Geometry - 1st Curriculum Exam
Converse of alternate exterior angles theorem:
If two lines in a plane are cut by a transversal and the alternate exterior angles are congruent, then the lines are parallel.
Converse of alternate interior angles theorem:
If two lines in a plane are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel.
Converse of corresponding angles theorem:
If two lines in a plane are cut by a transversal and the corresponding angles are congruent, then the lines are parallel.
Converse of same side interior angles theorem:
If two lines in a plane are cut by a transversal and the same side interior angles are supplementary, then the lines are parallel.
Perpendicular transversal theorem:
If two lines in a plane are perpendicular to the same line, they they are parallel to each other.
Flat plane postulate:
If two points are in a plane, then the line that contains these two points lies entirely in that plane.
Side-angle-side congruence theorem (SAS):
If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.
Isosceles triangle theorem:
If two sides of a triangle are congruent, then the angles opposite those sides are congruent. That is, the base angles of an isosceles triangle are congruent.
Hinge theorem:
If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second.
Converse of hinge theorem:
If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second.
Pythgorean Theorem
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
Location of centroid
Inside of all types of triangles
The point of concurrency of the angle bisectors of a triangle is the incenter.
The incenter is the center of the circle inscribed in a given triangle.
Geometric mean (altitude) theorem:
The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse.
Nonrigid transformation
a transformation that does not preserve the size, length, shape, lines, and angle measures of a figure -the pre-image and image are not congruent
Rigid transformation (isometry)
a transformation that preserves the size, length, shape, lines, and angle measures of a figure -the pre-image and image are congruent
Point existence postulate for planes:
A plane contains at least three noncollinear objects that are on the same plane are coplanar.
Adjecent angles
share a common vertex and a common side but do not overlap
If you are given sides a and b, side c is restricted to...
(a-b)<c<(a+b)
Meanings of the formula
- x1 and y1 are the coordinates of the endpoint at the start of the directed line segment - x2 and y2 are the coordinates of the endpoint at the start of the directed line segment - m:n is the ratio the segment needs to be divided into
A dilation, DQ,n, is a transformation of the plane where each point, P, of the plane is assigned a new point such that line length of QPprime = n * QP.
-Q is the center of dilation -n is the scale factor ◘A dilation moves each point, other than the center of dilation, a specific distance based on the value of n.
On a coordinate plane, you can represent a translation by a function that takes an input of (x,y) and maps to an output of (x+a,y+b).
-a is the number of units translated horizontally -b is the number of units translated vertically -the function rule is Ta,b(x,y) -the rule mapping is (x,y)→(x+a,y+b) -use the opposite operation to find the pre-image
Isosceles triangle
-a triangle with at least two sides that are congruent -has two base angles and one vertex angle
Parallel and perpendicular lines:
-all horizontal lines are parallel to each other -all vertical lines are parallel to each other -horizontal and vertical lines are perpendicular to each other
Reflections
-all points of the pre-image are reflected across a line of reflection to produce the image -the line of reflection is the perpendicular bisector of the line segment connecting a point and its corresponding image -any pre-image point that lines on the line of reflection is mapped to itself
Translations preserve:
-angle measures -length -shape -size -orientation
Rotations preserve:
-angle measures of figures -side length of figures 360 degrees-to itself
Rotations require:
-center of rotation -angle of rotation
A similarity transformation is one that preserves angle measures.
-dilations -rigid transformations compositions of dilations and rigid transformations
Reflections do not preserve:
-orientation
Transformation Notation
-prime notation is used to indicate that a figure has undergone a transformation -corresponding pre-image and image points are listed in the same order -a transformation maps a figure onto its image -an arrow is used to indicate a mapping
Reflectional symmetry
-reflectional symmetry of a geometric figure is the characteristic of a figure to map onto itself across a line of symmetry -the number of lines of reflectional symmetry will vary for each geometrical figure -the figure is said to have n-fold reflectional symmetry (n is the number of lines of symmetry)
Reflections preserve:
-segment lengths -angle measure
Rotational symmetry
-the rotational symmetry of a geometric figure is the characteristic of a figure to map onto itself through the center of rotation -the number of rotations that map a geometric figure onto itself is called the order of rotational symmetry
Why just two angles instead of three?
