1. ## Logic?

Explain how two triangles can have five parts (sides and angles) of one congruent to five parts of the other triangle but the two triangle is not congruent at all.

2. I haven't written down an example, but I believe the following would be possible:

(1) both triangles must be scalene (all sides, and thus all angles of the triangle must be different)

(2) the triangles are similar (thus they have the same angles)

(3) the side opposite the smallest angle of the first triangle is congruent to the side opposite the middle angle of the second triangle.

(4) the side opposite the middle angle of the first triangle is congruent to the side opposite the largest angle of the second triangle.

3. To have 5 parts of one triangle congruent to 5 parts of another, there just be only one angle or one side on each triangle that is not congruent to a part of the other triangle. If it is an angle, we have all three sides of one triangle congruent to all three sides of the other and the triangles are congruent by "SSS". It is a side, we have all three angles congruent to one another but there is no "AAA" theorem for congruence- the two triangles would be similar, as DrSteve says. There is a "SAS" congruence theorem so what must happen is that the angle between the two sides of one triangle must NOT be congruent to the angle between the corresponding congruent sides of the other triangle but congruent to some other angle in that triangle.

4. I believe that the following works. Let the two similar triangles have sides 1, 1.1, 1.21, and 1.1, 1.21, 1.331 respectively (we multiply each side of the first triangle by 1.1 to get each side of the second triangle, thus showing that the two triangles are similar).

Note that the triangle rule is satisfied for each of these triples, so that they both form triangles.