Math Is Fun Forum

CurlyBracket

Member

From: Your Maths Textbook

Registered: 2022-01-03

Posts: 155

Angles of a Parallelogram

Hi,
I have a question about Parallelograms:

      PQRS is a parallelogram. MP and NP divide ∠SPQ into three equal parts (∠MPQ >∠NPQ) and MQ and NQ divide ∠RQP into 3 equal parts (∠MQP > ∠NQP).
      If k(∠PNQ) = (∠PMQ) then k = ?

I'm not able to make a suitable diagram to agree with all the criteria of the figure.

Any help will be appreciated.

Last edited by CurlyBracket (2022-01-05 19:26:51)

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Filling the unforgiving minute with sixty seconds’ worth of distance run!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Angles of a Parallelogram

hi CurlyBracket

Welcome to the forum.

Here's my diagram for this.  I've used x and y to indicate the equal angles.

I can give you a solution hint if you need it.

Exceptio probat regulum!

Which 'rule' do you have in mind?

Bob

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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

CurlyBracket

Member

From: Your Maths Textbook

Registered: 2022-01-03

Posts: 155

Re: Angles of a Parallelogram

Hi Bob!

Thanks for the reply.

From your diagram, I can conclude that:

3x+3y=180
x+y=60

In Δ PMQ:
2x+2y+ ∠M=180
∠M = 180 - 2(x+y)
     = 180 - 120.          [from (x+y=60)]
     = 60

In Δ PNQ:
x+y+ ∠N=180
∠N = 180 - x - y
     = 180 - 60
     = 120

k N = M
k = M / N
   = 60/120
   =1/2

Hence, the value of k is  0.5

Exceptio probat regulum!

Which 'rule' do you have in mind?

I normally use this in a humorous situation, where something partially or completely untrue has been said. What about you?

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Filling the unforgiving minute with sixty seconds’ worth of distance run!

Mathegocart

Member

Registered: 2012-04-29

Posts: 2,226

Re: Angles of a Parallelogram

Hello guys,
I know this is a little off-topic, but as a response to CurlyBracket's Latin phrase question, a Latin phrase I find most shrewd is:
Amicus certus in re incerta cernitur.
— Ennius

It means: One's friends are known in the hour of need. Indeed, an adage that will stay true for eternity.

Last edited by Mathegocart (2022-01-06 11:10:44)

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The integral of hope is reality.
May bobbym have a wonderful time in the pearly gates of heaven.
He will be sorely missed.

CurlyBracket

Member

From: Your Maths Textbook

Registered: 2022-01-03

Posts: 155

Re: Angles of a Parallelogram

Hi Mathegocart,

That's a nice proverb. I hadn’t heard of it before. It's certainly very true.
Here's another:
Caesar non est supra grammaticos

We use it in modern context as "Knowledge before Power". However, the origin story is very different.

The Roman emperor had made a grammatical mistake in one of his speeches. He became very angry on being corrected, and declared that grammar should be modified in order to match with his statement. At this, the Council replied "Caesar non est supra grammaticos," meaning 'The Emperor is not above Grammar.'

The saying quickly became popular, and we understand it in a slightly modified form today.

Last edited by CurlyBracket (2022-01-07 01:43:55)

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Filling the unforgiving minute with sixty seconds’ worth of distance run!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Angles of a Parallelogram

hi CurlyBracket

That's the value of k that I got, well done!

I'd not seen your Latin phrase before so I googled and couldn't find 'regulum' but rather 'regulam'. 

It seems to be an abbreviated version of:

"exceptio probat regulam in casibus non exceptis" meaning the exception that proves the rule.

In maths that is bizarre because any exception (counterexample) means the rules hasn't been proved, so I investigated further.

Here's an example of what I think it really means:

A museum sign says "Entry free on Sundays".  It doesn't say what happens the rest of the week but most people would infer that there is a charge on the other 6 days.  So Sunday's rule is the exception that implies a rule that applies for the rest of the week.

A similar example would be "No parking here on market days".

Bob

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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

CurlyBracket

Member

From: Your Maths Textbook

Registered: 2022-01-03

Posts: 155

Re: Angles of a Parallelogram

Hi Bob,

Thanks again!

Yes, the difference in spelling is quite possible. I had fished the proverb out of a general dictionary from about 1935 ( to my surprise, the heading was ‘Common, daily life Latin Phrases one encounters each day in the Workplace: A Hand-guide’ although I had not heard of any), so either there was a typographical error or the spelling changed over the years. I suppose the former is far more probable.

Frankly, I never noticed the perspective you’ve given. It’s quite practical, and it makes more sense for it to be ‘Encountered in the Workplace’. I searched for it too, and turns out it’s also used legally!

A humorous phrase just took on a whole new meaning for me!

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Filling the unforgiving minute with sixty seconds’ worth of distance run!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Angles of a Parallelogram

hi

I've checked out the Latin. regula is the noun meaning rule.  In Latin, nouns take different endings depending on the status of the noun in the sentence (eg. cactus, cacti). regula is a 1st declension noun with ...am when it's in the accusative ie the object rather than the subject in a sentence.  There is no Latin word regulum but I have found internet sources where it is used with that spelling; so I suspect you picked it up from someone who just spelt it wrongly. ?? But do we really need a book of Latin commonly encountered in the workplace ?? Unless you're a doctor, lawyer or Latin teacher I'd have thought it unlikely you'd meet much Latin. 

probat comes from the verb probare meaning to test, prove, or approve.

exceptio is the Latin noun in the nominative (subject) case (3rd declension) meaning exception.

Thus we have a simple Latin sentence structured <subject> <verb> <object> and

my attempt at a translation is "The exception proves the rule"

If you truly want to make use of it in your signature then you need a 'rule' as well, hence my original question.

How about "MIF members are fun. Exceptio probat regulam!" But this would imply that those who are not members are not fun.  I'll leave you to decide if you want to go that far. 

Bob

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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

CurlyBracket

Member

From: Your Maths Textbook

Registered: 2022-01-03

Posts: 155

Re: Angles of a Parallelogram

Hi Bob,

Latin grammar sounds interesting, I have not had the occasion to encounter it until now. Yes, I do suspect a typographical error.

As for ‘commonly encountered in the Workplace’, I can only guess. Either these Latin proverbs WERE common 95 years ago or the publisher overestimated the number of Latin proverbs the common office-goer is aware of and uses frequently. If I remember rightly, the list included over 120 proverbs, both Latin and of languages other than English such as French and German.

The idea for the signature sounds promising. I’ll try thinking of something along those lines.

Last edited by CurlyBracket (2022-01-22 21:22:17)

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Filling the unforgiving minute with sixty seconds’ worth of distance run!