Jeremy Gibbs

Definitions

Modern Geometry

9-7-97

1. a. Given segment AB, the midpoint is the point which lies on segment AB halfway in between point A and point B thus making the following statement to be true: Segment AM = Segment MB.

b. Given segment AB, the perpendicular bisector is the line that passes through midpoint M and through point P that makes < AMP = <BMP.

c. Given four points: A, B, C, and D with point D in between A and C, ray BD is said to Bisect <ABC when <ABD = <CBD.

d. Given three points: A, B, and C. They are said to be collinear if the same line with the same slope passes through all of them.

e. Given three lines: l, m, and n, they are said to be concurrent if they have one point in common. If the three lines meet at one point, they are concurrent.

 

2. a. A triangle is made up of three points, A, B, and C, and three segments AB, BC, and CA.

b. For the same triangle, the sides are defined as the segments AB, BC, and CA. The vertices are points A, B, and C and the angles are made up of the vertices A, B, and C, and rays AB, BC, and CA. For example, angle A consists of vertex A and rays AB and AC.

c. For the same triangle, the sides opposite to vertex A is segment BC. The sides adjacent to vertex A are segments AB and CA.

d. For the same triangle, the medians of the triangle are those segments which pass through a vertex and the midpoint of the segment opposite to the original vertex. For example, one median would pass through vertex A and the midpoint of segment BC.

e. For the same triangle, the altitudes of the triangle consists of a segment that passes through a vertex and the perpendicular bisection of the opposite side. The bisection does not have to be at the Midpoint.. For example, one altitude would pass through vertex A and conclude at some point on segment BC that would perpendicular bisect segment BC.

f. An isosceles triangle is a triangle with two nonlinear congruent segments and a third segment not congruent to the other two. The base is that segment that is not congruent to the other two. The base angles are those that are made up of the base, the vertex, and one of the congruent segments. The two base angles are congruent to each other.

g. An equilateral triangle is a triangle with three nonlinear congruent segments. All of the angles for this triangle are congruent as well.

h. A right triangle is a triangle that contains one right angle.
  1. a. The angles of Quadrilateral ABCD are comprised of the vertex and two adjacent rays. For example, one such angle would consist of vertex A and rays AB and DA.

b. The adjacent sides of Quadrilateral ABCD are the sides around the angle. For example, for vertex A, the adjacent sides would be segments AB and DA.

c. The opposite sides of Quadrilateral ABCD are the sides that do not comprise the angle. For example, the opposite sides for angle <BAD are segments CD and BC.

d. The diagonals of Quadrilateral ABCD are segments with pass from one vertex to the opposite vertex. The vertices may not have any segments in common. For example, the two diagonals of quadrilateral ABCD are segments AC and BD.

e. A parallelogram is a quadrilateral for which the opposite sides are parallel and congruent to each other. As a result, the opposite angles will also be congruent. For example, segment AB will be congruent and parallel to segment CD. As a result, angle <DAB will be congruent to angle <BCD.