points, lines, planes, and angles. a point, line, and plane are three basic terms in geometry that...
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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