FULL GEOMETRY
Definition : A definition is the precise statement of the qualities of an idea, object or processexample. A donuts is a bagel with sugar on it usually and sometimes substances like jelly inside of it.
Postulates : Definition: A postulate or axiom is a statement that is assumed to be true without proofExample: Water is wetExample 2: A line segement can be drawn between any two points.
Evidence is : DefinitionspostulatesCommon notions
Indirect proof : A indirect proof is a proof when you assume the opposite is true. paragraph styleopposite is assumed truecontradiction is found.
Incenter : The incenter of a polygon is the center of the inscribed circle.
Circumcenter : The circumcenter of a polygon is the center of the circumscribed circle.
intercepted arcs : Parts of the circle (an arc) that are cut off from the rest of the circle's circumference by lines or segments intersecting the circle.
circle : A geometric figure consisting of all the points on a plane that are the same distance from a single point, called its center.
chord : Any line segment whose endpoints are on the circle.
How do you find the distance between the center point and a chord? : The distance is the measure of the perpendicular bisector between the center point and the chord.
diameter : A line segment that contains the center of the circle and has endpoints on the circle. This term also refers to the length of this line segment; the ______________of a circle is twice the radius.
arc : A part of the circumference of a circle.
measure of circle :
How could you show that two arcs are congruent? : Two arcs are congruent if they have the same length and belong to the same circle or two congruent circles.
inscribed angle : An angle formed by two chords of a circle that share an endpoint. It is not a central angle
intercepted arc : A part of the circle (an arc) that is cut off from the rest of the circle's circumference by lines or segments intersecting the circle.
vertical angles : A pair of opposite angles formed by intersecting lines. Vertical angles have equal measures.
What is the measure of angles formed by intersecting chords? : The measure of an angle formed by intersecting chords is half the sum of the measures of the intercepted arcs.
point of tangency : The point at which a tangent line meets a curve. In a circle, the radius ending at the point of tangency is always perpendicular to the tangent line..
What is the relationship between the radius and the tangent. : the radius is always perpendicular to the point of tangency.
Pi :
Radian : A unit of angular measure determined by the condition:The central angle of one radian in a circle of radius 1 produces an arc of length 1.
Area : the space taken up by a two dimensional figure or surface. area is measured in square units.
Sector : A part of the interior of a circle bounded by an arc and two radii that share the arcs endpoints.
the sector : The sector of a circle is bounded by a central angle of the circle and the intercepting arc.
how are the areas of a sector of a circle and the circle related? : the area of a sector is a fraction of the area of the circle.
what is the relationship between the circle and the vertices of an inscribed triangle? : The vertices of the triangle always touch the circle. many triangles can be inscribed in any one circle.
Inscribed triangle :
_____________ Triangles can be inscribed in a given circle. : several
Incenter : the center of the circle that can be inscribed in a given triangle.
Incenter : it has angle bisectors, the lengths of the angle bisectors are the same.
Acute circumcenter : Inside
convex : Having no indentations.
concave : Having one or more indentations.
How do you find the sum of the measures of exterior angels of a polygon? : the sum of the measures of the exterior angles of a polygon always equals 360 degrees
measure of each exterior angle = : 360 ÷ n
what is special about consecutive angels in a parallelogram? : the measure of consecutive angels always adds up to 180 degrees
what is special about diagonals in a parallelogram? : the diagonals intersect at the midpoint of each diagonal. they bisect each other.
When you make the Diagonals bisect each other, the quadrilateral becomes a : parallelogram. DIAGONALS
rectangle : A quadrilateral with four right angles. Rectangles are parallelograms; opposite sides are parallel and congruent.
Properties specific to rectangles : All angles are right anglesDiagonals are congruent
All properties of a parallelogram : Opposite sides are parallelOpposite sides are congruentOpposite angles are congruentConsecutive angles are supplementaryThe diagonals bisect each other.
square : A quadrilateral with four right angles and four congruent sides. Squares have all of the properties of parallelograms, rectangles, and rhombi.
