1. Point: An imaginary geometric that is undefined. A point is represented by a dot.
2. Line: An imaginary, undefined geometric term. A line is composed of connecting points. Think of the printout from your computer. If you draw a line and magnify it you will see connecting squares (points). A line is perfectly straight and extends indefinitely in two opposite directions.
3. Plane: An undefined geometric term composed of three non collinear points and the lines that connect them. Naturally two of these points are collinear, but the third point is part of a second line. The top of a desk represents part of a plane. Planes have length and width, but no depth; therefore they are imaginary. Like lines, planes extend indefinitely in all directions.
4. Collinear: Points that lie on a single line are collinear. "Co" means share or together.
5. Coplanar: Points that lie on a single plane are coplanar. Any three points are coplanar.
6. Segment: A segment is part of a line that begins at one point and ends at another.
7. Endpoints: Points on a line that define the beginning and end of a segment.
8. Ray: A ray is part of a line that begins at one point and extends indefinitely in one direction.
9. Angle: An angle consists of two rays that begin at a common endpoint. This endpoint is called the vertex of the angle. The rays are the sides of the angle. An angle divides a plane into two regions -- an interior region and an exterior region. Think of a clock. If the hour hand is on 12 and the minute hand is on 3, a 90 degree interior angle is formed and a 270 degree exterior angle is formed. If the hour hand is on 12 and the minute hand is on 11, a 330 degree interior angle is formed and a 30 degree exterior angle is formed.
10. Intersect: If geometric figures have one or more points in common, the figures intersect at the point or points they share. The set of points that they share (or have in common) are called their intersection. Think of the intersection of streets.
11. Postulates: Fundamental geometric ideas that are accepted as true without proof. Postulates are also called "axioms."
12. Length of segment AB: Let A and B be points on a number line, with coordinates a and b. The measure of segment AB is called its length and is determined by calculating the absolute value of the difference of a and b.
13. Number line: A line that has been constructed and marked at evenly spaced intervals that correspond with the real numbers.
14. Coordinate of a point: A real number that corresponds with a point on a number line.
15. Congruent: Figures that have the same size and shape are said to be congruent. One of the figures can be rotated so that it is a vertical or horizontal image of the other and if its size and shape does not change, the figures are still congruent.
16. Measure of an Angle: Angles are measured in degrees as shown by coordinates of a half circle ranging from 0 to 180 degrees. A protractor is used to measure angles. The formula for finding angles is to subtract the coordinate of a point on one ray from the coordinate of the other ray and take the absolute value of the difference.
17. Complementary angles: Two angles whose measures have a sum of 90 degrees. Each angle is called the complement of the other.
18. Supplementary angles: Two angles whose measures have a sum of 180 degrees. Each angle is called the supplement of the other.
19. Right angle: An angle whose measure is 90 degrees.
20. Acute angle: An angle whose measure is less than 90 degrees.
21. Obtuse angle: An angle whose measure is greater than 90 degrees, but less than 180 degrees.
22. Reflex angle: An angle whose measure is greater than 180 degrees
23. Straight angle: An angle whose measure is 180 degrees. This is a straight line.
24. Perpendicular lines are two lines that intersect to form a right angle.
25. Parallel lines are two coplanar lines that do not intersect.
26. Conjecture: A statement that is an educated guess based on observation. These statements have not been proven.
27. Segment bisector: A line that divides a segment into two congruent parts.
28. Midpoint: The point where a bisector intersects a segment.
29. Perpendicular bisector: A segment bisector that forms right angles with the line it bisects.
30. Angle bisector: A line or ray that divides an angle into two congruent angles.
31. Inscribed circle. A circle that is drawn inside another geometric figure and that touches it at one point on each of its sides. A circle inscribed in a triangle touches the triangle at one point on each of its three sides. "In" indicates inside.
32. Incenter. The center of an inscribed circle. It is the point where the angle bisectors of the geometric figure that contains the circle intersect.
