Question

A kite is a quadrilateral with two pairs of adjacent, congruent sides. The vertex angles are those angles in between the pairs of congruent sides. Prove the diagonal connecting these vertex angles is perpendicular to the diagonal connecting the non-vertex angles. Be sure to create and name the appropriate geometric figures. This figure does not need to be submitted.

Answer 1

@Vocaloid

Answer 2

I think I am mostly confused by the wording can you kind of explain what it is asking?

Answer 3

"two pairs of adjacent, congruent sides"

let's draw this

Answer 4

the vertex angles are the angles between congruent sides, so these two angles are the vertex angles:

Answer 5

the non-vertex angles are the other two angles

Answer 6

the question is asking you to prove that these two diagonals are perpendicular (90 degrees)

Answer 7

this might take me a moment, let me start working on the proof

Answer 8

Yah because it says not to include the diagram so I have to put that in word form I guess.

Answer 9

Take your time Voca

Answer 10

alright I think I have an idea

we can use SSS to show that each of the two halves (the left half and the right half) of the kite are equal, since the sides have been marked as equal

Answer 11

because the two triangles are congruent, their angles must also be congruent, meaning we can mark the angles at the top like so:

Answer 12

now, we can use the SAS theory to show that the two top triangles are equal, since we have two congruent sides and the angle in between them:

Answer 13

then, if those two triangles are equal, their corresponding angles must also be equal

Answer 14

now, those two angles in the middle (the ones marked with three lines) are both equal, and together they make up a straight line (180 degrees) so they must each be 90

Answer 15

Hmmm I think I get it

Answer 16

vertical angles are congruent, so the other two angles must also be 90 each

Answer 17

thus proving that each of the angles made by the intersecting angles are 90 thus they are congruent

qed

Answer 18

hope that helps, let me know if anything was unclear (the diagrams are a bit messy)

Answer 19

Thanks Voca you are def the best