Geometry Proofs (3 Viewers)

[Smile12345]

Hello All...

Could you please help me with the following question???? Diagram shown...

Given info: AD//BC, AX = DX, Angle ABC= Angle BCD



a) Show that Angle ABD = Angle ACD
b) Prove Triangle DAB is congruent to Triangle ACD
c) Show that AB = DC

Thanks heaps in advance, I appreciate your help...

 

[girlworld_club]

what's the question?

 

[Smile12345]

what's the question?

Sorry forgot to put it up... Please forgive me.

Would a) Be that they're the same angles on congruent triangles?

 

[girlworld_club]

a) For the first one, since Angle ABC= Angle BCD, and the lines are parallel. Then by alternate angles, Angle ABD = Angle ACD
b) By SAS. side: one side is shown to be equal, angle is the same (vertically opposite), and the other side but i'm not bothered to load the page again. Re: since angle abd and angle acd are equal then the two sides are equal (isosceles). anyone correct me if i'm wrong

 

[Smile12345]

a) For the first one, since Angle ABC= Angle BCD, and the lines are parallel. Then by alternate angles, Angle ABD = Angle ACD
b) By SAS. side: one side is shown to be equal, angle is the same (vertically opposite), and the other side but i'm not bothered to load the page again.

Thanks heaps.

 

[Smile12345]

For b) how do u use ver. opp. angles in Triangle's DAB and DAC? Wouldn't that be used more for C?

If I put up some others can you check to see I've done them right?

 

[HeroicPandas]

a) For the first one, since Angle ABC= Angle BCD, and the lines are parallel. Then by alternate angles, Angle ABD = Angle ACD
b) By SAS. side: one side is shown to be equal, angle is the same (vertically opposite), and the other side but i'm not bothered to load the page again. Re: since angle abd and angle acd are equal then the two sides are equal (isosceles). anyone correct me if i'm wrong

well done

 

[HeroicPandas]

HeriocPandas words: b) its not SAS, its AAS (using part a)

Thanks heaps Herioc... I've got AD is a common side, Angle ABD = ACD (Proven in a) and then what's the other angle?

please observe TRIANGLE AXD

 

[HeroicPandas]

lol sorry im BAD at 2U, angle ABC = angleBCD, i didnt read first part of question, good work girlwoodclub, i'll leave it to u

ignore everything i've said

 

[Smile12345]

lol sorry im BAD at 2U, angle ABC = angleBCD, i didnt read first part of question, good work girlwoodclub, i'll leave it to u

ignore everything i've said

No worries... Sorry I'm a bit confused!!

So a) is right... Then b)... It is SAS? I've got one angle and one side but need the last line of working... Please and thanks again...

Thanks for your time Girlworld_club...

 

[Smile12345]

hey girlworld_club.... Are you still there?? Can you please help me complete b please????

Thanks...

 

[girlworld_club]

side BX=CX

this can be proved by considering part a) which by alternate angles Angle ABD = Angle ACD. If these two base angles are equal, then the sides making the isosceles triangle are also equal. i hope it makes sense.

 

[Smile12345]

So I can use BX = CX even though there only part of the sides in Triangles DAB, ACD??

Yes, I understand why BX = CX. Thanks heaps.

 

[Drongoski]

Solution again (as requested)


a) ang XAD = ang ADX = @ say (base angles of isosceles triangle ADX)

ang ACB = ang CAD = @ (alt angles, AD//BC)

Similarly ang DBC = ang ADB = @

Now ang ABC = angle BCD (given)

.: ang ABC - ang DBC = ang BCD - ang ACB

i.e. ang ABD = ang ACD

b) In triangles DAB & ADC:

- ang ADB = ang DAC (shown in (a))

- ang ABD = ang DCA (shown in (a))

- AD = AD (common side)

.: triangles DAB & ADC (i.e. ACD) are congruent (AAS)

c) .: AB = DC (corresponding sides of congruent triangles DAB & ADC)



I've attempted to show the arguments in the above solution step by step. I hope you can now better follow the reasonings involved - an important requirement of all geometrical proofs.

 

[Smile12345]

I've attempted to carefully show the arguments in the above solution. I hope you can now better follow the reasoning - which is a key ingredient of geometrical proofs.[/QUOTE]

Thanks heaps. Yes, I can follow the reasoning perfectly.... Yes, I agree it is a key ingredient of geometrical proofs... It was part a that really stumped me ..

I really appreciate the time you spent on this... Thanks again.

 

[Smile12345]

If I put up some others, could you check to see if I've done them right? Please and thanks.

 

[Smile12345]

Find the vale of α....

Now I'll put my working...

Ang ECD = 120 degrees ( Co-interior angles with Ang FEC, CD // EF)

Ang ACD = 150 degrees (Co-interior angle with Ang BAC, AB // CD)

Ang ACD + Ang ECD + α = 360 degrees ( Angle sum of revolution)
150 deg + 120 deg + α = 360

Therefore α = 90 degrees....

How is this?

 

[HeroicPandas]

Find the vale of α.... View attachment 28355

Now I'll put my working...

Extend the line CD

 

[Smile12345]

Extend the line CD

Thanks HeriocPandas... Is my working correct???

 

[HeroicPandas]

Thanks HeriocPandas... Is my working correct???

of course!