Carnegie Learning: Honors Geometry: Chap 1-7 - Vocabulary
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New figure from the translation
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Original figure
proof
a logical series of statements and corresponding reasons that starts with a hypothesis and arrives at a conclusion
ray
a part of a line that begins with a single point and extends infinitely in one direction (2 Capital letters) ↦
line segment
a part of a line that includes two points all all collinear points between the two points ---------
paragraph proof
a proof in which the steps and corresponding reasons are written in complete sentences
flow chart proof
a proof in which the steps and reasons for each step are written in boxes. Arrows connect the boxes and indicate how each step is generated from one or more other steps and reasons
two-column proof
a proof in which the steps are written in the left column and the corresponding reasons are written in the right column. Each step and corresponding reason are numbered
construction proof
a proof that results from creating an object with specific properties using only a compass and a straightedge
right rectangular prism
a rectangle translated through space in a direction perpendicular to the plane containing the rectangle
translation
a rigid motion that "slides" each point of a figure the same distance and direction
straight edge
a ruler with no numbers
counterexample
a specific example that shows that a general statement is not true
theorem
a statement that can be proven
conditional statement
a statement that can be written in the form "If p, then q."
postulate
a statement that is accepted without a proof
Euclidean geometry
a system through postulates and logic to understand properties, equations, and reasons for mathematical purposes
truth table
a table that summaries all possible truth values for a conditional statement p→q
indirect measurement
a technique that uses proportions to determine a measurement when direct measurement is not possible EX: proportion to find height of flagpole
compass
a tool used to create arcs and circles
rigid motion
a transformation of points in space
right triangular prism
a triangle translated through space in a direction perpendicular to the plane containing the triangle
included angle
an angle formed by two consecutive sides of a figure
similar trianlger
are triangle that have all pair of corresponding angles congruent and all corresponding sides are proportional; have same shape, but not always same size
plane
described as a flat surface, infinite length and width but no depth and extends infinitely in all directions (1 italic letter)
point
described as a location, no size or shape (1 Capital letter) -
line
described as a straight, continuous arrangement of an infinite number of points, infinite length but no width (1 lowercase letter) ↔
altitude
is a line segment drawn from a vertex of a triangle perpendicular to the line containing the opposite side
median
is a line segment of a triangle drawn from a vertex to the midpoint of the opposite side
hemisphere
is half of a sphere bounded by a great circle
concurrent
lines, rays, or line segments are three or more lines, rays, or line segments intersecting at a single point
centriod
of a triangle is the point at which the medians of a triangle intersect
geometric mean
of two positive numbers a and b is the positive number x such that a/x=x/b
isometric paper
often used by artists and engineers to create 3D views of objects in 2D
construct
only a compass or straightedge
perpendicular
opposite reciprocal slopes, intersect to form right angles (⟂)
collinear points
points that are on the same straight line
endpoint of a line segment
points where the line segment begin and end
deduction
reasoning that uses a general rule to make a conclusion
induction
reasoning that uses specific examples to make a conclusion
parallel
same slope, never intersect (∥)
arc
the curve between two points on a circle (Named using the two endpoints)
legs of a trapeziod
the lines in a trapezoid connecting the parallel sides
transformation
the mapping, or movement, of all the points of a figure in a plane according to a common operation
propositional variables
the p and q values in propositional form
orthocenter
the point at which the altitudes of the triangle intersect
incenter
the point at which the angle bisectors of the triangle intersect
circumcenter
the point of a triangle at which the prependicular bisectors intersect
midpoint
the point that divides the line segment into two congruent segments
point of Concurentcy
the point where concurrent lines, rays, or line segments intersect
annulus
the region bounded by two concentric circles
sphere
the set of all points in space that are a given distance from a fixed point called the center of the sphere
disc
the set of all points on a circle and in the interior of a circle
bases of a trapeziod
the set of parallel sides
endpoint of a ray
the single point where the ray begins
supplementary angles
the sum of their angle measures is equal to 180 degrees
complementary angles
the sum of their angle measures is equal to 90 degrees
remote interior angles of a trianlge
the two angles that are not adjacent to the specified exterior angles
hypothesis
the variable p
conclusion
the variable q
converse
to interchange the hypothesis and conclusion "If q, then p"
protactor
tool used for measuring angles
linear pair
two angles that have noncommon sides that form a line; adjacent, supplementary
adjacent angles
two angles that share a common vertex and share a common side
vertical angles
two nonadjacent angles that are formed by two intersecting lines
congruent line segments
two or more lines of equal measure (≅)
coplanar lines
two or more lines that are located in the same plane
skew lines
two or more lines that do not intersect and are not parallel, do not lie on the same plane
undefined terms
we can only describe and create mathematical models to represent them
oblique cylinder
when a circle is translated through space in a direction that is not perpendicular to the plane containing the circle
oblique rectangular prism
when a rectangle is translated through space in a direction that is not perpendicular to the plane containing the rectangle
oblique triangular prism
when a triangle is translated through space in a direction that is not perpendicular to the plane containing the triangle
truth value
whether the statement is true or false, if it could be true then it is considered true, true or false not both
draw
with the use of a ruler, straightedge, compass, or protractor
sketch
without the use of tools
Angle Bisector/Proportional Side Theorem
"A bisector of an angle in a triangle divides the opposite side into two segments whose lengths are in the same ratio as the lengths of the sides adjacent to the angle." ... (AB/AC=BD/CD)
Right Angle Congruence Theorem
"All right angles are congruent"
Substitution Property
"If a and b are real numbers and a = b, then a can be substituted for b."
Reflexive Property
"If a is a real number, then a = a."
Converse of the Triangle Proportionality Theorem
"If a line divides two sides of a triangle proportionally to the third side, then it is parallel to the third side."
