An angle whose measure is less than 90°

Acute Angle

Two angles with a common vertex and a common side.

Adjacent Angles

Two non-adjacent angles that lie on the opposite sides of a transversal outside two lines that the transversal intersects. If the lines are parallel, then the angles are congruent.

2 and 81 and 7

Alternate Exterior Angles

Two non-adjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects. If the lines are parallel, then the angles are congruent.

3 and 64 and 5

Alternate Interior Angles

Formed by 2 rays (sides) with the same endpoint (vertex).

X

V Z

3

Angle

 

Angle Addition Postulate

If T is in the interior of ∠QRS, then m∠QRT + m∠TRS = m∠QRS.

A ray that divides an angle into two congruent angles.

A B

CD

Angle Bisector

 

Auxiliary LineA line (or ray or segment) added to a diagram to help in a proof or in determining the solution to a problem.

DE is an auxiliary line.

Biconditional

Two statements connected by the words “if and only if.”

CollinearPoints that are on the same line.

A B C DE

A, B, C, and D are collinear points. A, B, C, D, and E are non-collinear points.

Two angles whose measures have a sum of 90.

Complementary

Compound Statement

A statement formed when two or more simple statements are connected as either a conditional (if-then), a biconditional (if and only if), a conjunction (and), or a disjunction (or).

Conclusion

The “then” statement in an if-then statement.

A statement that tells if one thing happens another will follow.

Conditional Statement

Example: “If a polygon has three sides then it is a triangle.”

CongruentExactly equal in size and shape.

Congruent segments have the same length.

Congruent angles have the same measure.

Angles that have the same measure.

Congruent Angles

W

X

 

Segments that have the same length.

J KL M

Congruent Segments

 

An educated guess, opinion, hypothesis.

Conjecture

ConjunctionTwo statements joined by the word and, represented by the symbol ^.

Contrapositive

A version of a conditional statement formed by interchanging and negating both the hypothesis and conclusion of the statement.

Converse

A version of a conditional statement formed by interchanging the hypothesis and conclusion of the statement.

Lines that are in the same plane.

Coplanar Lines

Points that are in the same plane.

Coplanar Points

A

B E

C

D

F

A, B, C, D, and E are coplanar points.

A, B, C, D, E, and F are non-coplanar points.

Two non-adjacent angles that lie on the same side of a transversal, in “corresponding” positions with respect to the two lines that the transversal intersects. If the lines are parallel, then the angles are congruent.

1 and 52 and 4

3 and 84 and 7

Corresponding Angles

An example that shows that a conjecture is not always true.

Counterexample

The use of facts, definitions, rules and/or properties to prove that a conjecture is true.

Deductive Reasoning

DisjunctionThe symbol v represents a disjunction, you read it as “or.”

Distance Formula

The distance between can be found using the formula

A point at one end of a segment or the starting point of a ray.

Endpoint

Hypothesis

The “if” clause in an if-then statement.

The process of observing data, recognizing patterns, and making a generalization.

Inductive Reasoning

Inverse

A version of a conditional statement formed by negating both the hypothesis and conclusion of the statement.

LineA set of points that extends in 2 directions without end.

mA B

Line m or line AB or AB

Part of a line consisting of two endpoints and all points between them.

M

N

Line Segment

Segment MN or Segment NM or MN or NM

A pair of adjacent angles whose noncommon side are opposite rays.

Linear Pair

 

Logically Equivalent

When two statements have the same exact truth values.

A point that divides a segment into two congruent segments.

A C B

Midpoint

 

Midpoint Formula The midpoint of a segement with endpoints can be found using the formula

Negation of pThe symbol ~p is the negation of p and can be read as “not p.”

An angle whose measure is greater than 90° but less than 180°.

Obtuse Angle

Two collinear rays with the same endpoint. They always form a line.

FH

D

Opposite Rays

HF and HD are opposite rays.

Coplanar lines that do not intersect.

ac

Parallel Lines

a//c

Planes that do not intersect.

W

M

Parallel Planes

2 lines intersect to form right angles.

Perpendicular Lines

A

C B 

Planes intersect to form right angles.

