Assignment 2

Write clearly the solutions to the following problems, proofs and constructions and turn them in. Note that proofs can use the theorems we have proved in class and from Chapter 1 of BG.

2.1. Suppose ABC is an isosceles triangle with AB = AC.

(a) If M is the midpoint of BC, prove that triangle AMB = triangle AMC.

(b) Based on what you just proved, what is the strongest statement you can make about angle AMB?

2.2. SSA for triangles

Suppose that ABC and DEF are triangles, with AB = DE, AC = DF and angle B = angle E. This condition is called SSA (or you can reverse the order of letters if you like!).

(a) Show that SSA does not imply that triangles ABC and DEF are congruent in general.

(b) Prove that SSA does imply for right triangles: If angles B and E above are right angles, then triangles ABC and DEF are congruent.

(c) Given two lengths AB and AC, and an angle B, show in some examples how to construct of a triangle ABC with these 2 sides and this angle. More specifically show examples a construction method that works if the triangle ABC exists, and which leads in general to either no triangles, 1 triangle or 2 triangles, depending on what the sides and angle are.

2.3 SAA for triangles

Prove: Given triangles ABC and DEF, if angle A = angle D, angle B = angle E, and BC = EF, then triangle ABC is congruent to triangle DEF.

2.4 Concurrence of perpendicular bisectors of a triangle

Draw a general* triangle ABC and construct with straightedge and compass the perpendicular bisectors of the sides.

(a) Prove that the 3 perpendicular bisectors are concurrent at a point O for any triangle ABC.

(b) Construct the circle with center O through A.

(c) Explain why the circle with center O through A also passes through B and through C for any triangle ABC.

*General is a somewhat ill-defined term, but referring to an example, it means not special, so it should not be isosceles, equilateral, having a right angle or any other "special" angle such as 60 degrees. It means what one might call a "random" triangle if one could give a good definition of random.