**Chapter One: Foundations For Geometry **

**Section One: Understanding Points, Lines, and Planes**

__Undefined Terms__

Point: Names a location and has no size.

Line: A straight path that has no thickness and extends forever.

Plane: A flat surface that has no thickness and extends forever.

__Definitions__

Segment: The part of a line consisting of two points and all points between them.

Endpoint: A point on one end of a segment or the starting point of a ray.

Ray: The part of a line that starts at an endpoint and extends forever in one direction.

Opposite Rays: Two rays that share the same endpoint and extend in opposite directions to form a line.

__Postulates__

Postulate 1-1-1: Through any two points there is exactly one line.

Postulate 1-1-2: Through any three nonlinear points there is a plane containing them.

Postulate 1-1-3: If two points lie in a plane, then the line containing those points lies in the plane.

Postulate 1-1-4: If two lines intersect, then they intersect at exactly one point.

Postulate 1-1-5: If two planes intersect, then they intersect in exactly one line.

**Section Two: Measuring And Constructing Segments**

__Definitions__

Congruent Segments: Segments that have the same length.

Midpoint: A point that bisects a line segment in two equal segments.

Segment Bisector: Any ray, line, or segment that bisects a line segment at it’s midpoint.

__Postulates__

Postulate 1-2-1: The points on a line can be put in one-to-one correspondence with real numbers.

Segment Addition Postulate: If B is between A and C, then AB + BC = AC.

**Section Three: Measuring And Constructing Angles**

Figure A

_

Vertex: The common endpoint of an angle.

Acute Angle: An angle with a measure greater than 0 and less than 90.

Obtuse Angle: An angle with a measure greater than 90 and less than 180.

Right Angle: An angle with a measure of exactly 90.

Straight Angle: Formed by two opposite rays, it measures 180.

Angle Bisector: A ray that divides and angle in to two congruent angles.

Angle Addition Postulate: If point C is in the interior of <ABD, then the m<ABC+ m<CBD = m<ABD (Figure A).

__Definitions__

Angle: Two rays that share a common endpoint.Vertex: The common endpoint of an angle.

Acute Angle: An angle with a measure greater than 0 and less than 90.

Obtuse Angle: An angle with a measure greater than 90 and less than 180.

Right Angle: An angle with a measure of exactly 90.

Straight Angle: Formed by two opposite rays, it measures 180.

Angle Bisector: A ray that divides and angle in to two congruent angles.

__Postulates__

Protractor Postulate: Given line AB and point O on line AB, all rays drawn from point O can be put in to a one-to-one correspondence with real numbers 0 to 180.Angle Addition Postulate: If point C is in the interior of <ABD, then the m<ABC+ m<CBD = m<ABD (Figure A).

**Section Four: Pairs Of Angles**

Figure B

_

Linear Pair: A pair of adjacent angles whose non-common sides are opposite rays.

Complementary Angles: A pair of angles whose measures have a sum of 90 degrees.

Supplementary angles: A pair of angles whose measures have a sum of 180 degrees.

Vertical angles: A pair of non adjacent angles formed by two intersecting lines.

In Figure B:

A is an example of vertical angles.

B is a right angle.

C is an obtuse angle.

D is an acute angle.

E shows several things:

Angles 2 and 3 are complementary.

Angle 1 and the right angle formed by 2 and 3 are supplementary.

__Definitions__

Adjacent Angles: Two angles with a common side and a common vertex.Linear Pair: A pair of adjacent angles whose non-common sides are opposite rays.

Complementary Angles: A pair of angles whose measures have a sum of 90 degrees.

Supplementary angles: A pair of angles whose measures have a sum of 180 degrees.

Vertical angles: A pair of non adjacent angles formed by two intersecting lines.

In Figure B:

A is an example of vertical angles.

B is a right angle.

C is an obtuse angle.

D is an acute angle.

E shows several things:

Angles 2 and 3 are complementary.

Angle 1 and the right angle formed by 2 and 3 are supplementary.

**Section Five: Using Formulas In Geometry**

__Definitions__

Perimeter: The sum of the side lengths of a figure.

Area: The number of non overlapping square units that cover the interior of a figure.

Base: Any side of a triangle.

Height: A segment from a vertex that forms a right angle with the base.

Circumference: The distance around a circle.

Diameter: The length of a segment that passes through the center of a circle and has endpoints on the circle.

Pi: The ratio of the circumference of a circle to its diameter. It is often shown as 3.14 or 22/7.

Radius: The distance of a segment that has endpoints at the center of a circle and at any point on the circle.

__Formula__

Circumference of a circle: C=2(pi)r or C=(pi)d.

Area of a circle: A=(pi)r².

Area of a triangle: A=1/2bh.

Perimeter of a triangle: P=a+b+c.

Perimeter of a rectangle: 2w+2l.

Area of a rectangle: wl.

Perimeter of a square: 4s.

Area of a square: s².

**Section Six: Midpoint and Distance In The Coordinate Plane**

__Definitions__

Coordinate Plane: A plane that is divided into four quadrants by a horizontal line and a vertical line. The horizontal line is known as the x-axis. The vertical line is known as the y-axis.

Leg: The two sides that form the right angle.

Hypotenuse: The side across from the right angle that stretches from one leg to the other.

__Theorems__

Theorem 1-6-1: Pythagorean Theorem: In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. The formula is a squared+ b squared= c squared.

__Formulas__

**Section Seven: Transformations In The Coordinate Planes**

__Definitions__

Transformation: A change in the position, size, or shape of a figure.

Preimage: The image before a transformation.

Image: The result of a transformation.

Reflection: A flip across a line called the point of reflection.

Rotation: A transformation around a point P called the center of rotation. Each point and its image are the same distance from P.

Translation: All of the points of a figure move the same distance in the same direction.