Geometry Final Exam
Perimeter
(Of a polygon) is the sum of the length of its sides
Obtuse Angle
90<x<180 (x= the angle measure)
undefined terms
Definition that require terms that also need to be defined
Transitive Property of Congruence
EX: If <A is congruent to < and <B is congruent to <C, then <A is congruent to <C
Symmetric Property of Congruence
EX: If segment AB is congruent to segment CD, then segment CD is congruent to segment AB If <A is congruent to <B, then <B is congruent to <A
Reflexive Property of Congruence
EX: segment AB is congruent to segment AB <A is congruent to <A
Ruler Postulate
Every point on a line can be paired with a real number. This makes a one-to-one correspondence between the points on the line and the real numbers. The real number that corresponds to a point is called the COORDINATE of the point. Allows you to measure lengths of segments using a given unit and to find the distances between points on a number line.
Distance
The absolute value of the difference of their coordinates la-bl
P1
Through any two points there is exactly one line
Straight Angle
x=180 (x= the angle measure)
Right Angle
x=90 (x= the angle measure)
Truth Value
(Of a conditional) is either true or false. To show that a conditional is true, show that every time the hypothesis is true, the conclusion is also true.
Area
(Of a polygon) the number of squares units it encloses.
Perpendicular Bisector
(Of a segment) A line, segment, or ray that is perpendicular to the segment at its midpoint. The perpendicular bisector bisects the segment into congruent segments.
Acute Angle
0<x<90 (x= the angle measure)
Subtraction Property of Equality
If a=b, then a-c=b-c
Segment Addition Postulate
If three points A, B, and C are collinear and B is between A and C, then AB+BC=AC
Rectangle
P= 2b+2h or 2(b+h) A=bh
Conjecture
A conclusion you reach using inductive reasoning. It is important to gather enough data before you make a conjecture. Not all conjectures turn out to be true. You should test your conjecture multiple times.
Theorem
A conjecture or statement that you proe true
Proof
A convincing argument that uses deductive reasoning shows why a conjecture is true
Constuction
A geometric figure drawn using a straightedge and a compass
Transversal
A line that intersects two or more coplanar lines at distinct points
Linear Pair
A pair of adjacent angle whose non common sides are opposite rays. The angles of a linear pair form a straight angle.
Segment Bisector
A point, line, ray, or other segment that intersects a segment at its midpoint is said to bisect the segment. The point, line, ray, or segment is called a segment bisector.
Angle Bisector
A ray that divides an angle into two congruent angles. Its endpoint is at the angle vertex. Within the ray, a segment with the same endpoint is also an angle bisector. The ray or segment bisects the angle.
Good Definition
A good definition is a statement that can help you identify or classify an object. A good definition has several important components: - uses clearly understood terms. Commonly understood or already identified - precise. Good definitions avoid words such as large, sort of, and almost - reversible. You can write a good definition as a true biconditional How to find a bad one: find a counterexample
Conditional
If-then statement p ->q
Square
P= 4s A=s2 (squared)
Biconditional
A single true statement that combines a the conditional and its true converse. You can write a biconditional by joining the two parts of each conditional with the phrase: if and only if Ex: Converse- If two angles are supplementary, then the sum of the measures of the two angles is 180 The converse is true. You can form a true biconditional by joining the two parts of each conditional with the phrase: if and only if Biconditional- Two angles are supplementary if and only if the sum of the measures of the two angles is 180 p->q + q->p = p<->q
T2-4
All right angles are congruent
Law of Syllogism
Allows you to state a conclusion from two true conditional statements when the conclusion of one statement is the hypothesis of the other statement If p-> q is true and q->r is true, then p->r is true
postulate
An accepted statement of fact Building blocks of the logical system in geometry
Congruent Angles
Angles with the same measure. This means that if m<A=m<B, then <A is congruent to <B. You can mark angles with arcs to show that they are congruent. If there is more than one set of congruent angles, each set is marked with the same number of arcs.
Circle
C= pi x diameter or C= 2 x pi x diameter A= pi x radius squared
Equivalent Statements
Conditional and Contrapositive- either both true of false Converse and Inverse- either both true or false
Protractor Postulate
Consider ray OB and a point A on one side of line OB. Every ray of the form of ray OA can be paired one to one with a real number from 0 to 180. Allows you to find the measure of an angle. The MEASURE of the angle is the absolute value of the difference of the real numbers paired with the sides. Notice that the Protractor Postulate and the calculation of an angle measure are very similar to the Ruler Postulate and the calculation of a segment length.
Parallel Lines
Coplanar lines that do not intersect. The symbol ll means"is parallel to"
Converse
Exchange the hypothesis and the conclusion q ->p
Angle
Formed by two rays with the same endpoint. The rays are the SIDES of the angle. The endpoint is the VERTEX of the angle. When you name angles using three points, the vertex must go in the middle. The INTERIOR of an angle is the region containing all the points between the two sides of the angle. The EXTERIOR of an angle is the region containing all the points outside of the angle.