-triangle angle sum theorem -third angle theorem
Horizontal lines have a slope of...
0/zero
Dilation not centered at origin:
1. Draw new coordinate axes centered at G 2. Find the coordinates of the pre-image in the new coordinate system 3. Apply the mapping rule (x,y)→(nx,ny) 4. Plot the image 5. Read coordinate from the original coordinate axes
Dilation with center at origin:
1. Vertices: Do,n (x,y)→(nx,ny) 2. Segment lengths are proportional to n 3. Segments are parallel
Every rotation included which degrees?
360 degrees
The smallest angle of rotational symmetry =
360 degrees/n
Midsegment
A midsegment of a triangle connects the midpoints of two sides of a triangle.
Compositions
A composition of transformations is a combination of two or more transformations performed in a sequence.
Point existence postulate for lines:
A line contains at least two points (points on the same line are collinear).
The circumcenter location in each triangle-
Acute - inside Obtuse - outside Right - on
The location of the orthocenter depends on the type of triangle.
Acute - inside Obtuse - outside Right - on
Centroid ratio theorem:
Along each median in a triangle, the distance between the vertex and the centroid is twice the distance between the centroid and the side opposite the vertex.
Segment addition postulate:
If L is between K and M, then KL+LM=KM. If KL+LM=KM, then L is between K and M.
The scale factor, n, will determine whether the dilation is a reduction or an enlargement.
Congruent figure- n=1 Reduction- 0<n<1 Enlargement- n>1
Parallel line transitivity theorem:
If two line are parallel to a third line, then they are parallel to each other.
Function rule for dilation
Do,n (x,y)→(nx,ny)
Parallel postulate:
Given a line and a point not on the line, there is exactly one line in the same plane through the given point and is parallel to the given line.
Protractor postulate:
Given any angle, we can express its measure as a unique positive number from 0 to 180 degrees.
Midpoint theorem:
If P is the midpoint of TR, then PT is congruent to PR. Also, PT = PR = 1/2TR -have equal lengths -are congruent halves of a whole -each smaller segment of a bisected segment is half the length of the longer segment
Converse of the side-splitter theorem:
If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
Side splitter theorem:
If a line is parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.
Converse of the perpendicular bisector theorem:
If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
Alternate exterior angles theorem:
If a transversal intersects two parallel lines, then the resulting alternate exterior angles are congruent.
Alternate interior angles theorem:
If a transversal intersects two parallel lines, then the resulting alternate interior angles are congruent.
Corresponding angles theorem:
If a transversal intersects two parallel lines, then the resulting corresponding angles are congruent.
Same side interior angles theorem:
If a transversal intersects two parallel lines, then the resulting same side interior angles are supplementary.
Corollary of the converse of the isosceles triangle theorem:
If a triangle is equiangular, then it is equilateral.
Corollary of the isosceles triangle theorem:
If a triangle is equilateral, then it is equiangular.
Transitive property:
If a=b and b=c, then a=c
Side-angle-side similarity theorem:
If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.
Converse of triangle relationship theorem:
If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.
Triangle parts relationship theorem:
If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the short side.
Right triangle altitude theorem:
If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.
Side-side-side similarity theorem:
If the corresponding sides of two triangles are proportional, then the triangles are similar.
Hypotenuse-leg theorem (HL):
If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
Side-side-side congruence theorem (SSS):
If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
Angle-angle-side congruence theorem (AAS):
If two angles and a nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of another triangle, then the triangles are congruent.
Angle-side-angle congruence theorem (ASA):
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.
Congruent complements theorem:
If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent.
Congruent supplements theorem:
If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent.
