1. ## Congruent triangles

hi all attached is a question and answer from an exam paper which i am having trouble with.

firstly i do not see how they can state the ASA rule without further information.

secondly i would have drawn the perpendicular bisector of line YZ and labelled the point of intersection with YZ as W then stated

YW = WZ (common)
XW (common)
XY=XY (given XYZ isoscles triangle)

therefore triangle congruent SSS

2. Originally Posted by sammy28
hi all attached is a question and answer from an exam paper which i am having trouble with.

firstly i do not see how they can state the ASA rule without further information.

secondly i would have drawn the perpendicular bisector of line YZ and labelled the point of intersection with YZ as W then stated

YW = WZ (common)
XW (common)
XY=XY (given XYZ isoscles triangle)

therefore triangle congruent SSS

For the first thing -What more information were you expecting, can you be little specific

Second thing-You have proved that YXW is congruent to ZXW
I guess you wanted to add some more --
Can you be a little more specific

Are you trying to find the ways of proving it other than those given

3. thanks ardash, i made a mistake in proving wrong triangles congruent

if you label the intersection of segments NZ and YM, O; then
for ASA to work i am thinking that angle ZYM = angle ZYX/2. but without extra given angles such as the angle at YNZ and the angle OYN or the angle YOZ then i dont see how to prove.

even if i were to draw a circle radius OY centre O then extend line segment YM to the circumferance i get the circle thm. ' the angle of a semi circle is a right angle', but this is no use to this problem.

4. Hi

You want to prove that triangles YMZ and ZNY are congruent using ASA rule.

It means that you need to show that :
1) angle NYZ = angle MZY
2) angle NZY = angle MYZ
3) side YZ = side YZ

1) triangle XYZ is isosceles with XY = XZ therefore angle XYZ = angle XZY
But N is on the segment [XY] therefore angle XYZ = angle NYZ
and M is on the segment [XZ] therefore angle XZY = angle MZY
Therefore angle NYZ = angle MZY

2) is given by construction of M and N

3) is of course evident

ASA rule shows that triangles YMZ and ZNY are congruent

5. thanks rg. i see my mistake now you point it out so clearly