Proving that Angles OBC and CDO are Equal in a Parallelogram

In summary, the homework statement is that if you want to solve a parallellogram you have to think outside the box.
  • #1
dirk_mec1
761
13

Homework Statement



A parallellogram ABCD has an interior point O sucht that [tex]
\alpha + \beta = 180^o
[/tex]

http://img413.imageshack.us/img413/5636/post102741235319763.png [Broken]

Prove that:

[tex]
\angle{OBC}=\angle{CDO}
[/tex]

Homework Equations



Definitions of a parallellogram.

The Attempt at a Solution


I don't know how to start can some give me a hint?
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
Hi dirk_mec1! :smile:

Think outside the box …

Hint: what theorem do you know (nothing to do with parallelograms) about two triangles with angles adding to 180º?

move one of the triangles around :wink:
 
  • #3
tiny-tim said:
Hi dirk_mec1! :smile:
Hi!

Think outside the box …
Do you mean box or parallellogram? :tongue:

Hint: what theorem do you know (nothing to do with parallelograms) about two triangles with angles adding to 180º?
Sum of the angles in a triangle is 180o.

move one of the triangles around :wink:
I've thought about this but I don't understand what you mean.
 
  • #4
dirk_mec1 said:
Do you mean box or parallellogram? :tongue:

Both! :biggrin:
Sum of the angles in a triangle is 180o.

That's sum of three angles … how about sum of two angles?

Hint: circles are very un-boxlike … :wink:
 
  • #5
Do you mean: ''in a cyclic quadrilateral, opposite angles are supplementary (their sum is π radians)''?
 
  • #6
dirk_mec1 said:
Do you mean: ''in a cyclic quadrilateral, opposite angles are supplementary (their sum is π radians)''?

That's the one! :biggrin:

Now shift one of the triangles around so as to make that cyclic quadrilateral, and then draw a … ? :wink:
 
  • #7
tiny-tim said:
Now shift one of the triangles around so as to make that cyclic quadrilateral, and then draw a … ? :wink:

I'm sorry Tim I've looked at it and I can't find opposite angles for which the sum is 180 deg. I do know that the opposite angles in the parallelogram are equal.
 
  • #8
dirk_mec1 said:
I'm sorry Tim I've looked at it and I can't find opposite angles for which the sum is 180 deg.

move OAB up to the top :wink:
 
  • #9
tiny-tim said:
move OAB up to the top :wink:

Actually what do you mean by "shifting a triangle"? You can't switch angles so you probably mean something else.
 
  • #10
make a copy of OAB and and put it at the top
 
  • #11
Ok, I did that and thus:

[tex](\angle{OAB} + \angle{CDO}) +(\angle{OBA} + \angle{DCO}) =180^o[/tex]

but you mention drawing something I guess it's a circle but I'm not sure...
 

1. How do you prove that angles OBC and CDO are equal in a parallelogram?

In order to prove that angles OBC and CDO are equal in a parallelogram, we can use the property that opposite angles in a parallelogram are congruent. This means that angles OBC and CDO, which are opposite angles, will have equal measures.

2. What is the definition of a parallelogram?

A parallelogram is a quadrilateral with two pairs of parallel sides. This means that the opposite sides of a parallelogram are parallel and equal in length.

3. Can you use the properties of a parallelogram to prove that angles OBC and CDO are equal?

Yes, we can use the properties of a parallelogram to prove that angles OBC and CDO are equal. In addition to the property of opposite angles being congruent, we can also use the property that consecutive angles in a parallelogram are supplementary. This means that the sum of angles OBC and CDO will be 180 degrees, and since they are equal, each angle will be 90 degrees.

4. What other information do you need to prove that angles OBC and CDO are equal?

In order to prove that angles OBC and CDO are equal, we also need to know that the sides opposite these angles are equal in length. This is another property of parallelograms, known as the opposite sides being congruent.

5. Can you use any other methods besides using the properties of a parallelogram to prove that angles OBC and CDO are equal?

Yes, we can also use other methods such as using theorems like the Alternate Interior Angles Theorem or the Vertical Angles Theorem. However, these theorems are often used to prove that lines are parallel, which can then be used to prove the equality of angles OBC and CDO by using the properties of a parallelogram.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
17K
Replies
9
Views
3K
  • Precalculus Mathematics Homework Help
Replies
5
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
Replies
15
Views
3K
  • General Math
Replies
1
Views
2K
  • Introductory Physics Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Precalculus Mathematics Homework Help
Replies
1
Views
2K
Back
Top