Point

A location

Does a point have length?

NO

Does a point have width?

NO

Does a point have thickness?

NO

Line

An infinite set of points that extends in two directions

Does a line have length?

yes

Does a line have width?

NO

Does a line have thickness?

NO

Plane

An infinite set of points that creates a flat surface that extends without ending

Does a plane have length?

YES

Does a plane have width?

YES

Does a plane have thickness?

NO

Space

the set of all points

colilinear

points on the same line

noncollinear

points not on the same line

coplanar

points on the same plane

Noncoplanar

points not on the same plane

segment

a segment is named by giving its endpoints

ray

named by giving its endpoint and another point

segment addition postulate

If B is between A and C, then AB +BC = AC

congruent

two objects that have the same size and shape are congruent

midpoint of the segment

a midpoint divides a segment into two congruent segments

bisector of a segment

a segment bisector is a line, segment, ray, or plane that intersects a segment at its midpoint

angle

a figure formed by two rays with the same endpoint is an angle

vertex

…

measure of an angle

you can use a protractor to find the number associated with each side of an angle. To find the measure in degrees of an angle (m is greater than XYZ), compute the absolute value of the difference of these numbers

acute angle

angle is greater than 0 and less than 90 degrees

obtuse angle

greater than 90 degrees and less than 180 degrees

right angle

angles= 90 degrees

straight angle

angle = 180 degrees

angle addition postulate

if a point Y lies in the interior of angle XOZ, then measurement of angle XOY + measurement of YOZ = measurement of angle XOZ

congruent angles

two angles with equal measures are congruent: if measure of angle 5 = measure of angle 6, then angle 5 = angle 6

bisector of an angle

the ray that divides an angle into two congruent angles is the angle bisector

adjacent angles

coplanar angles with a common vertex and a common side but no common interior points are adjacent angles

postulate

a basic assumption accepted without proof

theorem

a statement that can be proved using postulates, definitions, and previously proved theorems

exists

there is at least one

one and only one

exactly one

determine

to define or specify

relationships between points

Two points must be collinear, three points may be collinear or noncollinear;Three points must be coplanar; three noncollinear points determine a plane; four points may be coplanar or noncoplanar;Four noncoplanar points determine space; space contains at least four non coplanar points

Three ways to determine a plane

three noncollinear points determine a plane (pos 7); a line and a point not on the line determine a plane (theorem 1-2), two intersecting lines determine a plane (theorem 1-3)

relationships between two lines in the same plane

either two lines are parallel, or they intersect in exactly one point

relationship between a line and a plane

either a line and a plane are parallel, or they intersect in exactly one point, or the plane contains the line

relationships between two planes

either two planes are parallel, or they intersect in a line

conditional statement (or conditional)

written in if-then

what letter do they use for the hypothesis?

p

what letter do they use for the conclusion?

q

conditional statement

if it is snowing, then it is cold?

p implies q

it is snowing implies it is cold

p only if q

it is snowing only if it is cold

q if p

it is cold if it is snowing

converse

of a conditional statement is formed by interchanging the hypothesis and the conclusion

conditional statement

if p, then q (if today is Tuesday, then tomorrow is Wednesday)

converse

if q, then p (if tomorrow is Wednesday, then today is Tuesday)

counterexample

one way of proving a statement false is to give this

biconditional statement

a statement combining a conditional and its converse

conditional statement

if p, then q (if today is Tuesday, then tomorrow is Wednesday)

converse

if q, then p (if tomorrow is Wednesday, then today is Tuesday)

biconditional

p if and only if q (Today is Tuesday if and only if tomorrow is Wednesday)

midpoint theorem

if M is the midpoint of AB, then AM = 1/2AB and MB =1/2AB

angle bisector theorem

if BX is the bisector of angle ABC, then m of angle ABX = 1/2 measurement of angle ABC and measurement of angle XBC = 1/2 measurement of angle ABC

complementary angles

two angles whose measure have the sum 90 are complementary

supplementary angles

two angles whose measures have the sum 180 are supplementary

are vertical angles congruent?

yes

perpendicular lines

two lines that intersect to form right angles; biconditional

(1) if two lines are perpendicular, then they form right angles

(2) if two lines form right angles, then the lines are perpendicular

(1) if two lines are perpendicular, then they form right angles

(2) if two lines form right angles, then the lines are perpendicular

Properties of perpendicular lines

1-if two lines are perpendicular, then they form congruent adjacent angles

2- if two lines form congruent adjacent angles, then the lines are perpendicular

3-if the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary

2- if two lines form congruent adjacent angles, then the lines are perpendicular

3-if the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary

if two angles are supplements of congruent angles (or of the same angle, then what does it say about the two angles?

