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