Rhombus Properties : All sides are congruentDiagonals are perpendiculardiagonals bisect opposite angles
what is special about an isosceles trapezoids legs? : the legs are congruent.
isosceles trapezoid : A trapezoid with two congruent legs. The base angles of an isosceles trapezoid are also congruent.
convex : Having no indentations.
concave : Having one or more indentations.
How do you find the sum of the measures of exterior angels of a polygon? : the sum of the measures of the exterior angles of a polygon always equals 360 degrees
measure of each exterior angle = : 360 ÷ n
what is special about consecutive angels in a parallelogram? : the measure of consecutive angels always adds up to 180 degrees
what is special about diagonals in a parallelogram? : the diagonals intersect at the midpoint of each diagonal. they bisect each other.
When you make the Diagonals bisect each other, the quadrilateral becomes a : parallelogram. DIAGONALS
rectangle : A quadrilateral with four right angles. Rectangles are parallelograms; opposite sides are parallel and congruent.
Properties specific to rectangles : All angles are right anglesDiagonals are congruent
All properties of a parallelogram : Opposite sides are parallelOpposite sides are congruentOpposite angles are congruentConsecutive angles are supplementaryThe diagonals bisect each other.
square : A quadrilateral with four right angles and four congruent sides. Squares have all of the properties of parallelograms, rectangles, and rhombi.
Rhombus Properties : All sides are congruentDiagonals are perpendiculardiagonals bisect opposite angles
How do you get an estimate of the circumference of a circle from the radius? : When given the radius of a circle, multiply that number by 6 to get a rough idea of its circumference.
chord : Any line segment whose endpoints are on the circle.
If a chord is bisected by a radius, is the radius perpendicular to the chord? : Yes
diameter : A line segment that contains the center of the circle and has endpoints on the circle. This term also refers to the length of this line segment; the ______________of a circle is twice the radius.
semicircle: : A 180° arc; half of a circle.
measure of circle :
When are two chords congruent? : two chords are congruent if and only if their associated central angles are congruent.
inscribed angle : An angle formed by two chords of a circle that share an endpoint. It is not a central angle
How are the angles formed by the chords and the intercepted arcs related? : Intersecting chords form a pair of congruent vertical angles. Each angle measure is half the sum of the intercepted arcs.
vertical angles : A pair of opposite angles formed by intersecting lines. Vertical angles have equal measures.
concave : Having one or more indentations.
regular polygons : Convex polygons in which all sides and angles are congruent.
measure of each exterior angle = : 360 ÷ n
parallelogram : A quadrilateral in which both pairs of opposite sides are parallel.
what is special about diagonals in a parallelogram? : the diagonals intersect at the midpoint of each diagonal. they bisect each other.
What is special about the lengths of parallel sides in a parallelogram? : they are congruent
rectangle : A quadrilateral with four right angles. Rectangles are parallelograms; opposite sides are parallel and congruent.
in 10 words or fewer, what is special about the angels of a rectangle : A rectangle has four right angels
All properties of a parallelogram : Opposite sides are parallelOpposite sides are congruentOpposite angles are congruentConsecutive angles are supplementaryThe diagonals bisect each other.
polygon : A closed plane figure.
diagonal : A line segment that connects two nonconsecutive vertices of a polygon..
How do you find the sum of interior angels in a polygon? : Sum=(N-2) 180Degrees
parallel : Lying in the same plane without intersecting. Two or more lines are ______________if they lie in the same plane and do not intersect.
What is special about the lengths of the sides of a parallelogram? : the lengths of the opposite sides are equal.
When you make both pairs of opposite angels congruent, does the quadrilateral become a parallelogram? : yes it makes a parallelogram
If the opposite angles of a quadrilateral are congruent, then the ___________________ : quadrilateral is a parallelogram.
convex : Having no indentations.
concave : Having one or more indentations.