33. Circumscribed circle. A circle drawn outside another geometric figure and touches the geometric figure on the outside of each of its corner points. "Circum" means around.
34. Circumcenter. The center of a circumscribed circle. It is the point where the perpendicular bisectors of the sides of the geometric figure inside the circle intersect.
35. Median. Segments drawn from the midpoint of each side of a triangle to its opposite vertex. (The vertex of a triangle is the point where two lines come together to form an angle.)
36. Centroid. The point in the center of the triangle where all three medians intersect.
37. Rigid Transformation: Changing the position of an object without changing the shape or size of the object.
Translation: Every point of a figure moves in a straight line, and all points move the same distance and in the same direction. The paths of the points are parallel.
Rotation: Every point of the preimage is moved by the same angle through a circle centered at a given fixed point known as the center of rotation.
Reflection: Every point of the preimage is moved across a line known as the mirror line so that the mirror is the perpendicular bisector of the segment connecting the point and its image.
1. Conditional: If - then statements.
If P then Q or (Read p implies q)
2. Hypotensis: The "if" part of the conditional.
3. Conclusion: The "then" part of the conditional. The process of drawing a conclusion is written as a logical argument called a syllogism. It has three parts. The first part is a conditional statement where the hypotenuse is a specific statement, and the conclusion is a general statement. The second part states a fact. The last part draws a conclusion based on the conditional.
If a car is a Mustang (specific make of car), then it is a Ford. (general make of car)
Sam's car is a Mustang. (Statement of fact.)
Therefore, Sam's car is a Ford. (conclusion).
4. Deductive reasoning or deduction: The process of drawing logically certain conclusions by using an argument.
5. Euler diagram: An illustration of the relationship between general and specific facts that lead to logical argument.
6. Converse: A statement made by interchanging the hypotenuse and conclusion of a conditional statement.
Conditional statement: If a person is seventy years old, then the person is a senior citizen.
Converse statement: If a person is a senior citizen, then the person is seventy years old.
Notice that the converse is not necessarily true.
7. Counterexample: An example that shows that the converse of a conditional is not always true. In the example above, a person who is sixty-five years old is also considered a senior citizen. Therefore, the converse is false.
8. Logical chain: A series of related conditional statements that are logically linked together.
9. Biconditional: The condition that exists when the conditional and its converse are both true. The statements are rewritten as one statement using "if and only if."
P if and only if Q or
A person is a teenager if and only if he or she is between 13 and 19 years old.
10. Inductive reasoning: The process of forming conjectures based on observations. Because some conjectures can be proven false, inductive reasoning is not acceptable in mathematical proofs.
11. Adjacent angles: Angles formed on the same side of two intersecting lines. They share a ray and a vertex, but they do not share any interior points..
Angles 1 and 2 above are adjacent angles.
12. Vertical angles: Angles formed on opposite sides of two intersecting lines. They share a vertex, but they do not share a common side or any interior points.
Angles 1 and 2 above are vertical angles.
1 Definition of slope: The slope of a non vertical line that contains point 1 (x, y) and point 2 ((x, y)) is equal to the ratio .
2. Isometric drawing: A drawing in which the horizontal lines of an object are represented by lines that form 30 degree angles with a horizontal line in the picture.
3. Orthographic projection: A view of an object in which points of the object are "projected" onto the picture plane along lines perpendicular to the picture plane.
4. Polyhedron: A closed spatial figure composed of polygons, call the faces of the polyhedron.
a. Vertices: The vertices of the faces are the vertices of the polyhedron
b. Faces: The polygons that make up the polyhedron
c. Edges: The place where two polygons meet.
c. Skew: Segments or rays of a polyhedron that are not parallel, yet would never meet if they were extended indefinitely.
d. Regular polyhedron: a polyhedron where all of the faces are congruent regular polygons, and the same number of polygons meet at each vertex. A cube is a regular polyhedron.
