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Math 335 — Homework 1 Solutions 1.a. A midpoint of a segment AB is a point M on the segment AB such that AM is congruent to MB. b. A perpendicular bisector of a segment AB is a line CD such that (1) the midpoint of AB lies on CD. (2) ∠AMC is a right angle. −→ ∼ ∠DBC. c. A ray BD bisects an angle ∠ABC if D is between A and C, and ∠ABD = ← → d. Points A, B, and C are collinear if C lies on the line AB. e. Line l, m, and n are concurrent if the is a point P which lies on each of them. 2.a. Let A, B, and C be noncollinear. Then the triangle △ABC consists of all points on at least one of the segments AB, AC, and BC. b. The vertices of the triangle △ABC are the points A, B, and C. The sides are the segments AB, AC, and BC. The angles are ∠ABC, ∠CAB, and ∠BCA. c. Given a triangle △ABC, the side opposite the vertex A is the side BC. The sides adjacent to A are AB and AC. d. Given a triangle △ABC, the medians are the segments VM, where V is a vertex of the triangle and M is the midpoint of the side opposite to V. e. The altitudes of the triangle △ABC are the line segments VP, where V is a vertex and P is a point on the side of △ABC opposite to V, such that the angle ∠VPW is right, where W is a vertex of △ABC distinct from V. f. An isoceles triangle is a triangle with two distinct, congruent sides. If AB and AC are the two congruent sides, then the base is BC. The base angles are ∠ABC and ∠BCA. g. An equilateral triangle is a triangle such that any two sides are congruent. h. A right triangle is a triangle such that at least one of the three angles is a right angle.