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.


@hartnn

Answer 1

@dan815

Answer 2

@cwrw238

Answer 3

Do you need a two-column proof, or just a paragraph proof?

Answer 4

A paragraph proof please.

Answer 5

@mathstudent55

Answer 6

Given: Kite ABCD with BA = BC & DA = DC
Prove: BD is perpendicular to AC

Answer 7

Ok so far?

Answer 8

Yes.

Answer 9

Ok, here we go.

Answer 10

Seg BA is congr to seg BC and seg DA is cong to seg DC by given
Seg BD is congr to seg BD by reflexive
Tr. BAD is congr to tr. BCD by SSS

Answer 11

<ABE & <CBE are congr. by CPCTC

Answer 12

Seg BE is congr to seg BE by reflexive.

Answer 13

Tr. ABE is congr to tr. CBE by SAS

Answer 14

Are you following so far?

Answer 15

Yes.

Answer 16

We're almost done.

Answer 17

<AEB is congr. to <CEB by CPCTC

Answer 18

Is there more?

Answer 19

Angles AEB and CEB are a linear pair so their measures add to 180 by the linear pair postulate. That means m<AEB + m<CEB = 180

Answer 20

By def of cong angles, m<AEB = m<CEB

Answer 21

Using m<AEB + m<CEB = 180 and m<AEB = m<CEB and substitution, we can arrive at:
m<AEB + m<AEB = 180 or 2m<AEB = 180 or m<AEB = 90.
By def of right angle, we can say that <AEB is a right angle.
That means lines BD and AC are perpendicular by def of perpendicular lines.

That is it.

Answer 22

Thank you!!

Answer 23

You're welcome.