they are congruent

If two angles are complements of congruent angles (or the same angle), what do you say aobut the two angles?

they are congruent

parallel lines

l is parallel to n if the lines are coplanar and they do not intersect

skew lines

p and q are skew lines if they are not coplanar -skew lines are neither parallel not intersecting

Are skew lines parallel?

no

Are skew lines intersecting?

no

parallel planes

plane N is parallel to plane M if they do not intersect. All planes are either parallel or intersecting.

all planes must be two things?

parallel or intersecting

if two parallel lines are cut by a transversal

then corresponding angles are congruent

if two parallel lines are cut by a transversal

then alternate interior angles are congruent

if two parallel ones are cut by a transversal

then same-side interior angles are supplementary

if a transversal is perpendicular to one of two parallel lines

then it is perpendicular to the other one also

if two parallel lines are cut by a transversal, then …

corresponding angles are congruent, alternate interior angles are congruent AND same-side interior angles are supplementary

if a transversal is perpendicular to one of two parallel ones, then

it is perpendicular to the other one also

if two lines are cut by a transversal and corresponding angles are congruent, then the lines are

parallel

if two lines are cut by a transversal and alternate interior angles are congruent, then the lines

are parallel

if two lines are cut by a transversal and same-side interior angles are supplementary, then the lines are

parallel

in a plane two lines perpendicular to the same line are

parallel

two lines parallel to a third line are

parallel to each other

Ways to prove two lines parallel

1. show that a pair of corresponding angles are congruent

2. show that a pair of alternate interior angles are congruent

3. show that a pair of same-side interior angles are supplementary

4. in a plane show that both lines are perpendicular to a third line

5. show that both lines are parallel to a third line

2. show that a pair of alternate interior angles are congruent

3. show that a pair of same-side interior angles are supplementary

4. in a plane show that both lines are perpendicular to a third line

5. show that both lines are parallel to a third line

triangle

the figure formed by three segments joining three noncollinear points

what are parts of a triangle

vertices, sides, angles

how do you classify a triangle?

the number of congruent sides

scalene triangle

no sides are congruent

isosceles triangle

at least two sides congruent

equilateral triangle

all sides congruent

triangles can be classified by angles too

…

acute triangle – how many angles?

three acute angles

right triangle – how many angles?

one right 90 degree angle

obtuse triangle – how many angles?

one obtuse angle

equiangular triangle – what about the angles?

all are congruent

the sume of the measures of the angles of a triangle =

180 degrees

exterior angle

the angle formed when one side of a triangle is extended

remote interior angle

the two angles of the triangle not adjacent to the exterior angle

the measure of an exterior angle of a triangle =

the sum of the measures of the two remote interior angles

polygons

have vertices, sides, angles, and exterior angles

how do you name a polygon?

by listing consecutive certices in order

diagonal

a segment connecting two nonconsecutive vertices of a polygon

convex polygons

polygons that no part of a diagonal is exterior to the polygon

3 sided polygon

triangle

4 sided polygon

quadrilateral

5 sided polygon

pentagon

6 sided polygon

hexagon

8 sided polygon

octagon

10 sided polygon

decagon

n sided polygon

n-gon

the sum of the measures of the angles of a convex polygon with n sides =

(n-2)180

the sum of the measures of the exterior angles of any convex polygon, one at each vertex =

360

regular polygon

a polygon that is both equiangular (all angles congruent) and equilateral (all sides congruent) is this kind of polygon

equiangular

all angles congruent

equilateral

all sides congruent

how many sides does a regular polygon have if the measure of each exterior angle is 60?

…

how many sides does a regular polygon have if the measure of each interior angle is 160?

