Geometry Review

Chapter OneA B This is an example of a line. A line is usually represent by 2 arrowheads to indicate the line extends without and end. The points on the line help to identify it. A line extends in one dimension.

Point A and Point B are collinear points, which are points that lie on the same line.

This an example of a plane. A plane is usually represented by a shape that looks like a table top or a wall. Even though the drawing makes it seem as though it has edges, you must imagine that it has no end. A plane extends in two dimensions.

K

L

NM

Line AB

Plane KLMN

Points K, L, M, and N are all coplanar points, which are points that lie on the same plane.

Chapter OneA

B

Point A(x ,y ) and Point B (x ,y ) are points in a coordinate plane, then the distance between A and B is:

1 1 2 2

Distance Formula

Perimeter, Circumference and Area Formulas

Square(side- s)

Rectanglelength-L width-W

P= 4sA=sˆ2

P= 2L + 2W

A= LWL

W

Triangle sides- a,b,c base-

bheight-h

s ha

b

c P= a +b +cA=½bh

r

Circleradius- r

C= 2rA= (rˆ2)

Chapter One

vertex sides

A

B

C

This is an example of an angle. It could be named either angle BAC or angle CAB. The middle letter in the angle name is always the vertex.

Classification of Angles

This is a right angle. A right angle has a measure of 90 degrees, exactly.

This is an obtuse angle. An obtuse angle has a measure more than 90 but less than 180 degrees.

This is an acute angle. An acute angle has a measure less than 90.

This is a straight angle. A straight angle has a measure of exactly 180 degrees.

Chapter OneThe Midpoint FormulaTo find the midpoint of a segment, you must take the average/mean of the x-coordinates and of the y-coordinates.

Vertical AnglesTwo angles are vertical angles when their sides form two pairs of opposite rays. In the figure to the left:

4 23 1

Linear PairsTwo adjacent angles are a linear pair if their non-common sides are opposite rays. In the figure to the left:

1 2

Chapter ThreeAlternate Exterior AnglesTwo angles are alternate exterior angles if they lie outside two lines on opposite sides of the transversal. In this figure, 1 & 7 are alternate exterior angles.

Alternate Interior Angles

Two angles are alternate interior angles if they lie between the two lines on apposite sides of the transversal. In this figure, 4 & 6 are alternate interior angles.

Chapter ThreeConsecutive Interior AnglesTwo angles are consecutive interior angles if they lie between two lines on the same side of the transversal. In this figure, 4 & 5 are consecutive interior angles.

Corresponding AnglesTwo angles are corresponding angles if they occupy corresponding positions. In this figure, 1 & 5 are corresponding angels.

Chapter ThreeFormula for the Slope of Parallel LinesIn a coordinate plane, two

non-vertical lines are parallel if and only if they have the same slope.

Any two vertical lines are parallel.

Formula for the Slope of Perpendicular Lines

In a coordinate plane, two non-vertical lines are perpendicular if and if the produce of their slopes is -1.

Vertical and horizontal lines are perpendicular.

Chapter FourClassification of Triangles by Their Sides

Equilateral Triangle

Isosceles Triangle

Scalene Triangle

3 congruent

sides

2 congruent

sides

no congruent

sides

Classification of Triangles by Their Angles

Equiangular Triangle

RightTriangle

Scalene Triangle

ObtuseTriangle

1 right angle

3acuteangles

3congruent

angles

1obtuseangle

Chapter Four

Chapter Four

A B

C Triangle Sum Theorem

m A + m B +m C = 180

1A B

C Exterior Angle Theorem

m1 = m A + m B

A B

C Corollary Theorem

m A + m B = 90

Chapter SixPolygon…is a plane figure that meets the following requirements:

1) It’s formed by three or more segments called sides, such that no two sides with a common endpoint are collinear.

2) Each side intersects exactly two other sides, one at each endpoint.

Convex Polygon…

if no line that contains a side of the polygon contains a point in the interior of the polygon.

Concave Polygon…

a polygon that is not convex.

•A regular polygon is both equilateral and equiangular.

Chapter SevenReflection

Rotation

Translation

Line of Reflection

Pre-image Image

Point of Rotation

Chapter EightIf a and b are two quantities that are measured in the same units, then the ratio of a to b is written as a:b or a/b

Cross Product Property

Reciprocal Property

AB = C

DIf Then AD = BC

If AB

CD= B

ADCThe

n=

Pythagorean Theorem

The sum of the square of each leg is equal to the square of the hypotenuse. The hypotenuse is only in a right triangle and is the leg opposite of the right angle.

a

b

a +b =c

2 2 2

cChapter Nine

45-45-90 Triangle TheoremIn a 45-45-90 triangle, the hypotenuse is √2 times as long as each leg.

x : x : x√2

30-60-90 Triangle Theorem

In a 30-60-90 triangle, they hypotenuse is two times as long as the shorter leg and the longer leg is √3 times longer than the shorter leg.

x : x√3 : 2x

Trigonometric Ratios

Sin = oppositehypotenuse

Cos =adjacent

hypotenuse

Tan = oppositeadjacent

Chapter TenCircle is the set of all points in a plane that are equidistance from a given point called the center of a circle.

.

Radius is the distance from a point on the circle to the center.•two circles are congruent when they have the same radius.

Diameter is the distance from one point on the circle to another and passes through the center.

Centerdiameter

radius

Chapter TenChord is a segment whose endpoints are points on the circle.

Secant is a line a that intersects a circle in two points.

Tangent is a line in the plane of a circle that intersects in the circle at exactly one point.

Chapter TenA minor arc of a circle is formed by a central angle less than 180.

A major arc of a circle is formed by a central angle more than 180.

A semicircle is formed by the endpoints of the diameter of a circle.

Chapter

ElevenPolygon Interior

AnglesTheorem 11.1: The sum of the measures of the interior angles of a convex n-gon is (n-2)180.Corollary: The measure of each interior angle of a regular n-gon is (1/n)(n-2)180.

Polygon Exterior Angles Theorem 11.2:

The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360.Corollary: The measure of each exterior angle of a regular n-gon is (1/n)360.

Chapter

Eleven

Area of an Equilateral TriangleThe area of an equilateral triangle is one fourth the square of the length of the side times the square root of three.

A = (¼)(√3)(sˆ2)

Area of a Regular Polygon

The area of a regular n-gon with side length s is half the product of the apothem a and the perimeter P, so

A= (½)aPP=number of sides x side

length

s

s

s

sa

Chapter

Eleven.

.

.

A

B

P

Arc Length

In a circle the ratio of length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360.

arc length of AB2r = m AB

360

Area of a Sector

The ratio of a the area A of a sector of a circle to the area of a circle is equal to the ratio of the measure of the intercepted arc to 360.

A rˆ2 = m

AB360

..

.

A

B

P

ChapterTwelvePolyhedron… is a sold that is bound

by polygons, called faces, that enclose a single region of space.

face

.An edge of a polyhedron is a line segment formed by the intersection of two faces.

A vertex of a polyhedron is a point where three or more edges meet.

THE

END Jaclyn A. DisharoonPd. 1