Triangle Proportionality Theorem
"If a line parallel to one side on a triangle intersects the other two sides, then it divides the two sides proportionally."
Transitive Property
"If a, b and c are real numbers, a = b, and b = c then a = c."
Addition Property of Equality
"If a, b, and c are real numbers and a = b, then a + c = b + c."
Subtraction Property of Equality
"If a, b, and c are real numbers and a = b, then a - c = b - c."
Cavalieri's Principle
"If all 1D slices of 2D figures have the same lengths, the the 2D figures have the same area." "given two solid figures included between parallel planes, if every plane had the same area in both solids, then the volumes of the solids are equal."
Side-Side-Side Similarity Theorem
"If all three sides corresponding sides of two triangles are proportional, then the triangles are similar."
Right Triangle/Altitude Similarity Theorem
"If an 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."
propositional form
"If p, then q." or p → q (read as "p implies q")
Segment Addition Postulate
"If point B is on the line AC and between points A and C, then AB+BC=AC
Angle Addition Postulate
"If point D lies in the interior of ∠ABC, then m∠ABD + m∠DBC = m∠ABC."
Right Triangle Altitude/Leg Theorem
"If the altitude is drawn to the hypotenuse of a right triangle, each leg of the right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg."'
Proportional Segments Theorem
"If three parallel lines intersect two transversals, then they divide the transversals proportionally."
Side-Side-Side Congruence Theorem
"If three side of one triangle are congruent to the corresponding sides of another triangle, the the triangles are congruent."
Angle-Angle-Side Congruence Theorem
"If two angles and a non-included side of one triangle are congruent to the corresponding two angles and the corresponding non-included side of a second triangle, then the triangles are congruent."
Angle-Side-Angle Congruence Theorem
"If two angles and the included side of one triangle are congruent to the corresponding two anlges oand the included side og another triangle, then the triangles are congruent."
Congruent Complement Theorem
"If two angles are complements of the same angle or of congruent angles, then they are congruent."
Congruent Supplement Theorem
"If two angles are supplements of the same angle or of congruent angles, then the angles are congruent."
Angle-Angle Similarity Theorem
"If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar."
Alternate Exterior Angle Converse Theorem
"If two lines intersected by a transversal form congruent alternate exterior angles, then the lines are parallel."
Alternate Interior Angle Converse Theorem
"If two lines intersected by a transversal form congruent alternate interior angles, then the lines are parallel."
Corresponding Angle Converse Postulate
"If two lines intersected by a transversal form congruent corresponding angles, then the lines are parallel."
Same-Side Exterior Converse Theorem
"If two lines intersected by a transversal form supplementary same-side exterior angles, then the lines are parallel."
Same-Side Interior Angle Converse Theorem
"If two lines intersected by a transversal form supplementary same-side interior angles, then the lines are parallel."
Side-Angle-Side Similarity Theorem
"If two of the corresponding sides of two triangles are proportional and the included angles are congruent, then the triangles are similar."
Alternate Exterior Angle Theorem
"If two parallel lines are intersected by a transversal, then alternate exterior angles are congruent."
Alternate Interior Angle Theorem
"If two parallel lines are intersected by a transversal, then alternate interior angles are congruent."
Corresponding Angle Postulate
"If two parallel lines are intersected by a transversal, then corresponding angles are congruent."
Same-Side Exterior Angle Theorem
"If two parallel lines are intersected by a transversal, then exterior angles on the same side of the transversal are supplementary."
Side-Angle Side Congruence Theorem
"If two side and the included angle of one triangle are congruent to the corresponding two sides and the included angle of a second triangle, then the triangles are congruent."
30-60-90 Triangle Theorem
"The length of the hypotenuse in a 30-60-90 triangle in two times the length of the shorter leg, and the length of the longer leg is √3 times the length of the shorter leg."
45-45-90 Triangle Theorem
"The length of the hypotenuse in a 45-45-90 triangle is √2 times the leg."
Exterior Angle Theorem
"The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles of the triangle"
Right Triangle Altitude/Hypotenuse Theorem
"The measure of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the measures of the two segments of the hypotenuse."
Triangle Midsegment Theorem
"The midsegment of a triangle is parallel to the third side of a triangle and is half the measure of the third side of the triangle."
Exterior Angle Inequality Theorem
"The sum of the lengths of any two sides of a triangle is greater than the lengths of the third side."
Triangle Inequality Theorem
"The sum of the lengths of any two sides of a triangle is greater that the length of the third side."
Triangle Sum Theorem
"The sum of the measures of the interior angles of a triangle is 180"
Vertical Angle Theorem
"Vertical Angles are congruent."
Linear Pair Postulate
"if two angles form a linear pair, then the angles are supplementary."
Same-Side Interior Angle Theorem
If two parallel lines are intersected by a transversal, then interior angles on the same side of the transversal are supplementary."
segment bisector
a line, line segment, or ray that divides a line segment into two line segments of equal measure
Volume of a Triangular Prism
V=(1/2)Bh
Volume of a Pyramid
V=(1/3)Bh
Volume of a Rectangular Prism
V=lwh
composite figure
a combination of multiple shapes
great circle of a sphere
a cross section of a sphere when a plan passes through the center of the sphere
right cylinder
a disc translated through space in a direction perpendicular to the plane containing the disc
angle
a figure that is formed by two rays that extend from a common point called a vertex
conjecture
a hypothesis that something is true, the hypothesis can later be proved or disproved
included side
a line segment between two consecutive angles of a figure
diameter of a sphere
a line segment with each end of the endpoint on the sphere that passes through the center of the sphere
radius of a sphere
a line segment with one endpoint on the sphere and one endpoint at the center
