(Scroll down if you want to see some explanations for this applet; [b]enjoy![/b])

In this applet, with given base segment AB[code][/code], you will get to [i]play around and create a pair of congruent triangles in four different ways:[/i][br][br][b][u]SSS (side-side-side):[/u][br][/b]In this case, we will construct two congruent triangles by drawing a circle centered at A with radius of r and another circle centered at B with radius of r[sub]2[/sub], placing points C and C' at the two intersections of these two circles, and then creating triangles ABC and ABC'. [b]Here, you will get to control the values of r and r'.[/b][br][br]Triangles ABC and ABC' are congruent to each other as these two triangles share side AB, AC = AC' = r, and BC = BC' = r[sub]2.[/sub]-- [b]making each of the three sides of triangle ABC congruent to the corresponding side of triangle ABC'.[br][br][br][/b][b][u]SAS (side-angle-side):[/u][br][/b]In this case, we will construct the two congruent triangles by drawing a circle centered at A with a radius of r, drawing two rays that each starts from A and forms an angle of value [math]\alpha[/math] from segment AB, and then placing points C and C' at the two intersections between the circle and one of the two rays -- forming triangles ABC and ABC' [b]Here, you will get to control the values of r and[/b][math]\alpha[/math][i].[/i][br][br]Triangles ABC and ABC' are congruent to each other as these two triangles share AB, AC = AC' = r, and m -- [b]making each of the two adjacent sides of triangle ABC congruent to the corresponding side of triangle ABC' & making the angle between those two sides in triangle ABC congruent to the angle between the corresponding sides in triangle ABC'.[br][br][br][u]AAS (angle-angle-side):[/u][br][/b]In this case, we will construct the two congruent triangles by drawing a pair of rays that each starts from A and forms an angle of value [math]\alpha[/math] from segment AB, then drawing a second pair of rays that each starts from a point along one of the first pair of rays (that is, C/C') and each forms an angle of value [math]\beta[/math]from segment AC/AC', and finally placing B at the intersection of the second pair of rays. [b]Here, you will get to control not only the values of [math]\alpha[/math] and [math]\beta[/math], but also the length of segment AB[/b] (in case you end up making the triangle too small or too big while playing around the value of the two angles).[b][br][/b][b][br][/b]In this case, two triangles ABC and ABC' are congruent to each other, as they share base side AB, m<BAC = m<BAC' = [math]\alpha[/math], and m<ACB = m<AC'B = [math]\beta[/math] -- [b]not only resulting in the two triangles sharing a congruent side, but also making an angle in contact with that side within triangle ABC congruent to the corresponding angle in triangle ABC' & making the angle not in contact with that side within triangle ABC congruent to the corresponding angle in triangle ABC'.[/b][br][br][br][b][u]ASA (angle-side-angle):[/u][br][/b]In this case, we will construct the two congruent triangles by drawing two rays that start from A and each forms an angle of value [math]\alpha[/math]from segment AB and then drawing another two rays that start from B and each forms an angle of value [math]\beta[/math]from segment AB. [b]Here, you will get to control not only the values of [/b][math]\alpha[/math][b]and [/b][math]\beta[/math][b], but also the length of segment AB[/b](in case you end up making the triangle too small or too big while playing around the value of the two angles).[br][br]Triangles ABC and ABC' are congruent to each other as these two triangles share AB, m<BAC = m<BAC' = [math]\alpha[/math], and m<ABC = m<ABC' = [math]\beta[/math][b] -- not only resulting in the two triangles sharing a congruent side, but also making each of the two angles in contact with that side within triangle ABC congruent to the corresponding angle in triangle ABC'.[br][/b][br][math]\alpha[/math] [br]