Mrs.Turner's Geometry test 2

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What are the three basic assumptions that we can make in geometry?

-Given any two distinct points, there is exactly one line that contains them -To every pair of points A and B, there corresponds a real number dist (A,B) less than or equal to 0, called the distance from A to B. -We can also assume that every line has a coordinate system.

What are the two most basic types of instructions in Geometric constructions?

-Given any two points A and B, a straight edge can be used to draw the line AB or segment AB. -Given any two points C and B, use a compass to draw the circle that has its center at C that passes through B

The sum of 3 angle measures of any triangle is

180

Given two right triangles ABC and A'B'C' if AB= A'B', m<B= m<B' and m<C=m<C' the triangles are congruent

AAS Triangle Congruency

Given two right triangles ABC and A'B'C' if m<CAB=m<C'A'B' AB= A'B' and m<CBA= m<C'B'A' the triangles are congruent

ASA Triangle Congruency

When one angle of the triangle is a right angle the sum of the measures of the other angles is 90

Acute angles in a right triangle theorem

If two values are equal,you can add the same value to both sides and the resulting values will remain equal A=B then A+C=B+C

Addition Property of Equality

The sum of the measures of all angles formed by three or more rays with the same vertex and whose interiors do not overlap is 360

Adjacent angles at a point

Given a sequence of n consecutive adjacent angles whose interiors are all disjoint such that the angle formed by the first n-1 angles and the last angle are a linear pair then the sum of all the angle measures is 180

Adjacent angles on a line theorem

If the C is the interior of <ADB then <AOC + <COB= <AOB

Angle Addition Postulate

If 3 or more values are added, the way that you group them does not change the value (a+b)+c=a+(b+c)

Associative Property of Addition

If there are 3 or more values multiplied the way that they're grouped does not change the product (a*b)c=a(b*c)

Associative Property of Multiplication

Five reasons to justify a proof

Axioms/Postulates Theorems Properties Given Facts Definitions

a transition, rotation, or reflection of the plane

Basic rigid motion

if we know that two triangles are congruent, we can conclude that all corresponding angles and all corresponding sides are congruent.

CPCTC Postulate

Point of concurrency of the medians

Centroid

sum remains the same regardless of the order in which they appear a+b=b+a

Commutative Property of Addition

product of numbers remains the same regardless of the order in which they appear a*b=b*a

Commutative Property of Multiplication

What theroem is this? -If three parallel lines intersect two transversals then the segments intercepted on the transversal are proportional

Corollary to Theorem

If C is the incenter of <AOB, and m<AOC= <COB, then OC bisects <AOB and OC is called the bisector of <AOB

Definition of Angle Bisector

A point A is said to be _____________________ from two different points B and C if AB=AC. A point A is said to be _____________________ from a point B and a line l if the distance between A and l is equal to AB

Definition of Equidistant

The interior of <BAC is the set of points in the intersection of the half-plane of AC that contains B and the half-plane of AB that contains C.

Definition of Interior

A point B is called a midpoint of AC if B is between A and C, and AB=BC

Definition of Midpoint

when all sides of a polygon have equal length and all interior angles have equal measure

Definition of Regular polygon

a ray and that measures 0 degrees.

Definition of Zero angle

Let line t be a transversal to lines l and m such that t intersects t at point P and intersects m at point Q. Let R be a point on line l and S be a point on line m such that the points R and S lie in opposite half planes to t. Then <RPQ and <PQS are called alternate interior angles of the transversal t with respect to line m and l.

Definition of alternate interior angles

The union of two non-colinear rays with the same endpoint

Definition of an angle

Subdivide the length around a circle into 360 arcs of equal length. A central angle for any of these arcs is called a one-degree angle and is said to have angle measure 1 degree. AN angle that turns through n one-degree angle is said to have an angle measure of n degrees.

Definition of degree

the distance between the image and images of the two points is always equal to the distance between the preimage and the two points

Definition of distance preserving

a straight angle is a line and measures 180 degrees

Definition of straight angle

If a value is multiplied by a polynomial the multiplication is distributed to each term in the polynomial a(b+c)= ab+ac)

Distributive Property

If two values are equal you can divide the same value from both sides and resulting value will remain same A=B then A/C=B/C

Division Property of Equality

All angles in an equilateral triangle have equal measure

Equilateral triangle theorem

THe sum of each exterior angle of a triangle is the sum of the measures of the opposite interior angles, or the remote interior

Exterior angles of a triangle theorem

a set of instructions for drawing points, lines, circles, and figures in the plane

Geometric Construction

Given two right triangles ABC and A'B'C' with right angles B and B' if AB= A'B' and AC= A'C' the triangle is congruent

HL Triangle Congruency

You can add zero to any number and the value will not change a+0=a

Identity Property of Addition

You can multiply any number by 1 and its value remains the same a*1=a

Identity Property of Multiplication

How can you find the center of a circle if the center is not shown?

Inscribe a triangle into the circle and construct the perpendicular bisectors of at least two sides. Where the bisectors intersect is the center of the circle

Any value added to its additive inverse will equal zero a+(-a)

Inverse Property of Addition

If any value is multiplied by its multiplied by its multiplicative inverse (also called reciprocal) it=1 a*(1/a)

Inverse Property of Multiplication

Base angles of an isosceles triangle are equal in measure

Isosceles triangle theorem

2 <s that form a linear pair are supplementary

Linear Pairs Theorem

_________ connects to the vertex of a triangle to the midpoint of the opposite side

Median

If two values are equal you can multiply the same value from both sides and resulting value will remain same A=B then A*C=B*C

Multiplication Property of Equality

The point of concurrency of the lines that contain the altitudes of the triangle

Orthocenter

Through any given point there exists exactly one line parallel to a given line though a given point.