Vertical angles theorem:
If two angles are vertical angles, then they are congruent angles.
Linear pair postulate:
If two angles form a linear pair, then they are supplementary.
Converse of the isosceles triangle theorem:
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
AA similarity theorem:
If two angles of one triangle are congruent to two angles of another, then the triangles are similar
Two figures are congruent if and only if one can be mapped onto the other one by one or more rigid transformation.
NO DILATION!!
Finding n
OA'/OA = OB'/OB = A'B'/AB
Points meaning
Point by itself is x1 and y1 Point with the arrow is x2 and y2
Dilation
Preserves: -angle measures -parallelism -collinearity of points -orientation of segments Creates: -proportional side lengths
Function rules for rotation
R(o,90 degrees)(x,y)→(-y,x) ~ -270 degrees R(o,180 degrees)(x,y)→(-x,-y) R(o,270 degrees)(x,y)→(y,-x) ~ -90 degrees
Basic function rule
R(o,degrees)(x,y)
Ratio vs Fraction
Ratio is part to part so you use (m/m+n) for it Fraction is part to while so leave as is
Triangles satisfying HL might be described as satisfying SSA, with the equal angle being a right triangle.
SSA is not a valid criteria for identifying congruent triangles when the angle is not a right triangle.
Ruler postulate:
The distance between any two points can be measured by finding the absolute value of the difference of the coordinates representing in the points.
Geometric mean (leg) theorem:
The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of a leg of this triangle is the geometric mean between the length of the hypotenuse and the segment of the hypotenuse adjacent to that leg.
Vertex angle of an isosceles triangle theorem:
The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.
The point of concurrency of the perpendicular bisectors of a triangle is the circumcenter.
The circumcenter is the center of the circle circumscribed around a given triangle.
Exterior angle inequality theorem:
The measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles.
Exterior angles theorem:
The measure of each exterior angle of a triangle equals the sum of the measures of the remote interior angles.
Angle addition postulate:
The measure of the larger angle is the sum of the measures of the two smaller ones.
Triangle midsegment theorem:
The midsegment of two sides of a triangle is parallel to the third side and is half as long.
Perpendicular bisector theorem:
The points on the perpendicular bisector of a segment are equidistant from the endpoints.
Shortest distance thoerem:
The shortest distance from a given line to a point not on the line is the length of the segment perpendicular to the line through the point.
Perpindicular lines
lines that intersect to form right angles
Triangle inequality theorem:
The sum of the lengths of any two sides of a triangle is greater than the length of the third side. - c<(a+b) ~ c-a<b or c-b<a
Triangle angle sum theorem:
The sum of the three angles of a triangle is 180 degrees.
Unique plane postulate:
Through any three noncollinear points there exists one and only one plane.
If two figures are congruent, then their corresponding parts are congruent.
Triangles - corresponding parts of congruent triangles are congruent (CPCTC) Figures - corresponding parts of congruent figures are congruent (CPCFC)
Compositions writing
You read from left to right
Point is represented by...
a capital letter
Angle
a figure formed by two rays with a common endpoint called the vertex
Directed line segment
a line segment that has direction and two endpoints
Transformation
a one-to-one mapping of all points of on a figure to another figure in a plane -the original figure, or input, is called the pre-image -the result of the transformation, or output, is the image
Linear pairs
a pair of adjacent angles whose non-common sides are opposite rays -add up to 180 degrees -can be formed by intersecting lines
Vertical angles
a pair of opposite congruent angles formed by intersecting lines
Line segment
a piece of a line with two endpoints and a specific length
An equilateral triangle is...
a special case pf an isosceles triangle
The geometric mean of two positive numbers, a and b, is the positive number x that satisfies...
a/x = x/b
Reflexive property:
a=a
Equilateral triangles must be...
acute - a<90 - isosceles at least 2 equal sides
Isosceles triangle can be...
