Math 367 Homework Assignment 6
due Thursday, March 24
1. Is there an AAA Congruence Theorem? That is, if 4ABC and 4DEF are triangles
for which ∠ABC ∼
= ∠DEF , ∠BCA ∼
= ∠EF D, and ∠CAB ∼
= ∠F DE, does it follow that
∼
4ABC = 4DEF ? If so, prove it, and if not, give a counterexample (e.g. by drawing a picture
of two triangles in the Cartesian plane that satisfy these hypotheses but are not congruent).
2. Is there a triangle with side lengths 8, 15, and 24? Justify your answer.
3. Prove the Converse to the Isosceles Triangle Theorem: If 4ABC is a triangle such that
∠ABC ∼
= AC. (Hint: Draw a picture. First use the ASA Theorem to
= ∠ACB, then AB ∼
show that 4ABC ∼
4ACB.)
=
4. An equilateral triangle is one in which all three sides have equal length. An equiangular
triangle is one in which all three angles have equal angle measure.
(a) Use the Isosceles Triangle Theorem and its Converse to show that a triangle is equilateral
if and only if it is equiangular.
In parts (b) and (c) below, you will show that the statement in part (a) can be false in models
where not all axioms of neutral geometry hold.
(b) Taxicab metric. Consider the model in which points, lines, half-planes, and angle measure
are all as usual for the Cartesian plane, but distance is given by the taxicab metric. Show
that the triangle with vertices (0, 0), (1, 1), (−1, 1) is equilateral but not equiangular. Find a
triangle that is equiangular but not equilateral.
(c) Square metric. Consider the model in which points, lines, half-planes, and angle measure
are all as usual for the Cartesian plane, but distance is given by the square metric. Find a
triangle that is equilateral but not equiangular. Find a triangle that is equiangular but not
equilateral.
5. Prove that if 4ABC is a triangle, DE is a segment such that DE ∼
= AB, and H is a
←→
half-plane bounded by DE, then there is a unique point F ∈ H such that 4DEF ∼
= 4ABC.
(Hint: Draw a picture. Use the Angle Construction Postulate to show that there is a point
F 0 ∈ H such that ∠F 0 DE ∼
= ∠CAB. Then use the Point Construction Postulate to obtain F .
Finally, apply the SAS Postulate.)
6. Prove that if A, B, and C are three points such that AB + BC = AC, then A, B, and C are
collinear. (Hint: Use the Triangle Inequality to prove the contrapositive of this statement.)
7. Prove that the hypotenuse is always the longest side of a right triangle. (Hint: Use the
Exterior Angle Theorem to show that in a triangle with a right angle, the other two angles
must be acute. Then use the Scalene Inequality.)