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