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simonmagusflies

Member

Registered: 2021-05-23

Posts: 32

Help with parallel angles please

I need help with 2 questions, please.
Here's the angle:

The questions are:
In the figure linked, is AB is parallel to CD...
1. If angle 5 = 30° and angle 3 = 150°?
A. Yes because angles 5 and 3 are same-side interior angles.
B. Yes because angles 5 and 3 are alternate interior angles
C. Yes because angles 5 and 3 are vertical angles.
D. No because angles 5 and 3 are vertical angles.
E. No because angles 5 and 3 are corresponding angles.
F. No because angles 5 and 3 are alternate interior angles

2. If angle 4 = 80° and angle 6 = 100°?
A. Yes because angles 4 and 6 are vertical angles.
B. Yes because angles 4 and 6 are same-side interior angles.
C. Yes because angles 4 and 6 are corresponding angles.
D. No because angles 4 and 6 are vertical angles.
E. No because angles 4 and 6 are alternate interior angles.
F. No because angles 4 and 6 are alternate exterior angles.

I'm pretty sure both answers are one of the "no"s, but I'm not sure of the reason.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Help with parallel angles please

hi simonmagusflies

Q1.  If the lines are parallel then angle 1 = angle 5 (corresponding angles) so 30 = angle 5 = angle 1. 

I then notice that angle 1 + angle 3 = 30 + 150 = 180 which is what we should get as these two angles male a straight line whatever the status with respect to parallelness.

So it seems to me that the lines are parallel.

Q2. My reasoning is similar here.  (6 and 2 would have to be equal and 2 and 4 add up to 180 , which they do)

So now I'm looking for a 'yes' answer in both cases. 

Have a look at this page and the ones linked from it:

https://www.mathsisfun.com/geometry/transversal.html

They're not alternates, nor corresponding.  Vertical angles are what I have always called vertically opposite angles and they're not that. Same-side interior angles are not mentioned by MIF and I've never met the term in 47 years of teaching geometry.  If you post all the definitions of these terms I'll give it a go and try to find sensible answers.

edit: tried googling same side interior angles (Mr Google seems to have heard of everything!) and up popped what we need to do this question.  Suggest you do the same. 

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

simonmagusflies

Member

Registered: 2021-05-23

Posts: 32

Re: Help with parallel angles please

Thanks for answering, here are the definitions:

1. Interior angles:
"No matter what the measurements are, angles 3, 4, 5 and 6 are called interior angles because they are in between lines AB and CD."

2. Exterior angles:
"Angles 1, 2, 7 and 8 are called exterior angles. Angles 1 and 8 are called alternate-exterior angles, as are angles 2 and 7."

3. Alternate-interior angles:
"Angles 3 and 6 are called alternate-interior angles, as are angles 5 and 4."

4. Corresponding angles:
"Angles 2 and 6 are called corresponding angles, as are angles 1 and 5, angels 3 and 7, and angels 4 and 8."

5. Vertical angles:
"Also regardless of the measurements, angles 1 and 4 will be the same size, as will angles 2 and 3, angles 5 and 8, and angles 6 and 7.  These are called vertical angles."

When AB is parallel to CD, the following statements are true:
- all corresponding angles are congruent, the same size. (And the corresponding angles are ONLY the same size if the lines are parallel.) Thus by knowing the measurement of 1 angle in the group, you could name the measurements of all the angles in the picture.
- alternate-interior angles are congruent
- corresponding angle are congruent
- alternate-exterior angles are congruent
- same-side interior angles will be supplementary. Angles 3 and 5 in the image above are same-side interior angles. Since angles 3 and 4 are supplementary and angles 4 and 5 are the same measure (alternate-interior angles are congruent), angles 3 and 5 are supplementary. Thus same-side interior angles are supplementary.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Help with parallel angles please

Thanks. That's really helpful.  I'll keep a note of those for future reference.  For example, what your course calls vertical angles, I was taught to call vertically opposite angles.

anyway back to the questions.  In Q1 angles 3 and 5 are supplementary and in Q2 angles 4 and 6 are supplementary, ie add up to 180.  Same side interior angles that add to 180 mean the lines are parallel; same reason for both questions.

Hope that helps,

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

simonmagusflies

Member

Registered: 2021-05-23

Posts: 32

Re: Help with parallel angles please

So, to both, the answer is "Yes because angles 5/4 and 3/6 are same-side interior angles."?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Help with parallel angles please

I'm hoping that is the case.  Fingers crossed!

I had a look at Euclid's Elements and did a word search.  What I have is a translation from the original Greek so there may be some loss in the translation but he uses 'vertically opposite', 'corresponding' and 'alternate'.  I also checked and he does not use the term 'ray' at all. 

https://en.wikipedia.org/wiki/Euclid

Mathematicians have worked with his version of geometry for over two thousand years.  I'm happy to follow the master.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

simonmagusflies

Member

Registered: 2021-05-23

Posts: 32

Re: Help with parallel angles please

Thanks, it was correct! You're terrific!