Math 367 Homework Assignment 8
due Thursday, April 21
1. Let 4ABC be a triangle. Prove that the interior angles of 4ABC measure 45◦ , 45◦ , 90◦ if
and only if the triangle is both right and isosceles.
2. Let 4ABC and 4DEF be two triangles such that 4ABC ∼ 4DEF and AB = DE.
Prove that 4ABC ∼
= 4DEF .
3. (a) (The ambiguous case.) Recall that there is no SSA Theorem. Show this by using a
protractor and ruler to construct the following example: Triangles 4ABC and 4DEF for
which µ(∠BAC) = µ(∠EDF ) = 30◦ , AB = DE = 6cm, BC = EF = 4cm, and 4ABC ∼
6
=
4DEF .
(b) Similarly, show by example, that there are two triangles 4A0 B 0 C 0 and 4D0 E 0 F 0 for which
µ(∠B 0 A0 C 0 ) = µ(∠E 0 D0 F 0 ) = 30◦ , A0 B 0 = 6cm, D0 E 0 = 3cm, B 0 C 0 = 4cm, E 0 F 0 = 2cm, and
4A0 B 0 C 0 6∼ 4D0 E 0 F 0 .
For #4–6, use the following definition: A rhombus is a quadrilateral that has four congruent
sides.
4. Prove that the diagonals of a rhombus intersect at a point that is the midpoint of each
diagonal.
5. Prove that the diagonals of a rhombus are perpendicular.
6. Prove that every rhombus is a parallelogram.