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CCSS.Math:

we're told that quadrilateral ABCD is a rhombus and what they want us to prove is that their diagonals are perpendicular that AC is perpendicular to B D now let's think about everything we know about a rhombus first of all a rhombus is a special case of a parallelogram and a parallelogram the sides opposite sides are parallel so that side is parallel to that side these two sides are parallel and in a rhombus not only are the opposite sides parallel it's a parallelogram but also all of the sides have equal lengths so this side is equal to this side which is equal to that side which is equal to that side right over there now there's other interesting things we know about the diagonals of a parallelogram which we know all rhombi are parallelograms the other way around is not necessarily true we know that for any parallelogram and a rhombus is a parallelogram then the diagonals bisect each other so for example let me label this point to the center let me label it point E we know that a e we know that AE is going to be equal to EC I'll put two slashes right over there and we also know that EB we also know that EB is going to be equal to EE D so this is all of what we know when someone just says that ABCD is a rhombus based on other things that we've proven to ourselves now we need to prove that AC is perpendicular to BD so an interesting way to prove it and you can kind of look at it just by eyeballing is if we can show that this triangle is is congruent to this triangle and that these two angles right over here correspond to each other then they have to be the same and they'll be supplementary and then there'll be 90 degrees so let's just prove it to ourselves so the first thing we see is we have a side aside and aside aside aside and aside so we can see the triangle let me write it here triangle let me just in a new color we see that triangle triangle a be a triangle a be e is congruent to triangle C be e C II and we know that by side-side-side congruence we have a side to side to the side a side a side and a side and then once we know that we know that all the corresponding angles are congruent in particular we know that a EB we know that angle a EB angle a EB is going to be congruent to angle so a EB to C EB C EB to angle C EB because they are corresponding angles of congruent corresponding angles of congruent triangles so this angle right over here is going to be equal to that angle over there and we also know that they are supplementary and so B if they're both supplementary so we also know also and let me write it this way they're congruent and they are supplementary supplementary so we have these two are going to have the same measure and they need to add up to 180 degrees so if I have two things that are the same thing and they add up to 180 degrees what does that tell me so that tells me that angle the measure of angle a a B is equal to the measure of angle C EB which is equal to which must be equal to 90 degrees they're the same measure and they are supplementary so this is a right angle and then this is a right angle and obviously if this is a right angle this angle down here is a vertical angle that's going to be a right angle if this is a right angle this over here is going to be a vertical angle and you see the diagonals intersect at a 90 degree angle so we've just proved so this is interesting a parallelogram the diagonals bisect each other for a rhombus where all the sides are equal we've shown that not only do they bisect each other but they're perpendicular bisectors of each other