141 terms

the points on a line can be paired with the real numbers in such a way that any two points can have coordinates 0 and 1

Ruler Postulate

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

Segment Addition Postulate

on line AB in a given plane, choose any point O between A and B. Consider ray OA and ray OB and all the rays that can be drawn from O on one side of line AB. These rays can be paired with the real numbers from 0 to 180 in such a way that: ray OA is paired with 0, and ray OB with 180, or, if ray OP is paired with X, and ray OQ with y, then angle POQ = l x-y l.

Protractor Postulate

If point B lies in the Interior of angle AOC, then angle AOB + angle BOC = angle AOC

Angle Addition Postulate

if angle AOC is a straight and and B is any point not on line AC, then angle AOB + angle BOC = 180

Angle Addition Postulate

a line contains at least two points; a plane contains at least three points not all in one line; space contains at least four points not all in one plane

Postulate 5

through any two points there is exactly one line

Postulate 6

through any three points there is at least one plane, and through any three non collinear points there is exactly one plane.

postulate 7

if two points are in a plane, then the line that contains the points is in that plane

Postulate 8

if two planes intersect, then their intersection is a line.

Postulate 9

If two lines intersect, then exactly one plane contains the lines

Theorem 1-1

through a line and a point not in the line there is exactly one plane

Theorem 1-2

If two lines intersect, then exactly one plane contains the lines

Theorem 1-3

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

Midpoint Theorem

If ray BX is the bisector of angle ABC, then angle ABX = 1/2angleABC and angle XBC = 1/2 angle ABC

Angle Bisector Theorem

Reasons used in Proofs

Given info, Definitions, Postulates (and properties from algebra), Proven Theorems

Vertical angles are congruent

Theorem 2-3

Complementary angles

two angles whose measures have the sum 90

Supplementary angles

two angles whose measures have the sum 180

if two lines are perpendicular, then they from congruent adjacent angles

Theorem 2-4

if two lines form congruent adjacent angles, then the lines are perpindicular

Theorem 2-5

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

Theorem 2-6

if two angles are supplements of congruent angles (or of the same angle) then the two angles are congruent

Theorem 2-7

if two angles are complements of congruent angles (or of the same angle), then the two angles are congruent

Theorem 2-8

Parallel lines

coplanar lines that do not intersect

Skew lines

noncoplanar lines

if two parallel planes are cut by a third plane, then the lines of intersection are parallel

Theorem 3-1

two nonadjacent interior angles on opposite sides of the transversal

Alternate interior angles

Same-Side Interior Angles

two interior angles on the same side of the transversal

two angles in corresponding positions relative to the two lines

Corresponding angles

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

Postulate 10

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

Theorem 3-2

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

Theorem 3-3

if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other ones also

Theorem 3-4

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

Postulate 11

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

Theorem 3-5

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

Theorem 3-6

in a plane two lines perpendicular to the same lines are parallel

Theorem 3-7

through a point outside a line, there is exactly one line parallel to the given line

Theorem 3-8

through a point outside a line, there is exactly one line perpendicular to the given line

Theorem 3-9

two lines parallel to a third line are parallel to each other

Theorem 3-10

Ways to prove two lines parallel

show that a pair of corresponding angles are congruent, show that both lines are parallel to a third line, show that a pair of alternate interior angles are congruent, show that a pair of same-side interior angles are supplementary, in a plane show that both lines are perpendicular to a third line

Scalene Triangle

no sides congruent

Isosceles Triangle

at least two sides congruent

Equilateral triangle

all sides congruent

Equiangular triangle

all angles congruent

the sum of the measures of the angles of a triangle is 180

Theorem 3-11

if two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent

Corollary 3-1

each angle of an equiangular triangle has measure of 60

Corollary 3-2

in a triangle, there can be at most one right angle or obtuse angle

Corollary 3-3

the acute angles of a right triangle are complementary

Corollary 3-4

the measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles

Theorem 3-12

Polygon

many angles

Convex Polygon

polygon such that no line containing a side of the polygon contains a point in the interior of the polygon

the sum of the measures of the angles of a convex polygon with n sides is (n-2)180

Theorem 3-13

the sum of the the measures of the exterior angles of any convex polygon, one angle at each vertex, is 360

Theorem 3-14

Inductive Reasoning

reasoning that is widely used in science and everyday life

Deductive Reasoning

conclusion based on accepted statements (definitions, postulates, previous theorems, corollaries, and given information) conclusion MUST be true if hypotheses are true

Inductive Reasoning

Conclusion based on several past observations. conclusion is PROBABLY true, but not necessarily true

Congruent

having the same size and shape

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

A way to prove two segments or two angles are congruent

identify two triangles in which the two segments or angles are corresponding parts, prove that the triangles are congruent, state that the two parts are congruent, using the reason CPCTC

