Unit 1 Notes(Meredith Derucki)
Points, lines, and planes are the basics of Geometry. Points define a definite point and are named by writing Point then followed by the point you are trying to express. Lines extend on forever and can be named by either saying Line (name of line, two points on the line) or the points of the line with the line symbol above them. Planes, which can contain points and lines, is a flat surface that extends on forever. You can name them by saying Plane (Name of plane.)
Collinear vs. Coplanar:
Collinear is when given points all fall on the same line. If points are collinear they cannot be coplanar. Coplanar means they are not collinear and can fall in the same plane.
Inductive Reasoning is showing that a statement is true through showing the truth value of other statements.
Conjectures: A statement that should be true based on inductive reasoning.
Counterexamples: Examples given that are then used to prove the conjecture is false.
Deductive Reasoning is the process of using logic reasoning with facts and other news to draw conclusions.
Conditional Statements are statements that can be written in “If p, then q” form. The p is the hypothesis and the q is the conclusion. They have the truth value of either true or false. p → q
Statements can also be changed around to form converse, inverse, and contrapositive statements. Negations are when you take for example, p and change it to ‘not p’ and is written as ~p.
Converse statements are formed by flipping around the hypothesis and the conclusion. q → p Said as “If q, then p.”
Inverse statements are changing the hypothesis and conclusion to the ‘nots’ or negating it. ~p → ~q Said as “If not p, then not q.”
Contrapositive statements are formed by flipping and negating the hypothesis and conclusion. ~q → ~p Said as “If not q, then not p.”
Biconditional Statements can be written as “p if and only if q.”
Proofs are arguments that use definitions, logic, properties, and other proven or given statements to show a conclusion is true.
Properties of Equality
Properties of Congruence are reflexive, symmetric, and transitive.
Algebraic Proofs work in the sense of you are solving an equation/expression and as you solve you apply a property of equality to each step.
Geometric Proofs work with unique properties to prove a statement given true with logic. Uses the Angle Addition Postulate and the Segment Addition Postulate. Which is adding angles or segments to then get a greater value of another segment.
Segments (Constructing and Measuring)
The distance between two points is the absolute value of the difference of the coordinates. Congruent segments are segments that have the same measure. A midpoint is a point that splits a segment into two congruent segments. Bisectors are rays/lines/etc. that create a midpoint and two congruent segments of the overall segment. To construct congruent segments you could measure using a ruler is the simplest way. To create a bisector of the segment you could measure or use a compass. When using a compass put the ‘pointy end’ on the endpoint of a line and draw a small curve that passes through the line. Do the same on the other side. Where the curves meet is where the bisector will be.
Fractional Distance is finding a fraction of a segment. To do this find rise/run. Then find whatever fraction of the segment you want, but find that fraction of the rise and run. You now will have a new rise/run. Move that way and the point you land on will make the fraction of the segment.
Angles
You can name and angle with either 3 points with the vertex in the middle, or name it just the vertex itself.
The types of angles are acute (less than 90 degrees), right (90 degrees exact), obtuse (over 90 degrees up to 179 degrees), straight angle (180 degrees).
Congruent angles are angles that have the same measure.
An angle bisector is a line/ray/etc. that splits the angle into two congruent angles.
Adjacent angles are angles that share a common edge and vertex, but do not overlap.
Vertical angles are angles that share a common vertex and no common sides and are opposite from each other.
Linear pair of angles are angles that are adjacent and the non common side is an example of opposite rays.(Rays that share a common endpoint and form a straight line.)
Complementary angles add up to 90 degrees.
Supplementary angles add up to 180 degrees.
To copy an angle and bisect it.
To copy an angle draw a straight line and then get a compass and draw a semicircle on the original angle and then without moving the measure of the compass redraw the semicircle figure on the straight line drawn before. Measure the length between the intersections of the semicircle and the angle on the original angle. Find that same measurement on the new angle being constructed. Now that the semicircle on the new angle has 2 intersection points draw from the edge of the segment to the intersection point that doesn’t have a line through it. This copies the angle.To bisect, take the compass and place the pointed part of the semicircle intersection points and draw a curved line not very circular like. Copy for the other side. Where the two curves meet is where the bisector will be. This bisects the angle.
Postulates are statements that are always true without having to be proven.
The Postulates for this unit are:
Points, lines, and planes are the basics of Geometry. Points define a definite point and are named by writing Point then followed by the point you are trying to express. Lines extend on forever and can be named by either saying Line (name of line, two points on the line) or the points of the line with the line symbol above them. Planes, which can contain points and lines, is a flat surface that extends on forever. You can name them by saying Plane (Name of plane.)
Collinear vs. Coplanar:
Collinear is when given points all fall on the same line. If points are collinear they cannot be coplanar. Coplanar means they are not collinear and can fall in the same plane.