BD

Perpendicular Planes

PlaneA flat surface that extends in all directions without end. It has no thickness.

WA

B CPlane W or Plane ABC

Point

A location in space

•A

Postulate

A statement that is accepted without proof.

Proof

An argument that transforms a conjecture to a theorem through the application of logical reasoning or deductive reasoning.

Pythagorean Theorem

For sides a, b, and c in a right triangle, a2 + b2 = c2.

Pythagorean Triple

Three integers a, b, and c such that a2 + b2 = c2

Part of a line consisting of one endpoint and all points of the line on one side of the endpoint.

R

S

Ray

RS not SR

An angle whose measure is exactly 90°.

Right Angle

Two angles that lie on the same side of a transversal and outside the lines cut by the transversal. If the lines are parallel, then the angles are supplementary.

1 and 82 and 7

Same Side Exterior Angles

Two angles that lie on the same side of a transversal and between the lines cut by the transversal. If the lines are parallel, then the angles are supplementary.

3 and 54 and 6

Same Side Interior Angles

Segment Addition Postulate

If B is between A and C, then AB + BC = AC.

If AB + BC = AC, then B is between A and C.

A line, segment or ray that intersects a segment at its midpoint

A C B

Segment Bisector

 

SlopeThe ratio of the vertical change of a line to the horizontal change of the line.

Slope-Intercept Form

A line with a slope m and y-intercept b can be written in the form y = mx + b.

An angle whose measure is exactly 180°.

Straight Angle

Two angles whose measures have a sum of 180.

Supplementary

Tautology

A statement that is always true.

Theorem

A result that has been proved to be true (using facts that were already known).

Transversal

A line that intersects two or more coplanar lines at different points.

Transversal

Truth TableTruth tables are used to determine the conditions under which a statement is true or false.

Truth Value

The truth value of a statement is the truth or falsity of that statement.

The common endpoint of the sides of the angle.

Vertex

Vertex

The non-adjacent angles formed by two intersecting lines.

Vertical Angles

 

y-Intercept

The y coordinate of the point where a graph intersects the y-axis.

A triangle with three acute angles.

AcuteTriangle

A segment from a vertex and perpendicular to the opposite side or the line containing the opposite side.

Altitude of a Triangle

Angle-Angle-Side (AAS) If two angles and a non-included side of a triangle are congruent to two angles and the corresponding non-included side of another triangle, then the two triangles are congruent.

so

Angle-Side-Angle (ASA)If two angles and the included side of a triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

so

The perpendicular segment from the center of a regular polygon to the midpoint of a side.

Apothem

Base AnglesThe two congruent angles in an isosceles triangle.

Bases (of a trapezoid)

The parallel sides of a trapezoid.

The point where the medians of a triangle intersect.

Centroid

The point where the three perpendicular bisectors of a triangle intersect.

Circumcenter

Circumcenter

CircumscribeTo draw on the outside of, touching as many points as possible.

Circumscribed CircleA circle that contains all the vertices of a polygon.

Concave PolygonA polygon that has one or more interior angles that are greater than 180˚.

Two or more triangles whose side lengths and angle measures are congruent.

Congruent Triangle

 

Convex Polygon

A polygon in which all interior angles have measures less than 180˚.

Corresponding Parts

The angles, sides and vertices that are in the same location in congruent or similar figures.

A segment that connects two non-consecutive vertices of a polygon.

Diagonal

Diagonal

Equiangular

A geometric figure in which all angles are equal.

A triangle with three congruent angles.

Equilangular Triangle

EquilateralA geometric figure in which all sides are equal.

A triangle with three congruent sides.

Equilateral Triangle

Exterior AngleThe angle formed by extending a side of a polygon.

Hinge TheoremIf two sides of one triangle are congruent to two sides of another triangle, and then the larger third side is across from the larger included angle.

Hypotenuse-Leg (HL)If the hypotenuse and leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, then the two triangles are congruent.

so

Incenter

The point of concurrency of the angle bisectors of a triangle.

Incenter

Interior AngleAn angle inside a shape.

A trapezoid with congruent legs.

Isosceles Trapezoid

A triangle with two or more congruent sides and angles.