Isosceles Triangle Theorem

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

an equilateral triangle is also equiangular

Corollary 4-1

an equilateral triangle has three 60° angles

Corollary 4-2

the bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint

Corollary 4-3

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

Theorem 4-2

an equiangular triangle is also equilateral

Corollary 4-4

AAS theorem

if two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the triangls are congruent

HL theorem

if the hypotenuse and a leg of one right triangle

Ways to prove triangles congruent

SSS, SAS, ASA, AAS

Ways to prove right triangles congruent

HL

Median

segment from a vertex to the midpoint of the opposite side in a triangle

Altitude

perpendicular segment from a vertex to the line that contains the opposite side in a triangle

Perpendicular bisector

line (or ray or segment) that is perpendicular to the segment at its midpoint

if a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment

Theorem 4-5

if a point is equidistant from the endpoints of a segment, then the point is on the perpendicular bisector of the segment

Theorem 4-6

if a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle

Theorem 4-7

if a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle

Theorem 4-8

opposite sides of a parallelogram are congruent

Theorem 5-1

opposite angles of a parallelogram are congruent

Theorem 5-2

diagonals of a parallelogram bisect each other

Theorem 5-3

if both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram

Theorem 5-4

if one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram

Theorem 5-5

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

Theorem 5-6

if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram

Theorem 5-7

Ways to prove that a quadrilateral is a parallelogram

show that BOTH pairs of opposite sides are parallel, show that BOTH pairs of opposite sides are congruent, show that ONE pair of opposite sides are both congruent and parallel,show that both pairs of opposite angles are congruent, show that the diagonals bisect each other

if two lines are parallel, then all points on one line are equidistant from the other line

Theorem 5-8

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

Theorem 5-9

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

Theorem 5-10

the segment that joins the midpoints of two sides of a triangle is parallel to the third side and is half as long as the third side

Theorem 5-11

Rectangle

quadrilateral with four right angles

Rhombus

quadrilateral with four congruent sides

Square

quadrilateral with four right angles and four congruent sides

the diagonals of a rectangle are congruent

Theorem 5-12

the diagonals of a rhombus are perpendicular

Theorem 5-13

each diagonal of a rhombus bisects two angles of the rhombus

Theorem 5-14

the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices

Theorem 5-15

if an angle of a parallelogram is a right angle, then the parallelogram is a rectangle

Theorem 5-16

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

Theorem 5-17

base angles of an isosceles trapezoid are congruent

Theorem 5-18

the median of a trapezoid is parallel to the bases and has a length equal to the average of the base lengths

Theorem 5-19

Exterior Angle Inequality Theorem

the measure of an exterior angle of a triangle is greater than the measure of either remote interior angle

if not p, then not q

Inverse

if not q, then not p

Contrapositive

if p, then q

Given statement

if q, then p

Converse

How to write an Indirect Proof

Assume temporarily that the conclusion is not true

How to write an Indirect Proof

reason logically until you reach a contradiction of a known fact

How to write an Indirect Proof

point out that the temporary assumption must be false, and that the conclusion must then be true

if one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle opposite the second side

Theorem 6-2

if one angle of a triangle is larger than a second angle, then the side opposite the first angle is longer than the side opposite the second angle

Theorem 6-3

the perpendicular segment from a point to a line is the shortest segment from the point to the line

Corollary 6-1

the perpendicular segment from a point to a plane is the shortest segment form the point to the plane

Corollary 6-2

the sum of the lengths of any two sides of a triangle is greater than the length of the third side

Triangle Inequality

SAS Inequality Theorem

if two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first trangle is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle

SSS Inequality Theorem

if two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second, then the included angle of the first triangle is larger than the included angle of the second

Ratio

quotient when the first number is divided by the second

Property of Proportions

a/b = c/d is equivalent to ad = bc

Property of Proportions

a/b = c/d is equivalent to a/c = b/d

Property of Proportions

a/b = c/d is equivalent to b/a = d/c

Property of Proportions

a/b = c/d is equivalent to a+b/b = c+d/d

Property of Proportions

if a/b = c/d = e/f =..., then a + c +e +.../b + d + f + ... = a/b = ...

AA ~

if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar

SAS Similarity Theorem

if an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar

SSS Similarity Theorem

if the sides of two triangles are in proportion, then the triangles are similar

Triangle Proportionality Theorem

if a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally

Triangle Angle-Bisector Theorem

if a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides

points all in one line

collinear points

points all in one plane

coplanar points

set of points that are in both figures

intersection

two lines that intersect to form right angles

Perpendicular lines

equation stating that two ratios are equal

proportion

corresponding angles are congruent and corresponding sides are in proportion

similar

ratio of similar polygons

scale factor

Corollary 7-1

if three parallel lines intersect two transversals, then they divide the transversals proportionally

auxiliary line

line (or ray or segment) added to a diagram to help in a proof

Transitive property

if a = b and b = c, then a = c

Substitution property

if a = b, then either a or b may be substituted for the other in any equation