Question

Given triangle ABC with right angle at C and altitude CD, draw the picture and explain why two triangle are congruent. Triangle ABC= triangle CBD

Answer 1

@mtbender74

Answer 2

Triangle ABC and Triangle CBD are congruent?

Answer 3

yup.

Answer 4

I can get them similar...but not congruent

Answer 5

I'm pretty sure they mean similar...since it's pretty impossible for them to be congruent

Answer 6

The equation is past #50. It's #1.

Answer 7

Since Triangle CBD must be smaller than Triangle ABC

Answer 8

Couldn't it look like this?

Answer 9

no, because it said that CD was the altitude of triangle ABC

Answer 10

Well then this makes no sense lol

Answer 11

i'm pretty sure they meant to say similar...this is a common question asked...showing that the altitude of a right triangle splits the triangle into 2 similar triangles

Answer 12

let's work that and maybe you can get clarification from your teacher?

Answer 13

Yeah okay

Answer 14

So for congruent triangles, you would prove with things like SSS, SAS, etc...

For similar triangles, we use AAA. Show that the two triangles have all the same angle measures and you're done.

Answer 15

Looking at my picture, what can you say about angle CDB?

Answer 16

Angle? you mean triangle?

Answer 17

nope...just the angle CDB

Answer 18

Then I don't really know.

Answer 19

that's ok...we know that CD is the altitude of triangle ABC...altitude make what kind of angle?

Answer 20

90 degree angle?

Answer 21

lol...confident :)

correct. altitudes make 90 degree angles. So the angle CDB is a right angle...so we've established...

angle ACB = angle CDB as they are both right angles.

Answer 22

every step, we need to compare an angle in the big triangle with it's similar counterpart in the little triangle

Answer 23

with me so far?

Answer 24

Yup

Answer 25

good. let's look at the picture again...this time, we'll look at angle ABC in the big triangle ad angle CBD in the small one...i've marked them in the drawing. What can we say about these two and why?

Answer 26

It's the same angle so it's going to be the same for each triangle.

Answer 27

right. angle ABC = angle CBD because they are the same angle.

so as of now we have that...

angle ACB = angle CDB (both right)
angle ABC = angle CBD (same angle)

all that's missing is angle BAC = angle BCD

for that, i ask, how many degrees are there total in a triangle?

Answer 28

180

Answer 29

right...so if

ACB + CBA + BAC = 180 and
CDB + DBC + BCD = 180

and we've already established that the first two in each of those equations are the same, what must be the case with the third?

Answer 30

Well then they would both have to be the same as well to each =180

Answer 31

Exactly!

So, angle BAC = angle BCD.

The technical reason is that equal parts taken from equal amounts have equal remainders, but saying that the sum of all of the angles in a triangle is 180 and from what we've already established, angle BAC = angle BCD should be sufficient. :)

so...in conclusion,

angle ACB = angle CDB
angle ABC = angle CBD
angle BAC = angle BCD

so triangle ACD is *similar* to triangle CDB by AAA similarity :)

Answer 32

Ohh...okay I get it haha. Well I have to eat dinner now, but I have two more afterwards :)

Answer 33

i might be around...look for me

Answer 34

Mmkay.