Teaching Models

Plane Figures and Geometric Concepts

Basic Terms
Plane geometry is the study of plane 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 a ray that also passes through point B, then ray AB is written ray AB. If B is the endpoint of a ray that also passes through point A, then ray BA is written Ray AB.

Two lines in a plane are either intersecting or parallel. To show that lines AB and CD are parallel, write Line AB || Line CD. Lines that intersect at right angles are perpendicular lines. To show that lines AB and CD are perpendicular, write Line AB perpendicular line Line CD.

Angles
An angle is determined by two rays with a common endpoint called the vertex of the angle. However, each pair of rays with a common endpoint determines two angles, ∠n and ∠m as shown at right. angle consisting of two 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. A complete rotation corresponds to 360°. An angle that corresponds to a quarter of a complete rotation is called a right angle and has a measure of 90°. An angle that corresponds to half of a complete rotation has a measure of 180° and is called a straight angle. The measure of angle a is denoted by m∠a, or simply ∠a. An angle is an acute angle if 0° < m∠a < 90°. An angle is an obtuse angle if 90° < m∠a < 180°.

Polygons
A simple, closed figure formed by three or more line segments meeting only at their endpoints is called a polygon. A regular polygon has all sides and all angles congruent. The vertex of a polygon is a point common to two sides of the polygon. Figures that are the same size and shape are congruent.

Corresponding parts of congruent figures are also congruent. 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. 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, polygons with 4 sides, are classified according to the properties of their sides and angles. Parallelograms have opposite sides parallel, so their opposite sides and opposite angles are 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.

Circles
CircleA circle is a plane figure consisting of all points that are the same distance from a given point called the center. Sometimes “circle” means the region within the circle. 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. If a radius is rotated completely about a circle, the rotation measures 360°.

Constructions
A construction is completed with only a safe drawing compass and a straightedge. However, a ruler may be used to check a construction. In this chapter, students learn how to construct perpendicular lines, parallel lines, congruent rectangles, and congruent triangles.

Symmetry
In this chapter, students learn to identify line symmetry and rotational symmetry. A figure has line symmetry if it can be folded in half and the two halves are congruent. A figure has rotational symmetry if it looks exactly the same after having been rotated about a fixed point less than a full turn.


Teaching Model 15.9: Symmetry


Houghton Mifflin Math Grade 5