Question

Prove the Converse of the Pythagorean Theorem using similar triangles. The Converse of the Pythagorean Theorem states that when the sum of the squares of the lengths of the legs of the triangle equals the squared length of the hypotenuse, the triangle is a right triangle. Be sure to create and name the appropriate geometric figures.

Answer 1

@ash2326 @aaronq @andriod09 @BulletWithButterflyWings @BostonBlue @e.mccormick @Erin001001 @stabar @iiamentertainment @jhonyy9 @goformit100 @hba @Haunted9 @lncognlto @rizwan_uet can someone help

Answer 2

i am not good at math so idk

Answer 3

ok thx anyways :)

Answer 4

http://www.qc.edu.hk/math/Junior%20Secondary/Converse%20of%20Py%20th.htm

Answer 5

I hate to just post that, but I cant draw on my tablet

Answer 6

tell me if you dont understand something

Answer 7

ok but thank you :)

Answer 8

EF = BC = a
ÐF is a right angle.
FD = CA = b
triangle
EF = BC = a
angle F is a right angle.
FD = CA = b
In triangle DEF,
By Pythagoras Theorem,
a2 + b2 = c2
the given
AB=c= a^2 + b^2 square root

Theorefore AB = DE
But by construction, BC = EF
and CA = FD
triangle ABC congruent to DEF (S.S.S.)

Answer 9

like this?

Answer 10

yep, as long as you understand it.

Answer 11

and i have this question

Answer 12

i do and thanks to you :D
the link explains everything :)

Answer 13

@skullpatrol @phi @Flisk

Answer 14

use asa

Answer 15

can you show and explain ? :)

Answer 16

you are given angle B= < F

<DCE = < DEC

any guesses what angle-side-angle we are going to use ?

Answer 17

prove two angles are the same and a side?

Answer 18

yes, but what angles and side are we going to use ?

Answer 19

<B= < F

is given. so we should use those angles

they tell us <DCE = < DEC

those angles are "vertical angles" to an angle in the triangle we are trying to show are congruent. that looks promising.

so it seems we want to use the side between <B and <ACB and the corresponding side in the other triangle

Answer 20

so thats given to

Answer 21

ok :)

Answer 22

almost. DF = DB, but we need to show EF= CB

Answer 23

but notice EF= DF - DC and CB= DB-DE

if you know DC= DE, and DF= DB then the difference will be the same

Answer 24

so you need to know why DC= DE
I would use isosceles triangle idea... if base angles are equal, then the opposite sides are equal

Answer 25

now i have to put the proof in order

Answer 26

yes, lots of details but the big picture is angle - side -angle

Answer 27

given: angle B= < F
<B= < F is given
this is what i wrote til now

Answer 28

<DCE = < DEC are verical angles
EF= DF - DC and CB= DB-DE
DC= DE opposite sides are equal
triangle ABE congruent to triangle GFE
by (ASA)

Answer 29

I hope you mean <DCE = < DEC given
then you say <ACB = <DCE vertical angles
also <FEG= < DEC vertical angles

now you want to say <ACB = <FEG
why is this true ?

Answer 30

we are doing the angles first. why is angle ACB = angle FEG ?

you start by saying

<DCE = < DEC given
<ACB = <DCE vertical angles
<FEG = < DEC vertical angles

Answer 31

if <DCE = <DEC you could say <ACB= <DCE means <ACB= <DEC
things equal to the same thing (in this case both <DEC and <ACB equal <DEC) are equal to each other so <ACB= <DEC

now you have
<ACB= <DEC things equal to the same thing are equal to each other
<FEG = < DEC vertical angles

now why is <ACB = <FEG ?

Answer 32

angle sides angle?

Answer 33

I guess this does not make sense to you.
notice that <ACB = <DEC and <FEG = *the same thing* < DEC
do you agree <ACB and <FEG both equal the same thing <DEC ?
why is <ACB = <FEG ?

Answer 34

well yah your rite it does not make sense but i get the first part up to vertical angles

Answer 35

really ok that looks simple !

Answer 36

you need the idea that if two different things equal the same thing then they equal each other.

for example,
if A = C and B=C then A=B

this is common sense, so it should make a little sense.

For this problem you say
<DCE = < DEC given
<ACB= <DCE vertical angles
<ACB= <DEC things equal to the same thing are equal to each other

<FEG = < DEC vertical angles

<ACB = <FEG things equal to the same thing are equal to each other

Answer 37

that is the whole sequence to show <ACB=<FEG

Answer 38

now tackle the side part.

First step.
<DCE = < DEC given
side DC= side DE why ?

Answer 39

but isnt there at the end ASA?

Answer 40

vertical angles

Answer 41

the stuff to show <ACB=<FEG is part I. the first angle. now we need part II, the side

Answer 42

corresponding sides so..

Answer 43

First step.
<DCE = < DEC given
side DC= side DE why ?

sides are not vertical angles.
see http://www.regentsprep.org/Regents/math/geometry/GP6/Lisosceles.htm
and look carefully for "converse theorem"

Answer 44

If two angles of a triangle are congruent, the sides opposite them are congruent. this

Answer 45

yes. Do you follow what it says? we are given the base angles are =, so using this theorem (which has been proven by some smart person) we *know* the opposite sides (DC and DE) are equal

Answer 46

yahh i get it now :) thank you for that! ^_^

Answer 47

now we know
DC +CB= DB

Answer 48

ok

Answer 49

now we know
DC +CB= DB sum of parts equal the whole
and DE+EF= DF sum of parts equal the whole

DB= DF given
so we can say DC +CB= DF (replace DB with DF)
and DC+CB = DE+EF things equal to the same thing are equal to each other
we just proved DC= DE so we can say
DC+CB= DC+EF (replace DE with DC)
CB= EF (subtracting equal amounts keeps the equality)

Answer 50

that proves the sides

the last part, is <B= <F given

so you have shown angle-side-angle are congruent, so the two triangles ABC= GFE

Answer 51

wow that was a long answer and im sure your tired of all this typing im sorry and thank you so so so much for all you help!!

Answer 52

yes, lots of details.

Answer 53

@bayan are u still in geometry can u help me