Teaching Models

Plane Figures

Basic Terms
Plane geometry is the study of figures in the plane such as points, lines, line segments, rays, angles, circles, triangles, quadrilaterals, and other polygons.

Understanding geometry depends on the understanding of some basic terms that are used to define many other terms.

A point is a location in space. A line is a straight, continuous, and unending collection of points. A plane is a collection of points that forms a flat, continuous, and unending surface. A line segment is the part of a line between two points including the endpoints. A line segment from point A to point B is written Segment AB. Segment AB is part of line AB, which is written Line AB. Two line segments are congruent if they have the same length. If line segments AB and CD are congruent, then Segment AB Congruent Segment CD. The symbol Congruent means “is congruent to.” A ray is part of a line that has one endpoint and continues in one direction without end. If A is the endpoint of ray AB, then the symbol for ray AB is Segment CD.

Angles
An angle consists of 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.

angle consisting of two rays

A complete rotation corresponds to 360°. An angle that corresponds to a quarter of a complete rotation is called a right angle and measures 90°. An angle that corresponds to half of a complete rotation measures 180°. It is called a straight angle. The measure of angle a is denoted by m∠a. An angle is acute if 0° < m∠a < 90° and obtuse if 90° > m∠a < 180°. Angles that have the same measure are congruent angles. If ∠a and ∠b are congruent, we write ∠a Congruentb.

two crossing rays with noted angles

Two angles are complementary if the sum of the measures is 90°. Two angles are supplementary if the sum of the measures is 180°. In the figure at right, ∠a and ∠c are supplementary, as are ∠c and ∠b, ∠b and ∠d, and ∠a and ∠d. When two straight lines intersect, four angles are formed. Any two adjacent angles in this case are supplementary. The opposite angles are called vertical angles. Vertical angles are congruent since they are each supplements to the same angle. In the figure, ∠a is supplementary to both ∠c and ∠d. Furthermore, ∠a Congruentb and ∠c Congruentd because ∠a and ∠b are vertical angles as are ∠c and ∠d.

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 segment that has one endpoint at the center of the circle and the other endpoint on the circle. A chord is any segment with both endpoints on the circle. A diameter is a chord that passes through the center of the circle. A central angle is any angle with its vertex at the center of the circle and sides that intersect the circle. The part of the circle formed by the interior of a central angle is called a sector of the circle. A complete rotation of a radius about the circle measures 360°.
Circle

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 the number of sides. A regular polygon has all sides congruent and all angles congruent. The vertex of a polygon is where the sides meet.

Triangles are classified according to the measures of their sides or their angles as equilateral (3 congruent sides), isosceles (2 congruent sides), scalene (no congruent sides), acute (3 acute angles), right (one right angle), or obtuse (one obtuse angle). An equilateral triangle is also equiangular, which means all angles are congruent. An equilateral triangle is a regular triangle. The sum of the angle measures in any plane triangle is 180°.

Quadrilaterals are classified according to the properties of their sides and 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 rectangles with all sides congruent. A square is a regular quadrilateral. Trapezoids are quadrilaterals with one pair of parallel sides.

The angle sum in a quadrilateral is 360° because any quadrilateral can be partitioned into two triangles by drawing a diagonal from opposite vertices. In general, any polygon can be partitioned into triangles. Adding a side to a given polygon increases the angle sum by 180° because the resulting polygon can be partitioned into one more triangle. For a polygon with n sides, the angle sum is equal to (n − 2) × 180°. In a parallelogram, opposite angles are congruent and consecutive angles are supplementary. In later grades, students will study parallel lines and learn why these angle relationships are true.


Teaching Model 14.3: Transversals and Angles


Houghton Mifflin Math Grade 6