If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Geometry (all content)>Unit 5

Lesson 2: Quadrilateral proofs & angles

# Proof: Diagonals of a parallelogram

Sal proves that a quadrilateral is a parallelogram if and only if its diagonals bisect each other. Created by Sal Khan.

## Want to join the conversation?

• are their areas (<ABE - <DCE) equal?
• Yes because if the triangles are congruent, then corresponding parts of congruent triangles are congruent.
• At , he says that DEC is congruent to BAE but wait a minute, how can they be congruent? I don't get this... Because Angle E should be > than Angle BAE because DEC is obtuse and BAE is acute right? Or am I missing something?
• He's wrong over there. It should be DCE.
• Is there a nutshell on how to tell the proof of a parallelogram? I doubt it.
• `1.Both pairs of opposite sides are parallel2.Both pairs of opposite sides are congruent3.Both pairs of opposite angles are congruent4.Diagonals bisect each other5.One angle is supplementary to both consecutive angles (same-side interior)6.One pair of opposite sides are congruent AND parallel`
• how do you find the length of a diagonal?
• there can be many ways for doing so you can prove the triangles formed by the diagonals congruent and then find its value or you can use herons formula to do so.
• As a minor suggestion, I think it is clearer to mark the diagram with information we know will be true (subject to our subsequent proofs). In this case, when writing the proofs, there is a stronger visual connection between the diagram and what is being written.

The way it is done in the video, each time an angle is referred to in the proof, I find myself looking at the diagram and following the 3 letters to see the angle, as opposed to sighting a symbol already marked on the diagram identifying the angle.
• I think you are right about this. But I think Sal was trying to save time like he said with the abbreviations.
• What's alternate Interior angles?
• Lets say the two sides with just the < on it where extended indefinitely and the diagonal he is working on is also extended indefinitely just so you can see how they are alternate interior angles. if two lines are both intersect both a third line, so lets say the two lines are LINE A and LINE B, the third line is LINE C. the intersection of LINE A with LINE C creates 4 angles around the intersection, the same is also true about the LINE B and LINE C. There is a quadrant/direction for each of the 4 corners of the angles. So there would be angles of matching corners for each of the two intersections. Now alternate means the opposite of the matching corner. So it's one angle from one intersection and the opposite corner angle from the matching corner on the other intersection.

http://www.mathsisfun.com/geometry/alternate-interior-angles.html
• In all was there 2 diagonals in that parallelogram ?
• in a parallelogram there are maximum 2 diagonals to be drawn
(1 vote)
• Is there an easier way to understand proofs? For some reason, I just struggle to concentrate when it comes to the videos.
• Opposite sides are parallel, If you have a parallelogram ABCD, you can check if opposite sides like AB and CD, as well as BC and AD, are parallel by comparing their slopes. If the slopes of these pairs of sides are the same, then they're parallel.

Opposite sides are equal in length: You can prove this by comparing the lengths of opposite sides. For example, if the distance between A and B equals the distance between C and D, and the distance between B and C equals the distance between A and D, then the opposite sides are equal.

Diagonals bisect each other: The point where the diagonals intersect divides each diagonal into two equal parts. You can prove this by finding the midpoint of each diagonal and showing they meet at the same point.

Consecutive angles are supplementary: The total of consecutive angles in a parallelogram is always 180 degrees. This can be proved by knowing that opposite angles are equal, and the sum of angles in any quadrilateral is always 360 degrees.

Hope this helped SharpDepressedPenguin