Inductive Reasoning is showing that a statement is true through showing the truth value of other statements.
Conjectures: A statement that should be true based on inductive reasoning.
Counterexamples: Examples given that are then used to prove the conjecture is false.
Deductive Reasoning is the process of using logic reasoning with facts and other news to draw conclusions.
Conditional Statements are statements that can be written in “If p, then q” form. The p is the hypothesis and the q is the conclusion. They have the truth value of either true or false. p → q
Statements can also be changed around to form converse, inverse, and contrapositive statements. Negations are when you take for example, p and change it to ‘not p’ and is written as ~p.
Converse statements are formed by flipping around the hypothesis and the conclusion. q → p Said as “If q, then p.”
Inverse statements are changing the hypothesis and conclusion to the ‘nots’ or negating it. ~p → ~q Said as “If not p, then not q.”
Contrapositive statements are formed by flipping and negating the hypothesis and conclusion. ~q → ~p Said as “If not q, then not p.”
Biconditional Statements can be written as “p if and only if q.”
Proofs are arguments that use definitions, logic, properties, and other proven or given statements to show a conclusion is true.
Properties of Equality
- Addition Property of Equality: If a=b, then a+c=b+c.
- Subtraction Property of Equality: If a=b, then a-c=b-c.
- Multiplication Property of Equality: If a=b,then ac=bc.
- Division Property of Equality: If a=b and c does not equal 0, then a/c=b/c.
- Reflexive Property of Equality: a=a.
- Symmetric Property of Equality: If a=b, then b=a.
- Transitive Property of Equality: If a=b and b=c, then a=c.
- Substitution Property of Equality: If a=b then b can be substituted for a in any expression.
Properties of Congruence are reflexive, symmetric, and transitive.
Algebraic Proofs work in the sense of you are solving an equation/expression and as you solve you apply a property of equality to each step.
Geometric Proofs work with unique properties to prove a statement given true with logic. Uses the Angle Addition Postulate and the Segment Addition Postulate. Which is adding angles or segments to then get a greater value of another segment.
Segments (Constructing and Measuring)
The distance between two points is the absolute value of the difference of the coordinates. Congruent segments are segments that have the same measure. A midpoint is a point that splits a segment into two congruent segments. Bisectors are rays/lines/etc. that create a midpoint and two congruent segments of the overall segment. To construct congruent segments you could measure using a ruler is the simplest way. To create a bisector of the segment you could measure or use a compass. When using a compass put the ‘pointy end’ on the endpoint of a line and draw a small curve that passes through the line. Do the same on the other side. Where the curves meet is where the bisector will be.
Fractional Distance is finding a fraction of a segment. To do this find rise/run. Then find whatever fraction of the segment you want, but find that fraction of the rise and run. You now will have a new rise/run. Move that way and the point you land on will make the fraction of the segment.
Angles
You can name and angle with either 3 points with the vertex in the middle, or name it just the vertex itself.
The types of angles are acute (less than 90 degrees), right (90 degrees exact), obtuse (over 90 degrees up to 179 degrees), straight angle (180 degrees).
Congruent angles are angles that have the same measure.
An angle bisector is a line/ray/etc. that splits the angle into two congruent angles.
Adjacent angles are angles that share a common edge and vertex, but do not overlap.
Vertical angles are angles that share a common vertex and no common sides and are opposite from each other.
Linear pair of angles are angles that are adjacent and the non common side is an example of opposite rays.(Rays that share a common endpoint and form a straight line.)
Complementary angles add up to 90 degrees.
Supplementary angles add up to 180 degrees.
To copy an angle and bisect it.
To copy an angle draw a straight line and then get a compass and draw a semicircle on the original angle and then without moving the measure of the compass redraw the semicircle figure on the straight line drawn before. Measure the length between the intersections of the semicircle and the angle on the original angle. Find that same measurement on the new angle being constructed. Now that the semicircle on the new angle has 2 intersection points draw from the edge of the segment to the intersection point that doesn’t have a line through it. This copies the angle.To bisect, take the compass and place the pointed part of the semicircle intersection points and draw a curved line not very circular like. Copy for the other side. Where the two curves meet is where the bisector will be. This bisects the angle.
Postulates are statements that are always true without having to be proven.
The Postulates for this unit are:
- Through any two points there is exactly one line.
- Through any three noncollinear points there is exactly one plane.
- If two points lie in a plane, then the line containing those points lies in the plane.
- If two unique lines intersect, then they intersect at exactly one point.
- If two unique planes intersect, then they intersect at exactly one line.
- Segment Addition: If B is between A and C, then Ab+BC=AC. In order for this to be true they must be on the same line.