Chapter Outline

Geometry

Geometry: the branch of mathematics that studies points, lines, angles, surfaces, and solid figures

Common Geometric Shapes

Points and lines:

• A point has a position in space.
• A ray is a straight path of points that begins at one point and continues in one direction.
• A line is a straight path of points that continues in two directions.
• A line segment has definite length; every line segment has two endpoints.
• A horizontal line runs left to right.
• A vertical line runs up and down.
• Parallel lines run in the same direction.
• Two lines meet (or cross) at a point of intersection.
• Perpendicular lines intersect to form right angles.

Angles:

• An angle is formed by two rays extending from the same point. This point is called the vertex.
• The size of an angle depends on the amount of rotation of the sides. One complete rotation is 360°.
• An acute angle has less than 90°.
• A right angle has exactly 90°.
• An obtuse angle has more than 90° and less than 180°.
• A straight angle has exactly 180°.
• A reflex angle has more than 180°.

Angle relationships:

• Two angles that add up to 180° are called supplementary angles.
• Two angles that add up to 90° are called complementary angles.
• Adjacent angles are two angles that share a side.
• Vertical angles are two angles opposite (across from) each other when two lines intersect. Vertical angles are equal.

Plane figures:

• A polygon is a closed figure with three or more line segments. (Closed means that the line segments meet.)
• A triangle is a closed, plane figure with three sides.
• A quadrilateral is a closed, plane figure with four sides.

Types of quadrilaterals:

• A rectangle has four right angles. The sides opposite each other are parallel and equal in length.
• A square has four right angles and four equal sides. The sides opposite each other are parallel.
• A rhombus has four equal sides.
• A parallelogram has two pairs of parallel sides.
• A trapezoid has one pair of parallel sides.

A circle is a figure in which every point is the same distance from the center.

• The distance around a circle is called the circumference.
• The distance from the center to a point on the circle is called the radius.
• The distance across the circle through its center is called the diameter.
• The ratio of the circumference of a circle to its diameter is called pi. The value of pi (or π) is approximately 3.14, or ²²/₇.

Perimeter and Circumference

Perimeter is the distance around a plane figure.

• Perimeter is measured in units such as inches, feet, yards, centimeters, and meters.
• To find the perimeter of a figure whose sides are all line segments, add the length of each side.

The perimeter of a circle is called the circumference.

Area

Area is a measure of the amount of surface within the perimeter of a plane figure.

• Area is measured in square units such as square inches, square feet, and square meters.
• There are different formulas for the area of different figures.

Solid Figures

Solid geometry is the study of three-dimensional figures:

• A cube is a box whose dimensions are all the same. Each of the six faces is a square.
• A rectangular solid is a box each of whose corners is a right angle. Each face is a rectangle.
• A square pyramid is a solid figure whose base is a square and whose four triangular faces meet at a common point called the vertex.
• A cylinder is a solid figure whose top and bottom are parallel circles. The height of a cylinder is the perpendicular distance between the top and bottom.
• A cone is a solid figure with a circular base and a vertex. The perpendicular distance from the vertex to the center of the base is the height.
• A sphere is a solid figure of which every point is the same distance from the center. The distance from any point on the surface of a sphere to the center is called the radius.

Volume

Volume is a measure of the amount of space inside a three-dimensional figure.

• Volume is measured in cubic units such as cubic inches or cubic meters.
• There are different formulas for the volume of different figures.

Triangles

A triangle is a plane (flat) figure with three sides.

• Each of the three points where the sides meet is called a vertex.
• The sum of the three angles of a triangle is 180°.

Types of triangles:

• An equilateral triangle has three equal sides and three equal angles.
• An isosceles triangle has two equal sides and two equal angles (called base angles). The angle with a different measurement is called the vertex angle.
• A scalene triangle has no equal sides and no equal angles.
• A right triangle has one right angle. The side opposite the right angle is called the hypotenuse. The other two sides are called legs.

Similarity and Congruence

Similar figures have the same shape but different sizes. The corresponding (matching) sides of similar figures can be written as a proportion. Two triangles are similar if the following conditions are met:

• Two angles of one triangle are equal to two angles of the other triangle
• The sides of one triangle are proportional to the sides of the other triangle.

Geometric figures are congruent if they have the same shape and the same size. Two triangles are congruent if they meet any one of these three conditions:

• The ASA requirement—two angles and a corresponding side of one triangle are the same as two angles and a corresponding side of the other triangle.
• The SAS requirement—two sides and a corresponding angle of one triangle are the same as two sides and a corresponding angle of the other triangle.
• The SSS requirement—the three sides of one triangle equal the three sides of the other triangle.

The Pythagorean Relationship

The Pythagorean relationship, or Pythagorean theorem, states that the square of the hypotenuse of a right triangle is equal to the sum of the square of the other two sides.

• The formula for the Pythagorean relationship is a² + b² = c², where a and b are the legs of a right triangle and c is the hypotenuse.