Mrs.Turner's Geometry test 2
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 $If two lines intersect they intersect at exactly one point$