Question

Prove Triangle ABC is an isosceles

Answer 1

Well first you need to know what makes a triangle isosceles in terms of angles. Then you'd wanna look at the angles using rules for the angles inside triangles.

Answer 2

but how will I make use of the special segments to prove abc is an isosceles? :(

Answer 3

Segments inside the triangle? Well, you know the ones with those small boxes are 90 degrees. And the total interior angle of a triangle is....? And I see you have those equal signs on one of the lines inside the triangle. You should have another one on another line segment, signifying that they are of equal lengths. Triangles with same sides have the same angles.

Answer 4

i need to write a two-column proof for this and I still dont know how to get started :(

Answer 5

start wid whats given,

Answer 6

and use @wolfe8 steps, prove the base angles are congruent first, then conclude sides opposite to them are congruent

Answer 7

give it a try...

Answer 8

Just always remember, the prrof on isoceles triangles is that it has 2 or more sides congruent and therefore, its base angles must be equal as well :)

Answer 9

But I cant use ITT as a reason. Probably I should use CPCTC for this :(

Answer 10

yes ! u must use CPCTC to prove base angles are congreuent,
put down statements first... then u can tweak them :)

Answer 11

hey @kelceyroche , maybe you can isolate the two right triangles there and you can prove that they are congruent, with than you can prove that the triangle is isosceles

Answer 12

oh yes got it, so triangle ACD is congruent to triangle ABE by AAS theorem then Angle B is Congruent to angle C by CPCTC? :D

Answer 13

Although I do think your drawing is missing something. I can't tell (although from experience I can guess) which lines have the same length.

Answer 14

there are Given statements, below the drawing, which say altitudes are congruent @wolfe8

Answer 15

and then Angle D is congruent to Angle E by perpendicularity?

Answer 16

yes both are right angles, so they're equal

Answer 17

can I use perpendicularity as a reason? :)

Answer 18

I am pretty sure the proof I outlined in the previous post is what they want you to do.

Answer 19

you familiar wid Leg Hypotenuse theorem ?

Answer 20

have a look at phi's pervious post :)

Answer 21

Do i still lack anything in my two-column proof?

Answer 22

@ganeshie8

Answer 23

yes, its bit off on 5th row,

Answer 24

what else do I lacK? :(

Answer 25

1. \(\large CD \perp AB \) 1. Given

2. \(\large BE \perp AC \) 2. Given

3. \(\large BE \cong CD \) 3. Given

4. \(\large \triangle DBC \cong \triangle ECB \) 3. By Leg Hypotenuse congruence

Answer 26

after that, you use CPCTC and to justify base angles are congruent

Answer 27

If you use the hypotenuse-leg theorem, then that shows the two right triangles are congruent.

You should put in a line saying you have a right triangle

Answer 28

1. \(\large CD \perp AB \) 1. Given

2. \(\large BE \perp AC \) 2. Given

3. \(\large BE \cong CD \) 3. Given

4. \(\large BC \cong CB \) 3. By reflexive property

5. \(\large \triangle DBC \cong \triangle ECB \) 3. By Leg Hypotenuse congruence

Answer 29

yes, add that statement also which explicitly says D and E are right angles

Answer 30

how am i suppose to reason that they are right angles? @phi
@ganeshie8

Answer 31

perpendicular BE forms a right angle

Answer 32

Thank you so much!!! and then I end with the SSS postulate?

Answer 33

1. \(\large CD \perp AB \) 1. Given

2. \(\large \angle BDC \cong 90 \) 2. definition of perpendicular

3. \(\large BE \perp AC \) 3. Given

4. \(\large \angle BEC \cong 90 \) 4. definition of perpendicular

5. \(\large BE \cong CD \) 5. Given

6. \(\large BC \cong CB \) 6. By reflexive property

7. \(\large \triangle DBC \cong \triangle ECB \) 7. By Leg Hypotenuse congruence

Answer 34

SSS must never enter this proof, your next step should be CPCTC

Answer 35

sss is the "long way" to show 2 right triangles are congruent. the hypotenuse-leg theorem is the short way: if you have a right triangle, and the hypotenuse and leg are congruent to another right triangle, the triangles are congruent. now you can do CPCT to show the base angles ABC and ACB are congruent. and then two congruent base angles mean you have an isosceles triangle

Answer 36

you would use SSS if you did not know the hypotenuse - leg theorem

Given: hypotenuse and one leg congruent with those of another right triangle, you can say by pythagoras that the 3rd side is congruent, and so have 3 congruent sides., so by sss you have congruent triangles.

Answer 37

here is the complete proof, see if it makes sense

1. \(\large CD \perp AB \) 1. Given

2. \(\large \angle BDC \cong 90 \) 2. definition of perpendicular

3. \(\large BE \perp AC \) 3. Given

4. \(\large \angle BEC \cong 90 \) 4. definition of perpendicular

5. \(\large BE \cong CD \) 5. Given

6. \(\large BC \cong CB \) 6. By reflexive property

7. \(\large \triangle DBC \cong \triangle ECB \) 7. By Leg Hypotenuse congruence

8. \(\large \angle DBC \cong \angle ECB \) 8. By CPCTC

9. \(\large AB \cong AC \) 9. Converse of isosceles triangle theorem

10. \(\large \triangle ABC \) is isosceles 10. By definition of isosceles triangle (two sides congruent)

Answer 38

hank you so much!! we didn't study HL yet but now I can explain it, haha thank you so so so so much!!

Answer 39

np :) if you didnt study HL, better take it all the way to SSS or SAS... or watever u familiae wid :)

Answer 40

But I think our teacher gives these homeworks bc she wants us to read about all these in advance :D so atleast I can be active in class tomw :D

Answer 41

that sound good :) but again, if u want to go SSS way, u just need to include two more statements between 6th and 7th statements above

Answer 42

so how Do i do it SSS?

Answer 43

1. \(\large CD \perp AB \) 1. Given

2. \(\large \angle BDC \cong 90 \) 2. definition of perpendicular

3. \(\large BE \perp AC \) 3. Given

4. \(\large \angle BEC \cong 90 \) 4. definition of perpendicular

5. \(\large BE \cong CD \) 5. Given

6. \(\large BC \cong CB \) 6. By reflexive property

6.i. \(\large CD \cong BE \) 6.i. By pythagorean theorem, if two leg pairs are same, third leg pair wud be same as well

7. \(\large \triangle DBC \cong \triangle ECB \) 7. By SSS congruence

8. \(\large \angle DBC \cong \angle ECB \) 8. By CPCTC

9. \(\large AB \cong AC \) 9. Converse of isosceles triangle theorem

10. \(\large \triangle ABC \) is isosceles 10. By definition of isosceles triangle (two sides congruent)

Answer 44

thats SSS. Enjoy.. :)

Answer 45

thank you so much!!! :D

Answer 46

yw :)