# GEOMETRY DEFINITIONS

**Perpendicular** Two lines are called perpendicular if
they form a right angle.

**Congruent Triangles** Two triangles ΔABC and ΔDEF are
congruent (written

ΔABC
ΔDEF) if all three corresponding angles and all three corresponding

sides are equal.

**Similar Triangles** Two triangles ΔABC and ΔDEF are similar (written

ΔABC ~ ΔDEF) if all three corresponding angles are equal.

**Parallel Lines** Two lines are parallel if they do not intersect.

**Midpoint** of a line segment The midpoint of a segment AB is the point M on the

segment for which MA = MB.

**Angle bisector** The bisector of an angle is the line that goes through the vertex
of the

angle and splits the angle into two equal parts.

**Parallelogram A** quadrilateral is a parallelogram if the opposite sides are
parallel.

**Rectangle A** quadrilateral is a rectangle if it has four right angles.

**Square A** quadrilateral is a square if it has four equal sides and four right
angles.

I**sosceles A** triangle with two equal sides is called isosceles.

**Distance from a point to a line** The distance from a point P to a line m is de ned

to be the length of the line segment from P to m which is perpendicular to m.

**Definition of concurrent lines** Three lines are concurrent if they meet at a
single

point.

**Definition of perpendicular bisector** The perpendicular bisector of a line segment

is the line that goes through the midpoint and is perpendicular to the segment.

**Definition of circumcenter** The point where the three perpendicular bisectors of
the

sides of a triangle meet is called the circumcenter of the triangle.

**Definition of incenter **The point where the three angle bisectors meet is called
the

incenter of the triangle.

**Definition of altitude **An altitude of a triangle is a line that goes through a
vertex of

the triangle and is perpendicular to the opposite side.

**Definition of orthocenter **The point where the three altitudes meet is called the
or-

thocenter of the triangle.

**Definition of median **A** **median of a triangle is a line that goes through a vertex
of

the triangle and through the midpoint of the opposite side.

**Definition of centroid** The point where the three medians meet is called the
centroid

of the triangle.

**Definition of collinear **Three points are said to be collinear if they all lie on
the same

line.

**Definition of signed ratio** Let ℓ be any line and let C", A and B be three points
on

ℓ. Make ℓ into a number line by choosing an origin and a positive direction and
let

c', a and b be the coordinates of C", A and B. We define

to be and we call this a signed ratio.

**Definition of circle A** circle consists of all of the points which are at a given
distance

(called the radius) from a given point (called the center).

**Definition of tangent line A** line is tangent to a circle
it intersects the
circle in

exactly one point.

**BASIC FACTS**

**BF 1** SSS: if two triangles have three pairs of corresponding sides equal, then
the tri-

angles are congruent.

**BF 2** SAS: if two triangles have two pairs of corresponding sides and the
included angles

equal, then the triangles are congruent.

**BF 3** ASA: if two triangles have two pairs of corresponding angles and the
included side

equal, then the triangles are congruent.

**BF 4** If two triangles are similar then their corresponding sides are
proportional: that

is, if ΔABC is similar to ΔDEF then

**BF 5** If two parallel lines ℓ and m are crossed by a
transversal, then all corresponding

angles are equal. If two lines ℓ and m are crossed by a transversal, and at
least one

pair of corresponding angles are equal, then the lines are parallel.

**BF 6** The whole is the sum of its parts; this applies to lengths, angles, areas
and arcs.

**BF 7** Through two given points there is one and only one line. (This means two
things.

First, it is possible to draw a line through two points. Second, if two lines
have

two or more points in common they must really be the same line).

**BF 8** On a ray there is exactly one point at a given distance from the endpoint.
(This

means two things. First, it is possible to find a point on the ray at a given
distance

from the endpoint. Second, if two points on the ray have the same distance from

the endpoint they must really be the same point.)

**BF 9** It is possible to extend a line segment to an infinite line.

**BF 10** It is possible to find the midpoint of a line segment.

**BF 11** It is possible to draw the bisector of an angle.

**BF 12** Given a line ℓ and a point P (which may be either on ℓ or not on ℓ) it is
possible

to draw a line through P which is perpendicular to ℓ.

**BF 13** Given a line ℓ and a point P not on ℓ, it is possible to draw a line
through P

which is parallel to ℓ.

**BF 14** If two lines are each parallel to a third line then they are parallel to
each other.

**BF 15** The area of a rectangle is the base times the height.