Points, Lines, Planes, and Angles

A point, line, and plane are three basic terms in geometry that are NOT given a formal definition, yet we recognize them when we see them.

A line is a set of points.

Any two distinct points determine a line.

Any point on a line separates the line into three parts: the point and two half lines.

A ray is a half line including the endpoint.

A line segment is part of a line between two points, including the endpoints.

Line segment AB

Ray BA

Ray AB

Line AB

SymbolDiagramDescription

A B

A

A

A

B

B

B

We can think of a plane as a two-dimensional surface that extends infinitely in both directions.

Any three points that are not on the same line (noncollinear points) determine a plane.

A line in a plane divides the plane into three parts, the line and two half planes.

Any line and a point not on the line determine a unique plane.

The intersection of two planes is a line.

An angle is the union of two rays with a common endpoint; denoted .

The vertex is the point common to both rays. The sides are the rays that make the angle. There are several ways to name an angle:

ABC, CBA, B

The measure of an angle is the amount of rotation from its initial to its terminal side.

Angles can be measured in degrees, radians, or, gradients.

Angles are classified by their degree measurement. Right Angle is 90° Acute Angle is less than 90° Obtuse Angle is greater than 90° but

less than 180° Straight Angle is 180°

Adjacent Angles-angles that have a common vertex and a common side but no common interior points.

Complementary Angles-two angles whose sum of their measures is 90°.

Supplementary Angles-two angles whose sum of their measures is 180°.

If ABC and CBD are supplementary and the measure of ABC is 110°. Determine the measure of CBD.

A B

C

D

110°

If ABC and CBD are supplementary and the measure of ABC is 6 times larger than CBD, determine the measure of each angle.

A B

C

D

A B

C

D

ABC = 154.2°CBD = 25.7°

Vertical angles are the nonadjacent angles formed by two intersecting straight lines.

Vertical angles have the same measure.

A line that intersects two different lines, at two different points is called a transversal.

5 6

1 2

4

87

3

One interior and one exterior angle on the same side of the transversal–have the same measure

Corresponding angles

Exterior angles on the opposite sides of the transversal–have the same measure

Alternate exterior angles

Interior angles on the opposite side of the transversal–have the same measure

Alternate interior angles

5 6

1 2

4

87

3

5 6

1 2

4

87

3

120°

Polygons

Polygons are named according to their number of sides (line segments).

Icosagon20Heptagon7

Dodecagon12Hexagon6

Decagon10Pentagon5

Nonagon9Quadrilateral4

Octagon8Triangle3

NameNumber of Sides

NameNumber of Sides

The sum of the measures of the interior angles of a triangle is 180°.

30° 80°

____°

Acute Triangle

All angles are acute.

Obtuse Triangle

One angle is obtuse.

Right Triangle

One angle is a right angle.

Isosceles Triangle

Two equal sides.

Two equal angles.

Equilateral Triangle

Three equal sides. Three equal angles (60º) each.

Scalene Triangle

No two sides are equal in length.

Two polygons are similar (~) if their corresponding angles have the same measure and the lengths of their corresponding sides are in proportion.

4

3

4

6

8 8

12

6

83

6

5y

x

Catherine Johnson wants to measure the height of a lighthouse. Catherine is 5 feet tall and determines that when her shadow is 12 feet long, the shadow of the lighthouse is 75 feet long. How tall is the lighthouse?

?

x

7512

5

If corresponding sides of two similar figures are the same length, the figures are congruent.

Corresponding angles of congruent figures have the same measure.

Quadrilaterals are four-sided polygons, the sum of whose interior angles is 360°.

Quadrilaterals may be classified according to their characteristics.

Trapezoid

Two sides are parallel.

Parallelogram

Both pairs of opposite sides are parallel. Both pairs of opposite sides are equal in length.

Rhombus

Both pairs of opposite sides are parallel. The four sides are equal in length.

Rectangle

Both pairs of opposite sides are parallel. Both pairs of opposite sides are equal in length. The angles are right angles.

Square

Both pairs of opposite sides are parallel. The four sides are equal in length. The angles are right angles.

Perimeter and Area

P = s1 + s2 + b1 + b2

P = s1 + s2 + s3

P = 2b + 2w

P = 4s

P = 2l + 2w

Perimeter

Trapezoid

Triangle

A = bhParallelogram

A = s2Square

A = lwRectangle

AreaFigure

12A bh

11 22 ( )A h b b

Find the perimeter and area of the trapezoid.

16 in

6 in

13 in 13 in12 in

Circles Circumference is the length around the

outside of the circle. Radius is a segment from the center of

the circle to any point on the edge. Diameter is a segment from edge to

edge through the center. (Twice the radius!)

Formulas for area and circumference involve the constant pi or π.

Circles

What is π?

We can just use the approximate value of 3.14, but…

π = 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632788659361533818279682303019520353018529689957736225994138912497217752834791315155748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488…

Terri is installing a new circular swimming pool in her backyard. The pool has a diameter of 27 feet. How much area will the pool take up in her yard? (Use π = 3.14.)

Volume and Surface Area

Volume is the measure of the capacity of a figure. It is the amount of material you can put inside a three-dimensional figure.

Surface area is sum of the areas of the surfaces of a three-dimensional figure.It refers to the total area that is on the outside surface of the figure.

Volume – the amount of physical space the object occupies

Surface Area – unfolding and finding the

area of each face

3 units

5 units 2 units

Sphere

Cone

V = r2hCylinder

V = s3Cube

V = lwhRectangular Solid

DiagramFormulaFigure

213V r h

343V r

Sphere

Cone

SA = 2rh + 2r2Cylinder

SA= 6s2Cube

SA = 2lw + 2wh +2lhRectangular Solid

DiagramFormulaFigure

SA r2

r r2

h2

SA 4r 2

Mr. Stoller needs to order potting soil for his horticulture class. The class is going to plant seeds in rectangular planters that are 12 inches long, 8 inches wide and 3 inches deep. If the class is going to fill 500 planters, how many cubic inches of soil are needed? How many cubic feet is this?

We need to find the volume of one planter.

Soil for 500 planters would be

where B is the area of the base and h is the height.

Example: Find the volume of the pyramid.

13V Bh

12 m

12 m

18 m