Isosceles Triangle

A quadrilateral who two distinct pairs of adjacent congruent sides.

Kite

Legs (of a trapezoid)

The nonparallel sides of a trapezoid.

A segment from a vertex to the midpoint of the opposite side.

Median of a Triangle

Midsegment

The segments whose endpoints are the midpoint of two sides of a triangle.

A triangle with one obtuse angle.

Obtuse Triangle

The point where the three altitudes of a triangle intersect.

Orthocenter

Orthocenter

A quadrilateral in which both pairs of opposite sides are parallel and congruent.

Opposite Angles are congruent and consecutive angles are supplementary.

Parallelogram

A line or segment that is perpendicular to the side of a triangle at its midpoint.

Perpendicular Bisector of a Triangle

Perpendicular Bisector

Point of Concurrency

The point where three or more lines intersect.

A closed plane figure formed by segments that only intersect at their endpoints.

Polygon

Quadrilateral

A polygon with four sides.

A parallelogram with four right angles and congruent diagonals.

Rectangle

Regular PolygonA polygon that is both equilateral and equiangular.

An interior angle in a polygon that is not adjacent to the exterior angle.

In a triangle the sum of the two remote interior angles is equal to the exterior angle.

Remote Interior Angles

A parallelogram with all sides congruent and diagonals that are perpendicular

Rhombus

A triangle with one right angle and two acute angles.

Right Triangle

A triangle with no congruent sides.

Scalene Triangle

Side-Angle-Side (SAS) If two sides and the included angle of a triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

so

Side-Side-Side (SSS)If the three sides of a triangle are congruent to the three sides of another triangle, then the two triangles are congruent.

so

A parallelogram with all sides congruent and four right angles.

A square has all the properties of a rectangle and a rhombus.

Square

Trapezoid

A quadrilateral with exactly one pair of parallel sides.

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Adjacent Leg

The leg that is closest to the included angle in a right triangle.

Angle-Angle Similarity (AA~)

In two triangles, if two pairs of corresponding angles are congruent, then the triangles are similar. and so

Angle Bisector Proportionality Theorem

An angle bisector of an angle of a triangle divides the opposite side in two segments that are proportional to the other two sides of the triangle.

Constant of Proportionality

The constant value of the ratio of two proportional quantities x and y; usually written y = kx, where k is the factor of proportionality

CosecantThe cosecant of an acute angle in a right triangle is the ratio of the length of the hypotenuse to the length of the opposite side.

CosineThe cosine of an acute angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse.

CotangentThe cotangent of an acute angle in a right triangle is the ratio of the length of the adjacent side to the length of the opposite side.

Geometric Mean 

Opposite Leg

The leg that is across from the included angle in a right triangle.

Parallel Proportionality Theorem

If three or more parallel lines intersect two transversals, then they divide the transversals proportionally.

so

SecantThe secant of an acute angle in a right triangle is the ratio of the length of the hypotenuse to the length of the adjacent side.

Side-Angle Side Similarity (SAS~)

If an angle of a triangle is congruent to an angle of another triangle and if the included sides of these angles are proportional, then the two triangles are similar.

and so

Side-Side-Side Similarity (SSS~)

If the corresponding sides of two triangles are proportional, then the two triangles are similar.

so

Similar PolygonPolygons with congruent corresponding angles and corresponding side lengths in proportion.

SineThe sine of an acute angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse.

Special Right Triangle 30-60-90 and 45-45-90 are called special right triangles because they have some regular feature that makes calculations on the triangle easier.

TangentThe tangent of an acute angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.

Triangle Proportionality Theorem

If two or more lines parallel to a side of a triangle intersect the other two sides of the triangle, then they divide them proportionally.

𝐴𝐶𝐶𝐸

=𝐴𝐵𝐵𝐷

Angles formed by Chords

If two chords intersect inside a circle, then the measure of the angle formed is one half the sum of the measure of the arcs intercepted by the angle and its vertical angle.

𝑚∠1=1 /2¿

Angles formed by SecantsIf two secant segments intersect outside a circle, then the measure of the angle formed is one half the difference of the measure of the intercepted arcs.

