Plane Figures

Basic Terms
A study of plane figures is the study of figures in the plane such as points, lines, line segments, rays, angles, circles, triangles, quadrilaterals, and other polygons. To understand geometry, students need to understand its basic terms, which are used to define many other terms. A plane is a collection of points that forms a flat, continuous, and unending surface.

A point is a location in space. A line is a straight, continuous, and unending collection of points. A line segment is the part of a line between two points including the endpoints. In geometry, the symbol names a line segment with endpoints at A and B. is part of line AB, which is written . A ray is part of a line that has one endpoint and continues without end in one direction. If A is the endpoint of a ray that also passes through point B, then AB is written . If B is the endpoint of a ray that also passes through point A, then ray BA is written . Notice that and name the same line segment, and that and name the same line. However, and name different rays.

Two lines in a plane are either intersecting or parallel. To indicate that and are parallel, write || . Lines that intersect at right angles are perpendicular lines. To indicate that and are perpendicular, write .

Angles
An angle is determined by two rays that have a common endpoint, called the vertex of the angle. Each pair of rays with a common endpoint determines two angles, ∠n and ∠m, as shown below.

Angles are classified by the size of the opening between the rays. The measure of an angle is determined by how much one ray of the angle must be rotated through the interior of the angle until it coincides with the other ray of the angle. An angle that forms a square corner is called a right angle. An angle that has a measure less than a right angle is an acute angle. An angle that has a measure greater than a right angle is an obtuse angle. Angles can be named in several ways.

Say Write
angle ABC ABC
The vertex of the angle is the middle letter.
angle DEF DEF
angle B B
A single letter can be used if there is no confusion as to the angle being considered.
angle E E
angle M m
A small letter or number written inside the angle can also be used to name an angle.
angle three ∠3

Polygons
A simple, closed plane figure formed by three or more line segments meeting only at their endpoints is called a polygon. Polygons are named according to their number of sides:

triangle, 3 sides
pentagon, 5 sides
hexagon, 6 sides
heptagon, 7 sides
octagon, 8 sides
nonagon, 9 sides
decagon, 10 sides
undecagon, 11 sides
dodecagon, 12 sides

At this grade level, only triangles, quadrilaterals, pentagons, hexagons, and octagons are considered.

Quadrilaterals are classified according to the properties of their sides or angles. Parallelograms have opposite sides parallel and opposite sides and opposite angles congruent. Rectangles are parallelograms with four right angles. Rhombuses are parallelograms with all sides congruent. Squares are regular quadrilaterals with all sides and all angles congruent. Squares are rhombuses with four right angles. Trapezoids are quadrilaterals with one pair of parallel sides.

Triangles are classified according to the measures of their sides or their angles:

equilateral—3 congruent sides
isosceles—2 congruent sides
scalene—no congruent sides
acute—3 acute angles
right—1 right angle
obtuse—1 obtuse angle

Circles
A circle is a plane figure consisting of all points that are the same distance from a given point called the center. A radius is any line segment that has one endpoint at the center of the circle and the other endpoint on the circle. A diameter is any line segment that passes through the center of a circle and has its endpoints on the circle.

A central angle is any angle with its vertex at the center of the circle. If a radius is rotated completely about the circle, the rotation measures 360°. An angle that corresponds to one quarter of one rotation is called a right angle and has a measure of 90°. An angle that corresponds to one half rotation has a measure of 180° and is called a straight angle. A three-quarter rotation corresponds to 270°.

Teaching Model 16.4: Quadrilaterals and Other Polygons