…

inductive reasoning

based on several past observations

conclusions based on inductive reasoning, are they always true?

sometimes, but not always true

deductive reasoning

based on accepted statements, including previous theorems, postulates, definitions, and given information

in deductive reasoning, the conclusion must be true if?

the hypotheses are true

congruent

if the same size and shape

two triangles are congruent if and only if …

their vertices can be matched up so corresponding parts (angles and sides) of the triangles are congruent

two polygons are congruent if and only if their vertices can be matched up so that corresponding parts are

congruent

SSS postulate

if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent

SAS postulate

if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent

ASA postulate

if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent

isosceles triangle theorem

defined as having at least two sides congruent. they have special names for their parts.

if two sides of a triangle are congruent, then the angles opposite those sides are congruent.

if two sides of a triangle are congruent, then the angles opposite those sides are congruent.

names of isosceles triangle parts

legs, base, vertex angle, base angles

legs

congruent sides of isosceles triangle

base

the third side of isosceles triangle

vertex angle

the angle opposite the base of isosceles triangle

base angles

the angles adjacent tot he base of the isosceles triangle

if two angles of a triangle are congruent, then the sides opposite those angles are:

congruent

If two angles of a triangle are congruent, then the sides opposite those angles are:

congruent

other methods of proving triangles are congruent

AAS theorem, HL theorem

AAS Theorem

if two angles and non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent

HL Theorem

If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent

All ways to prove two triangles are congruent (all triangles)

1. SSS postulate

2. SAS postulate

3. ASA postulate

4. AAS theorem

2. SAS postulate

3. ASA postulate

4. AAS theorem

WAys to prove two right triangles are congruent

HL Theorem

median of a triangle

a segment from a vertex of a triangle to the midpoint of the opposite side is a this

altitude of a triangle

a segment from a vertex of a triangle perpendicular to the line that contains the opposite side is this

how many medians does every triangle have?

3

how many altitudes does every triangle have?

3

where are the medians of a triangle located?

inside the triangle

where are the altitudes of a triangle located?

inside, outside or part of the triangle

perpendicular bisector

a line, segment, or ray perpendicular to a given segment at its midpoint is >>>> of the segment

a point lies on the perpendicular bisector of a segment if and only if the point is

equidistant from the endpoints of the segment

a point lies on the bisector of an angle if and only if

it is equidistant from the sides of the angle

distance from point to a line

…

parallelogram

a quadrilateral with both pairs of opposite sides parallel is this

opposite sides of a parallelogram are

congruent

opposite angles of a parallelogram are

congruent

diagonals of a parallelogram …

bisect each other

How to prove quadrilaterals are parallelograms?

…

If both pairs of opposite sides of a quadrilateral are …..then the quadrilateral is a parallelogram

congruent

if one pair of opposite sides of a quadrilateral are both …then the quadrilateral is a parallelogram

congruent and parallel

if the diagonals of a quadrilateral ……., then the quadrilateral is a parallelogram

bisect each other

if both pairs of opposite angles of a quadrilateral are …, then the quadrilateral is a parallelogram

congruent

if three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal

…

a line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side

if M is teh midpoint of XY and MN is parallel to YZ, then N is the midpoint of XZ

the segment that joins the midpoints of two sides of a triangle…

1) is parallel to the third side

2) is half as long as the third side

2) is half as long as the third side

rectangle

quadrilateral with four right angles

a parallelogram because both pairs of opposite angles are congruent

rectangle

rhombus

a quadrilateral with four congruent sides

rhombus

a parallelgram because both pairs of opposite sides are congruent

square

a quadrilateral with four right angles and four congruent sides

square

is a parallelogram, rectangle and a rhombus

rectangle

has four right sides, all the properties of a parallelogram and has congruent diagonals

If one angle of a parallelogram is a right angle, then the parallelogram…

is a rectangle

rhombus

has four congruent sides, has all teh properties of a parallelogram, has diagonals that are perpendicular, and has diagonals that bisect its angles

if two consecutive sides of a parallelogram are congruent, then the parallelogram is a

rhombus

square

1- four right angles and four congruent sides

2- has all the properties of the parallelogram

3- has diagonals that are both congruent and perpendicular

4-has diagonals that bisect its angles

2- has all the properties of the parallelogram

3- has diagonals that are both congruent and perpendicular

4-has diagonals that bisect its angles

the midpoint of the hypotenuse of a right triangle is …

equidistant from the three vertices

trapezoids

1- a quadrilateral

2- exactly one pair of parallel sides

2- exactly one pair of parallel sides

isosceles trapezoid

a trapezoid with congruent legs

trapezoid sides are called:

parallel sides: bases

other sides: legs

other sides: legs

base angles of an isosceles trapezoid are

congruent

median of a trapezoid

the segment that joins the midpoints of the legs of the trapezoid

the median of a trapezoid:

1-is parallel to the bases

2- has a length equal to the average of the base lengths

2- has a length equal to the average of the base lengths