alternate Exterior Angles

alternate exterior angles: 
angles in opposite positions outside two lines intersected by a transversal

Demonstration

Geogebra

Video Tutorial G

Example 1 Pages 202 - 203

Read the example problem and try to answer the question YOURSELF if you can.
Use the textbook solution or the Video Tutorial if you need help.

Video Tutorial

Example 2 Pages 203 - 204

Read the example problem and try to answer the question YOURSELF if you can.
Use the textbook solution or the Video Tutorial if you need help.

Video Tutorial

Build Your Skills Pages 204 - 207

These questions all come from your textbook.

If you need extra practice or want to maintain your skills,  see the NEXT section for more practice questions.

Doing Practice Problems

The problems in this section reinforce the math presented in this lesson, and give you opportunities to build on and practice your skills.

•	Work through the practice problems for this lesson.
	These problems are for practice, you don’t need to hand them in!
	(your teacher may assign specific questions for you to complete)

•	Use the answers in the back of the book to CHECK YOUR WORK!

•	If you make a mistake, TRY TO FIX IT!

•	If you need help or want more information, view the video solutions provided.

•	If you still need more help,  CONTACT YOUR TEACHER!

•	EXTEND YOUR THINKING problems will be more challenging than the others.

Video Solution Links

Check your work for every problem you do!
If you make a mistake,  try to fix it!
Use the video solutions for more information.
If a problem still doesn’t make sense, ASK YOUR TEACHER FOR HELP!

({{"Question 1", "Question 2", "Question 3", "Question 4"}})

Develop and Maintain Your Skills

Doing these problems

These questions are provided to help you develop and maintain your skills with the ideas of this lesson.

These questions are modelled after the questions in the textbook,  so there is nothing different here, just more questions if you need them.

These questions are completely optional (unless your teacher specifically assigns them)

Each question has an answer AND a video solution.

DO your best to answer the question YOURSELF first!

Check your work with the answer provided.

If you make a mistake,  try to fix it!

Use the video solutions for more information.

If a problem still doesn't make sense, ASK YOUR TEACHER FOR HELP!

Practice Problems

Problem

In the following diagram, identify each of the following, and specify which lines and transversals you are using.

a) 	an interior angle on the same side of the transversal as ∠6

b) 	an angle corresponding to ∠2

c) 	an angle corresponding to ∠4

d) 	an alternate interior angle to ∠4

Answer

a) Using (Subscript[ℓ, 1])   and (Subscript[ℓ, 2]) , with transversal (Subscript[ℓ, 3]),  ∠2 and ∠6 are interior angles on the same side of the transversal.

b) Using (Subscript[ℓ, 1])   and (Subscript[ℓ, 2]) , with transversal (Subscript[ℓ, 3]) , ∠1 corresponds to ∠2.

c) Using (Subscript[ℓ, 1])   and (Subscript[ℓ, 2]) , with transversal (Subscript[ℓ, 4]) , ∠7 corresponds to ∠4.

d) Using (Subscript[ℓ, 3])   and (Subscript[ℓ, 4]) , with transversal (Subscript[ℓ, 2]) , ∠4 and ∠9 are alternate interior angles.

Video Solution Link

Problem

In the diagram below, identify the relationship between each pair of angles.

a) ∠7 and ∠8

b) ∠2 and ∠7

c) ∠1 and ∠6

d) ∠5 and ∠7

Answer

a) 	∠7 and ∠8 are alternate interior angles.

b) 	∠2 and ∠7 are corresponding angles.

c) 	∠1 and ∠6 are exterior angles on the same side of the transversal.

d) 	∠5 and ∠7 are interior angles on the same side of the transversal.

Video Solution Link

Problem

Given the diagram below, identify the following angles.

a) an alternate exterior angle to ∠2

b) an interior angle on the same side of the transversal as ∠7

c) an alternate interior angle to ∠4

d) an angle corresponding to ∠5

Answer

a) 	∠6 is the alternate exterior angle to ∠2.

b) 	∠3 is an interior angle on the same side of the transversal to ∠7.

c) 	∠3 is an alternate interior angle to ∠4.

d) 	∠3 is a corresponding angle to ∠5.

Video Solution Link

Problem

Identify each of the following angles. 
Name the two lines and the transversal you are using.

a) two angles corresponding to ∠1

b) an interior angle on the same side of the transversal as ∠10

c) an alternate interior angle to ∠5

d) two interior angles on the same side of the transversal as ∠8

Answer

a) 	Angles corresponding to ∠1:
	∠7, using lines  (Subscript[ℓ, 3])   and (Subscript[ℓ, 4]) with transversal (Subscript[ℓ, 1])
	∠3, using lines (Subscript[ℓ, 1])and (Subscript[ℓ, 2]) with transversal ℓ3 

b) 	Interior angle on the same side of the transversal as ∠10:
	∠4, using lines  (Subscript[ℓ, 3])   and (Subscript[ℓ, 4])  with transversal (Subscript[ℓ, 2])

c) 	Alternate interior angle to ∠5:
	∠10, using lines   (Subscript[ℓ, 3])   and (Subscript[ℓ, 4]) with transversal (Subscript[ℓ, 2])

d) 	Two interior angles on the same side of the transversal as ∠8:
	∠5, using lines  (Subscript[ℓ, 3])   and (Subscript[ℓ, 4]) with transversal (Subscript[ℓ, 2]) 
	∠7, using lines (Subscript[ℓ, 1]) and (Subscript[ℓ, 2]) with transversal (Subscript[ℓ, 4])

Video Solution Link

Problem

In the diagram below, where t is the transversal, identify two pairs of each of the following angles.

a) alternate interior angles

b) corresponding angles

c) interior angles on the same side of the transversal

Answer

a)   Alternate interior angles:
	∠3 and ∠5 
	∠2 and ∠8 

b) 	Corresponding angles:
	∠1 and ∠5 
	∠2 and ∠6 
	∠3 and ∠7 
	∠4 and ∠8 

c)	Interior angles on the same side of the transversal:
	∠2 and ∠5 
	∠3 and ∠8

Video Solution Link

Problem

A flashlight shines down onto a floor as shown in the diagram below. If the outer rays are considered to be two lines and the floor is a transversal, name a pair of corresponding angles.

Answer

There are two pairs of corresponding angles: ∠3 and ∠5, and ∠4 and ∠6.

Video Solution Link

Problem

In the diagram below, calculate the sizes of each of the angles.

Answer

∠1 = 95°
∠2 = 85°
∠3 = 95°
∠4 = 68°
∠5 =112°
∠6 = 68°

Video Solution Link

Problem

Calculate the sizes of the six angles indicated in the figure.

Answer

∠1 = 60°
∠2 = 120°
∠3 = 60°
∠4 = 110°
∠5 =70°
∠6 = 110°

Video Solution Link