Alternate Interior Angles Theorem
If a transversal intersects two parallel lines, then alternate interior angles are congruent
Same-side Interior Angles Postulate
If a transversal intersects two parallel lines, then same-side interior angles are supllementary
Transitive Property of Equality
If a=b and b=c, then a=c
Division Property of Equality
If a=b and c is not equal to 0, then a/c = b/c
Multiplication Property of Equality
If a=b, the a x c= b x c
Addition Property of Equality
If a=b, then a+c=b+c
Substitution Property of Equality
If a=b, then b can replace a in any expression
Symmetric Property of Equality
If a=b, then b=a
Angle Addition Postulate
If point B is in the interior of <AOC, then m<AOB+m<BOC=m<AOC
Law of Detatchment
If the hypothesis of a true conditional is true, then the conclusion is true. If p->q is true and p is true, then p is true To use the Law of Detachment, identify the hypothesis of the given true conditional. If the second given statement matches the hypothesis of the conditional, then you can make a valid conclusion
Congruent Complements Theorem
If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent <1 and <2 are complements and <3 and <2 are complements, then <1 is congruent to <3
T2-5
If two angles are congruent and complementary, then each is a right angle
Congruent Supplements Theorem
If two angles are supplements of the same angle (or o ongruent angles), then the two are congruent If <1 and <3 are supplements and <2 and <3 are supplements, the <1 is congruent to <2
Linear Pair Postulate
If two angles form a linear pair, then they are supplementary
P2
If two distinct lines intersect, then they intersect in exactly one point
P3
If two distinct planes intersect, then they intersect in exactly one line
Congruent Segments
If two segments have the same length, then the segments are called this. This means that if AB=CD, then segment AB is congruent to segment CD. You can also say that if segment AB is congruent to segment CD, then AB+CD. You can mark segments alike to show that they are congruent. If there is more than one set of congruent segments, you can indicate each set with the same number of marks.
Point
Indicated a location and has no sides Undefined term
Same-side Interior Angles
Interior angles that lie on the same side of the transversal
ray
Is part of a line that consists of one endpoint and all the points of the line on one side of the endpoint
segment
Is part of a line that consists of two endpoints and all points between them
Hypothesis
Is the part p following the if part of if-then
Conclusion
Is the part q following the then part of if-then
Corresponding Angles
Lie on the same side of the transversal and in corresponding positions
Two-Column Proof
Lists each statement of the left and the justification, or the reason for each statement, is on the right. Each stetemnt must follow logically from the steps before it.
Inverse
Negate both the hypothesis and the conclusion of the conditional -p -> -q
Contrapositive
Negate both the hypothesis and the conclusion of the converse -q -> -p
Skew Lines
Non coplanar; they are not parallel and do not intersect
Alternate Exterior Angles
Nonadjacent exterior angles that lieon opposite sides of the transversal
Alternate Interior Angles
Nonadjacent interior angles that lie on opposite sides of the transversal
Midpoint Formulas
On a number line- the coordinate of the midpoint is the average or mean of the coordinates of the endpoints. A+B/2 In the coordinate plane- the coordinates of the midpoint are the average of the x-coordinates and the average of the y-coordinates of the endpoints (x1+x2/2 , y1 + y2/2)
Triangle
P= a+b+c A= 1/2bh
Parallel Planes
Planes that don't intersect
coplanar
Points and lines that lie on the same plane
collinear points
Points that lie on the same line
Inductive Reasoning
Reasoning based on patterns you observe. You can observe patterns in some number sequences and some sequences of geometric figures to discover relationships
plane
Represented by a flat surface that extends without ends and has no thickness. A plane contains indefinitely many lines Undefined term
line
Represented by a straight path that extends in two opposite directions without end and has no thickness. A line contains infinitely many points Undefined term
Area Addition Postulate
The area of a region is the sum of the areas of its non overlapping parts
Distance Formula
The distance between two points d= the square root of (x2-x1)2 + (y2-y1)2 The distance formula is based on the Pythagorean Theorem. When you use the distance formula, you are really finding the length of the side of a right triangle.
External Angles
The exterior angles lie outside of the lines and include the transversal
Interior Angles
The interior angles lie inside of two lines and include the transversal
Midpoint
The midpoint of a segment is the point that divides the segment into two congruent segments.
Negation
The negation of a statement p is the opposite of the statement. the symbol is -p and is read "not p." The negation of the statement "The sky is blue" is "The sky is not blue." You can use negations to write statements related to a conditional. Every conditional had three related conditional statements.
Deductive Reasoning
The process of reasoning logically from given statements or facts
Finding Information From a Diagram
There are some relationships you can assume to be true from a diagram that has no marks or measures. there are others measures you cannot assume directly.
Segments and rays can also be parallel or skew
They are parallel if they lie in parallel lines and skew if they lllie in skew lines
P4
Through any three non collinear points there is exactly one plane
Supplementary Angles
Two angles whose measures ave a sum of 180. Each angle is called the supplement of the other
Complementary Angles
Two angles whose measures have a sum of 90. Each angle is called the complement of the other.
Vertical Angles
Two angles whose sides are opposite rays
Adjacent Angles
Two coplanar angles with a common side, a common vertex, and no common interior points.
Perpendicular Lines
Two lines that intersect to form right angles
opposite rays
Two rays that share the same endpoint and form a line
Distributive Property
Use multiplication to distribute to each term of the sum or difference within the parenthesis Sum: a(b+c)=ab+ac Difference: a(b-c)= ab-ac
Vertical Angles Theorem
Vertical angles are congruent
intersection
When you have two or more geometric figures, their intersection is the set of points the figures have in common
Paragraph Proof
Written as sentences in a paragraph
Counterexample
You can prove a conjecture wrong is false by finding one counterexample. An example that shows that a conjecture is incorrect A counterexample can help you determine whether a conditional with a true hypothesis is true. If you find one counterexample for which the hypothesis is true and the conclusion is false, then the truth value of the conditional is false.
Reflexive Property of Equality
a=a
