Question 1182026
<pre>

if two sides of one triangle are congruent to two sides of another triangle, then
the third side of the triangle with the larger included angle is longer. 

{{{drawing(400,350,-8,8,-7,7,

locate(-.1,5.6,G),locate(-6,0,A), locate(6,0,B), locate(-2,0,X),locate(2,0,Y),

triangle(-6,0,0,5,-2,0),  triangle(6,0,0,5,2,0),triangle(-2,0,2,0,0,0,0), 

green(line(-2,0,-4,-5),line(2,0,-4,-5),locate(-4,-5,E)) 


)}}}

GA congruent to GB
angle A congruent to angle B
AX congruent to YB
GX congruent to GY
triangle XGY is isosceles
GX congruent to GY
AX congruent to XY
angles GXY and GYX are acute (base angles of an isosceles triangle)
angle GXA is obtuse (supplementary to an acute angle)
GA > GX   (by the SAS inequality theorem.
GA > GY   (since GX is congruent to GY    <-----------STEP M   

For contradiction, assume angle XGY congruent to angle XGA  

Extend GX to twice its length to E such that GX congruent to EX.
Draw YE 
AX congruent to XY (given)
GX congruent to EX (by construction)
angle GXA congruent to angle EXY  (vertical angles are congruent)
triangle GAX and triangle EYX are congruent by SAS
GA congruent to EY  (c.p.c.t)      <----------------- STEP N 
angle XGA congruent to angle YEX (c.p.c.t.)
angle YEX congruent to angle XGY
Triangle GYE is isosceles  (base angles congruent)
GY congruent to EY  (c.p.c.t)     <------------------ STEP P
 
From steps N and P above,
GA is congruent to GY              <----------------- Step Q

Step Q contradicts Step M.

Therefore, the assumption that angle XGY congruent to angle XGA
is false, so the student did not trisect angle AGB. 

Edwin</pre>