Parallel Lines through a given point theorem

What theorem is this? -If two or more parallel lines are cut by two transversals then they divide the transversal proportionally

Parallel proportions theorem

Any value is equal to itself a=a

Reflexive Property

Given two right triangles ABC and A'B'C' so that AB=A'B' m<A=m<A' and AC=A'C' the triangles are congruent

SAS Triangle Congruency

Given two right triangles ABC and A'B'C' if AB=A'B' AC=A'C' BC= B'C' the triangles are congruent

SSS Triangle Congruency

What are the five ways to prove two triangles congruent?

SSS,SAS,ASA,AAS,HL

If two lines are intersected my a transversal then exterior angles on the same side of the transversal are supplementary

Same side exterior angles theorem

If C lies on AB then AC+CB=AB

Segment Addition Postulate

What theorem is this?: -If a line is parallel to one side of the triangle and intersects the other two sides then it divides those sides proportionally

Side-Splitter Theorem

If two values are equal, they can be substituted for each other in an expression or equation A=D and C*A=D C*B=D

Substitution Property of Equality

If two values are equal you can subtract the same value from both sides and resulting value will remain same A=B then A-C=B-C

Subtraction Property of Equality

If 2 values are equal it doesn't matter which side of the equal they're on if a=b then b=a

Symmetric Property

Why are circles so important in constructions

The radius of equal-sized circles, which must be used in construction of an equilateral triangle, does not change. This consistent length guarantees that all three side lengths of the triangle are equal.

If two values are = to a 3rd value then they are = to each other A=B and B=C then A=C

Transitive Property

What theorem is this? 0-if a ray bisects an angle of a triangle then it divides the opposite side into two segments that are proportional to the other two sides of the triangle

Triangle Angle Bisector theorem

The sum of the three angle measures of any triangle is 180

Triangle Angle Sum Theorem

What is the center of gravity/balance in the triangle

centroid

When three r more lines intersect in a single point, they are ______________, and the point of intersection is the ______________________________________________________________

concurrent, point of concurrency

Let line t be a transversal to lines t and m. If <x and <y are alternate interior angles and <y and <z are vertical angles, then <x and <z are corresponding angles

corresponding angles

2 adjacent angles that form a line and one supplementary angle

definition of Linear Pair

If two rays with the same vertex are distinct and collinear, then the rays form a line called a ___________________________

definition of Straight Angle

Two angles <AOC and <COB with a common side OC are _________________

definition of adjacent angles

the image of any angle is again an angle and for any given angle, the angle measure of the image of that angle is equal to the angle measure of the pre-image of that angle

definition of angle preserving

Every triangle ABC determines three angles. Namely <BAC, <ABC, and <ACB

definition of angles of a triangle

something useful when solving unknown angles

definition of auxillary line

Given a point C in the plane and a number r>o, the circle with the center C and the radius r is the set of all points in the plane that are distance r from point C

definition of circle

the center of the circle that circumscribes that triangle

definition of circumcenter

Let <ABC be an interior angle of a triangle ABC and let D be a point on AB such that B is between A and D. Then <CBD is an exterior angle of the triangle ABC

definition of exterior angles of a triangle

A figure that has undergone the transformation

definition of image

the triangle is the center of the circle that is inscribes in the triangle

definition of incenter

A point lies in the _______________ if it lies in the interior of each of the angles of the triangle. In any triangle, the measure of the exterior angle is equal to the sum of the measures of the opposite interior angles These are also sometimes called ________________________________________

definition of interior of a triangle. remote interior angles

Transformations that preserve length of segment and measures of angles

definition of isometry

a triangle with at least two sides of equal length

definition of isosceles triangle

The line segment between points A and B is the set consisting of A,B, and all points on the line A and B

definition of line segment

Two lines are parallel if they lie in the same plane and do not intersect. Two segments or rays are parallel if the lines containing them are parallel

definition of parallel

Two lines are ____________________ if they intersect in one point and if any of the angles formed by the intersection of the lines is a 90 degree angle. Two segments are _____________________ if the lines containing them are ________________ lines

definition of perpendicular

A figure that is about to undergo a transformation

definition of preimage

A segment from the center of a circle to a point in the circle

definition of radius

A six-sided figure with congruent sides and angles. Each interior angle has a measure of 120°.

definition of regular hexagon

An angle is called a __________________ if it measures 90 degrees

definition of right angle

2 <'s whose sum is 180 degrees

definition of supplementary <s

Two angles are __________________ (or vertically oppisite angles) if their sides form two pairs of opposite rays

definition of vertical angles

A ___________ is an example of a transformation that preserves angle measures but not the length of segments

dilation

any point on the perpendicular bisector of a segment is ______________________ from the endpoints of the segment

equidistant

Opposite sides of a rectangle are congruent if it has ______________________________

four angles

name three transformations that are isometries

translation, reflection, rotation

Vertical angles formed by intersecting lines are congruent

vertical angles theorem

What are the four basic constructions and the axioms to justify these?

~1~ You can connect any two points with a straight edge $Through any Two Points there is exactly one line $ ~2~ You can extend any part of a line $Any point of a line can be extended indefinitely $ ~3~ Draw a circle from any given point of a line $From any given point, a circle can be drawn with a given Radius of R $ -4- Label intersection points $If two lines intersect they intersect at exactly one point$


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