How do you find the sum of the measures of exterior angels of a polygon? : the sum of the measures of the exterior angles of a polygon always equals 360 degrees
measure of each exterior angle = : 360 ÷ n
what is special about consecutive angels in a parallelogram? : the measure of consecutive angels always adds up to 180 degrees
what is special about diagonals in a parallelogram? : the diagonals intersect at the midpoint of each diagonal. they bisect each other.
regular polygons : Convex polygons in which all sides and angles are congruent.
diagonal : A line segment that connects two nonconsecutive vertices of a polygon..
Which law of cosines formula do you use when you know a side an angle and a side? : c= √ a² + b² -2abcos(c)
SSS step 1. : SSS: Find the cosine of the missing angel by using cos (c) = a² + b² -c²_______________________2ab
SAS Step 3. : SAS: Plug all given information into the formula c= √ a² + b² -2abcos(c)(SOLVE IN CHUNKS THEN PLUG IN)
Law of sines :
isosceles triangle theorem : A theorem stating that if two sides of a triangle are congruent, then the angles opposite those sides are congruent.
What is the ratio of the hypotenuse of a 45-45-90 triangle? : the hypotenuse of a 45-45-90 triangle will always be x√2
hypotenuse: : The side across from the right angle in a right triangle. It is the triangle's longest side.
HL congruence theorem : A theorem stating that if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the right triangles are congruent
HA and LA (leg not included) are really just : AAS
LA (leg included) Is really just : ASA
leg : Either of the two shorter sides of a right triangle.
hypotenuse : The side across from the right angle in a right triangle. It is the triangle's longest side.
incenter : The center of the circle that can be inscribed in a given triangle. the point where the 3 angle bisectors intersect.
perpendicular bisector: : A line, ray, or line segment that bisects a line segment at a right angle.
altitude of a triangle : The line segment from a vertex of a triangle that is perpendicular to the opposite side or to the line containing the opposite side.
centroid : The point at which the three medians of a triangle intersect. The centroid of any triangle is inside the triangle..
Longest Side and Largest Angle Theorem : The longest side of a triangle is always opposite the angle with the largest measure.
Shortest Side and Smallest Angle Theorem : the shortest side of a triangle is always opposite the angle with the smallest measure.
SAS similarity theorem: : A theorem stating that if an angle of one triangle is congruent to an angle of another triangle, and if the lengths of the sides including these angles are proportional, then the triangles are similar.
postulates : Statements that are assumed to be true without proof. They are also called axioms.
AAS theorem : A theorem stating that if two angles and a nonincluded side of one triangle are congruent to the corresponding two angles and side of another, then the triangles are congruent.
Does the theorem AAA work? : No
congruence transformation : An action that can be performed on a geometric object without changing its size or shape.
CPCTC : An abbreviation that stands for "corresponding parts of congruent triangles are congruent." If two triangles are congruent, each side or angle of one triangle is congruent to the corresponding side or angle of the other triangle.,
Rigid Transformation : a transformation that does not change the shapes angles and lengths
Non Rigid transformations : Transformations that change lengths, and angles.
triangle : A polygon with three sides.
sides : The three line segments that form a triangle
right triangle : A triangle that contains a right angle.
obtuse triangle : A triangle that has one angle measuring more than 90°.
Exterior angle = ________________ :
similar : Having exactly the same shape. corresponding angles are congruent and corresponding sides are proportional in length.
proportional : Having equal ratios..
ratio : A comparison that shows the relative size of one quantity with respect to another. The ratio a to b is often written with a colon (a:b) or as a fraction ().
parallel lines : Lines lying in the same plane without intersecting. Two or more lines are _______ if they lie in the same plane and do not intersect.
What does the symbol II mean? : parallel
corresponding angles theorem : All pairs of corresponding angels are congruent.
alternate interior angles : Two angles formed by a line (called a transversal) that intersects two parallel lines. The angles are on opposite sides of the transversal and inside the parallel lines.