𝑚∠𝐸=1/2¿

Angles formed by Tangents

𝑚∠𝐶=1/2¿

If two tangent segments intersect outside a circle, then the measure of the angle formed is one half the difference of the measure of the intercepted arcs.

A continuous part of a circle. The measure of the arc is the measure of the angle formed by the two radii with endpoints at the endpoints of the arc.

Arc

An angle whose vertex is at the center of a circle and whose sides are radii of the circle

Central Angle

Central Angle

A line segment on the interior of a circle with both endpoints lying on the circle.

Chords

 

CircleThe set of all points in a plane that are a given distance (the radius) from a given point (the center) in the plane.

Congruent Arc

Arcs of a circle that have the same length.

A chord that contains the center of the circle

Diameter

Equation of a Circle

The equation of a circle with center (h,k) and radius r is

(x – h)2 + (y – k)2 = r2

External Secant Segment

The parts of a secant segments that are outside the circle.

EF and EH are external secant segments

An angle whose vertex is on the circle and whose sides are chords of the circle.

Inscribed Angle

Inscribed Angle

Major ArcAn arc with a measure greater than 180˚.

Minor Arc

An arc with a measure less than 180˚.

The point where the tangent line intersects a circle. A radius is perpendicular the tangent at the point of tangency.

Point of Tangency

A segment from the center of a circle to a point on the circle.

Radius

A line/segment that intersects a circle exactly twice.

Secant Line/Segment

SectorA region formed by two radii and an arc of a circle.

Segments Lengths in Circles formed by Chords

If two chords of a circle intersect, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.

𝑃𝐴 ∙𝑃𝐷=𝑃𝐶 ∙𝑃𝐵

Segments Lengths in Circles formed by Secants

If two secants intersect at a point outside a circle, the product of one secant segment and its external secant segment equals the product of the other secant segment and its external secant segment.

𝑃𝐴 ∙𝑃𝐷=𝑃𝐶 ∙𝑃𝐵

Segments Lengths in Circles formed by a Secant and a Tangent

If a secant and a tangent intersect at a point outside a circle, the product of the length of the secant segment and its external secant segment equals the square of the length of the tangent segment.

𝑃𝐴2=𝑃𝐶 ∙𝑃𝐵

A line or segment that intersects a circle exactly once.

Tangent Line/Segment

DilationA transformation in which the image is similar (but not congruent) to the pre-image.

ReflectionA transformation in which a figure is flipped over a line, called a line of reflection.

RotationA transformation in which each point of the pre-image travels clockwise or counterclockwise around a fixed point a certain number of degrees.

TessellationA covering of a plane consisting of one or more types of shapes such that there are no overlaps or gaps between the shapes.

TranslationA transformation that moves each point of a figure the same distance and in the same direction.

A solid bounded by a circular base and a curved surface with one vertex

Cone

Cross SectionThe intersection of a solid figure and a plane.

A solid bounded by two congruent and parallel circular bases joined by a curved surface.

Cylinder

EdgesThe line segment formed by the intersection of two faces of a polyhedron.

Faces

One of the polygons that make up a three dimensional solid figure.

HemisphereA half-sphere.

Isometric DrawingA drawing on isometric dot paper the represents a three-dimensional figure and shows the top, side, and front views.

Lateral Area

The surface area of a solid excluding the base(s).

Lateral FacesThe nonparallel bases, or bases, of a solid.

NetA two-dimensional drawing used to represent or form a three-dimensional object or solid.

ObliqueNot perpendicular.

PolyhedronA closed three-dimensional figure consisting of polygons that are joined along their edges.

A polyhedron that has two congruent parallel faces (bases) that are joined by faces that are parallelograms.

Prism

A polyhedron with three or more triangular faces that meet at a point (vertex) and one other polygonal face called the base.

Pyramid

It is the shortest distance from the vertex of a cone or pyramid to the edge of the base.

Slant Height

Sphere

The set of all points (x, y, z) that are a given distance, the radius, from a point, the center.

Surface Area

The total area of all the surfaces of a three-dimensional figure.

Volume

The number of cubic units in a three-dimensional figure.