## 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 . is part of line AB, which is written . Two line segments are **congruent** if they have the same length. If line segments AB and CD are congruent, then . The symbol 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 .

**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.

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 ∠b.

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 ∠b and ∠c ∠d because ∠a and ∠b are vertical angles as are ∠c and ∠d.

### Circles

**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°.

**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