Vertical angels : Angels that share a vertex and are congruent.
angle of incidence : The angle between a ray of light meeting a surface and the line perpendicular to the surface at the point of contact.
linear pairs : Pairs of adjacent angles whose measures add up to 180°. _________________ of angles are supplementary.
congruent : Having the same size and shape.
collinear : Lying in a straight line. Two points are always ___________. Three or more points are ____________if a straight line can be drawn through all of them. Linear sounds like line
coplanar : Lying in the same plane. Four or more points are _____________ if there is a plane that contains all of them.
two-dimensional : Having length and width but no height.,
zero-dimensional : Having no length, width, or height. A point
What is the flat plane postulate? : If you are given two points in a plane, then the line that goes through both points is also in that same plane. In other words, a plane is flat.
Three points are collinear if they : fall on the same line.
Line segment : A part of a line with endpoints at both ends. The symbol AB means "the_________________ with endpoints A and B."
End point : The point at the very end of a ray or line.
Line : The set of all points in a plane that are equidistant from two points.
A____ Has no length, width or height : point
zero angle : An angle that has a measure of zero degrees and whose sides overlap to form a ray.
straight angle : An angle whose sides form a line.
adjacent angles : Angles that share a vertex and one side.
supplementary angles : Having angle measures that add up to 180°. If two ____________ angles are adjacent, they form a straight angle.
Perpendicular lines form _______ angles. : right
Hashmarks are? : lines that point out whither things are congruent.
QED : quod Erat demonstrandum
Statements are : Given 1stDeduction or conclusion
Theorem : a statement that has already been proven to be true.has been provengenerally accepted in the world of math
Corollaries : A corollary is a statement that naturally follows, or makes sense, based on something you have already proven.Has been provendirectly ties to a previously proven statementgenerally accepted in the world of math.
conditional statement : A statement that has the form "If A, then B," where A is what you assume is true and B is the conclusion.
contrapositive : A statement in the form "If not B, then not A," given the statement "If A, then B."
intersect : To cross over
plane : A flat surface that extends forever in all directions. A _________ has no thickness, so it has only two dimensions.
What symbols means parallel in diagrams? : >> Or II
transversal : A line, ray, or segment that intersects two or more coplanar lines, rays, or segments at different points.
consecutive interior angles : Two angles formed by a line (called a transversal) that intersects two parallel lines. The angles are on the same side of the transversal and are inside the parallel lines.
Consecutive interior angles postulate : If two parallel lines are cut by a transversal, the _____________ angles are supplementary.
law of reflection : A law stating that the angle of incidence is congruent to the angle of reflection.
vertical angles: : A pair of opposite angles formed by intersecting lines. Vertical angles have equal measures
perpendicular bisector : A line, ray, or line segment that bisects a line segment at a right angle.
vertical angles made by perpendicular lines are ___________ : Congruent.
Midpoint : The point in the very center of a line segment. They separate a line segment into two equal line segments.
Segment Addition Postulate : Point C is between points A and B if AC + CB = AB
Point C is between points A and B if AC+CB _______ AB : =
What are the three ways to name an angle : By vertex and two other points. by vertex. and by a number
right angle : An angle that measures 90°. ________ angles are often marked with a small square symbol.
acute angle : An angle that measures less than 90°. An _____ angle is smaller than a right angle.
linear pair : A pair of adjacent angles whose measures add up to 180°. _________ pairs of angles are supplementary.
complementary angles : Having angle measures that add up to 90°. If two ____________ are adjacent, they form a right angle.
collinear : Lying in a straight line. Two points are always ___________. Three or more points are ____________if a straight line can be drawn through all of them. Linear sounds like line
coplanar : Lying in the same plane. Four or more points are _____________ if there is a plane that contains all of them.
two-dimensional : Having length and width but no height.,
zero-dimensional : Having no length, width, or height. A point
What is the flat plane postulate? : If you are given two points in a plane, then the line that goes through both points is also in that same plane. In other words, a plane is flat.
Three points are collinear if they : fall on the same line.