acute, right, or obtuse
Circle
all coplanar points equidistant from a given point
Incenter in triangles
always inside
Reflex angle
an angle greater than 180 degrees but less than 360 180 degrees < x < 360 degrees
Obtuse angles
an angle whose measure is greater than 90 degrees and is less than 180 degrees 90 degrees < x < 180 degrees
When parallel lines are cut by a transversal...
angle pairs have special relationships
Right angles
angles that measure 90 degrees x = 90 degrees
Acute angles
angles that measures greater than 0 degrees and less than 90 degrees 0 degrees < x < 90 degrees
Rigid transformations in the coordinate plane...
can be described using mapping rules
A plane is represented by...
capital, italicized letter
Theorem
conditional statement containing a hypothesis and conclusion
Distance formula
d = √[( x₂ - x₁)² + (y₂ - y₁)²]
Slopes of parallel lines are...
equal
Exterior angles are created by...
extending only one side of the triangle and should be supplementary to the interior angle
Line is represented by...
lower case letter or 2 capital letter with line symbol on top
Symmetric property:
if a=b, then b=a
Third angle theorem:
if two angles of one triangle are congruent to two angles of another triangle, then the third angles of the triangles are also congruent.
Substitution property:
if x=3 and x+y=12, then 3+y=12
Parallel lines
lines in the same plane that never intersect
Skew lines
lines that are noncoplanar and do not intersect
Postulate
mathematical statement taken as true but not proven (also called axiom)
Straight angles
measures exactly 180 degrees and forms a straight line x = 180 degrees
Translations
moves every point of the pre-image of an object the same distance and direction to create the image
Rotations
moves every point of the pre-image through an angle of rotation about the center of rotation to create an image ~ counterclockwise-positive ~ clockwise-negative
Regular polygons with n sides will have...
n-fold reflectional symmetry
Slopes of perependicular lines are...
negative reciprocals
Point
no dimension, location on coordinate plane designated by an ordered pair (x,y)
For regular polygons, the order of rotational symmetry =
number of equal sides (n)
Line
one-dimensional set of infinite points, has no beginning or end
Ray
part of a line with one endpoint that continues without end in one direction
How to write a/x = x/b
segment 1/altitude - altitude/segment 2
The concurrency of angle bisectors of a triangle theorem states that...
the angle bisectors of a triangle intersect at a point equidistant from the side of the triangle.
The concurrency of perpendicular bisectors of a triangle theorem states that...
the perpendicular bisectors of a triangle intersect at a point equidistant from the vertices of the triangle.
Triangle consist of...
three interior angles and three exterior angles
Supplementary angles
two angles whose measures have a sum of 180 degrees -don't need to be a linear pair
Complementary angles
two angles whose measures have a sum of 90 degrees -don't have to be adjacent
Plane
two-dimensional set of all points, flat or level surface, has no beginning or end
Vertical lines have an _______ slope.
undefined
Midpoint:
use 1/2 ad (m/m+n) so 1:1
To find pre-image:
use reciprocal of scale factor
Orthocenter
where the altitudes of a triangle intersect
Centroid
where the medians of a triangle intersect
A point of concurrency is...
where three or more lines intersect
How to write it
x+y/a = a/y and x+y/b = b/y hyp./leg = leg/adj. seg. of hyp.
Rules for reflection:
x-axis : rx-axis(x,y)→(x-y) y-axis : ry-axis(x,y)→(-x,y) y=x : ry=x(x,y)→(y,x) y=-x : ry=-x(x,y)→(-y,-x)
The section formula can be used to find the coordinates of the point that partitions a directed line segment into a given ratio.
x=(m/m+n)(x2-x1)+x1 y=(m/m+n)(y2-y1)+y1
Criteria for lines of symmetry:
•When crossing from an angle to an angle: -it is an angle bisector of the vertex •When crossing a vertex to a side: -it is an angle bisector of the vertex -it is a perpendicular bisector of the side of the figure •When crossing from a side to a side: -it is a perpendicular bisector of both sides
