The System of Mathematics Unit 1 Starting Vocabulary Geometry Mr. Ferguson collinear Points are collinear iff they lie on the same line. Points B and C are collinear.

Points A and C are noncollinear. coplanar Points are coplanar iff they lie in the same plane. Points A, B, and C are coplanar. between A point X is between points A and B iff A, X,

and B are collinear in that order (or in the order B, X, A). A- X- B The notation for between is Examples & Non-examples * * *

* Point D is between points B and C. B-C-E E is not between B and C. F is not between B and E. line segment A set of points is a line segment iff it consists of two points (called endpoints)

and all the points between them. GH consists of the endpoint G and H and all points between them. ray A set of points is a ray iff it consists of segment AB and all points X such that A-BX.

Point A is called the endpoint. We always name a ray by starting with its endpoint this is ray intersect Two objects intersect iff they share at least one point in common. angle

Two rays form an angle iff they share a common endpoint. The common endpoint is called the vertex of the angle. WXY Notation: YXW

W or *always write the vertex as the middle letter X Y

midpoint A point M is the midpoint of segment AB iff M is between A and B, and AM = MB. M is the midpoint of segment AB. X is not the midpoint of segment PQ congruent segments Two segments are congruent iff their measures (lengths) are equal.

[email protected] Segment AB is not congruent to segment EF. We denote that segments are congruent by using the same number of tick marks. congruent angles Two angles are congruent iff their measures are equal.

[email protected] mSTU = 53.85 S mMNO = 43.66 O

T U P M Q *Angle MNO is not congruent to angle N STU

Pay close attention to notation! mPQR = 43.66 R acute angle An angle is an acute angle iff its measure is greater than 0 degrees and less than 90 degrees.

Angle VWX is acute since right angle An angle is a right angle iff it measures exactly 90 degrees. Angle ABC is a right angle since A

B C We denote a right angle by drawing a small box at the vertex. obtuse angle An angle is an obtuse angle iff its measure is greater than 90 degrees but

less than 180 degrees. Angle DEF is an obtuse angle since straight angles An angle is a straight angle iff its measure is exactly 180 degrees. G H

I Angle GHI is a straight angle since complementary angles Two angles are complementary iff the sum of their measures equals 90 degrees. Angle JKL and angle MNO are complementary since

It is not necessary that two angles share the same vertex and one common side in order to be complementary. supplementary angles Two angles are supplementary iff the sum of their measures equals 180 degrees. Angle PQR and angle STU are supplementary since

It is not necessary that two angles share the same vertex and one common side in order to be supplementary. circle A set of points forms a circle iff it consists of all points equidistant from a point called the center. *Circles are named by their center. C

radius A segment is a radius of a circle iff one endpoint is the center of the circle and the other endpoint lies on the circle. is a radius CWof circle C CXof circle C is a radius

diameter A segment is a diameter of a circle iff its endpoints lie on the circle and the center of the circle lies between the endpoints. is a diameter of circle C. WY is not a

AZdiameter of circle C. perpendicular Two lines are perpendicular iff the lines are coplanar and their intersection forms a right angle. E B C D

F This definition also applies for rays and segments. parallel Two lines are parallel iff they are coplanar and they never intersect. This definition also applies to rays and segments,

so long as it is used in the same context. adjacent angles Two angles are adjacent iff they share a common vertex, one common side, and no other points. Example: Angle ABC and angle CBD are adjacent. Angle ABD and angle ABC are not adjacent.

opposite rays Two rays are opposite rays iff they have a common endpoint, and all points on both rays are collinear. Ray BA and ray BC are opposite rays. linear pair Two angles form a linear pair iff the angles are adjacent and their non-common rays are

opposite rays. Angle ABD and angle CBD are a linear vertical angles Two angles are vertical angles iff the angles are not adjacent and their sides are opposite rays. Angle AED and angle CEB are vertical

segment bisector A line is a segment bisector of a segment iff it intersects the segment at its midpoint. line PQ is a bisector of segment AB A ray or segment can also be a segment bisector. angle bisector

A line is an angle bisector of an angle iff the line intersects the angle at its vertex and it divides the angle into two congruent angles. S Ray TV is the angle bisector of angle STU. V

T A ray or segment can also be an angle bisector. U vertical angles polygon

A figure is a polygon iff it consists three or more segments and each segment intersects two other segments, one at each endpoint. triangle A polygon is a triangle iff its has exactly 3 sides.T R

S quadrilateral A polygon is a quadrilateral iff it has exactly 4 sides. valid An argument is valid iff when the premises are all true, then the conclusion is true.

A valid argument does not need to have true premises it just has to be the case that each premise implies the next, all the way to the conclusion. That is, there is a logical chain between all premises and the conclusion. sound An argument is sound iff the argument is

valid and all the premises are true. A sound argument must be valid (the logical chain exists), but it also must have true premises, meaning that the conclusion must be true. In Geometry, we want all of our proofs to be sound.