5. Parallel planes
Two planes are parallel if and only if they do not intersect (page 864).
6. A line perpendicular to a plane
A line is perpendicular to a plane if and only if it is perpendicular to every line in the plane that _______.
7. A line parallel to a plane
A line that is not contained in a given plane is parallel to the plane if and only if it is parallel to a line contained in the plane.
8. Dihedral angle: A dihedral angle is the figure formed by two half-planes with a common edge. Each half plane is called a face of the angle, and the common edge of the half-plane is called the
edge of the angle.
(Notice the half-plane includes the line.)
9. Measure of a dihedral angle.
The measure of a dihedral angle is the measure of an angle formed by two rays that are on the faces and that are perpendicular to the edge.
10. Right prism: A prism in which all the lateral faces are rectangles. Remember that all the angles of a rectangle are right angles.
11. Oblique prism: A prism that has at least one non rectangular face. (See "prism" below)
Prism: A polyhedron that consists of a polygonal region and its translated image on a parallel plane, with quadrilateral faces connecting corresponding edges.
Base: The polygonal region and its translated image.
Lateral edges: The "line" where two of the quadrilateral faces of a prism meet.
Lateral faces: Quadrilateral-shaped sections of planes that connect the bases of a prism.
Triangular prism: A prism that has congruent bases shaped like triangles that are connected by quadrilaterals
Rectangular prism: A prism that has congruent bases shaped like rectangles that are connected by quadrilaterals.
Pentagonal prism: A prism that has congruent bases shaped like pentagons (five-sided figures) that are connected by quadrilaterals.
Hexagonal prism: A prism that has congruent bases shaped like hexagons ( six-sided figures) that are connected by quadrilaterals.
cube: A regular prism that contains six congruent squares-- two are the bases and the other four are the lateral faces.
12. Diagonal of a right rectangular prism:
In a right rectangular prism with dimensions l x w x h, the length of a diagonal is given by:
13. Right-handed system: The arrangement of the x, y, and z axis of three-dimensional coordinate planes.
14. Octants (think about quadrants on a coordinate plane.) The x, y, and z axis of three-dimensional coordinate planes divide space into eight octants.
15. Distance formula in three dimensions
The distance, d, between the first point (x, y, z) and the second point (x, y, z) is given by
16. Midpoint formula in three dimensions. (This is just like the midpoint formula for two dimensions, but the z coordinates are included.)
The midpoint of a segment with endpoints at point 1 (x, y, z) and point 2 (x, y, z) is determined by taking one-half the sum of each corresponding coordinate point -- ((x+x)/2, (y + y)/2, (z + z)/2).
17. Equation of a plane.
Remember the standard form for the equation of a line is Ax+ By = C. The equation of a plane is similar with the z coordinate added. Ax + By + Cz = D where A, B, and C are real numbers and A is positive and is a whole number.
18. Parametric equations -- the instructions have different rules for each coordinate. They cannot be written in one equation, but in three separate equations for each coordinate.
19. Perspective drawings
a. Perspective (Latin origin)
20. Vanishing point
Unit 7: Surface Area and Volume
Directions: Define the following terms, formulas, etc. and illustrate when possible.
surface area and Volume Formulas (page 430)
Altitudes of prisms (page 437)
Height of prisms.
Surface area of right prisms (page 438)
Cavalieri's Principle (page 440)
Volume of a Prism (page 440)
Pyramids (page 446)
vertex of the pyramid
altitude of a pyramid
height of a pyramid
Surface area of a regular pyramid
Volume of a pyramid
Altitude of a cylinder
Height of a cylinder
Axis of a cylinder
Volume of a cylinder
Surface area of a right cone
Volume of a cone
Volume of a sphere
Area of the circle in a sphere
Area of the annulus
Surface area of a sphere (page 472)
Revolutions in Coordinate Space