ACE Geometry PACE 1109 Handbook

Acute angle

Acute angle

An angle measuring less than 90 degrees

Acute triangle

A triangle which has three acute angles

Adjacent angles

Two angles that have both a vertex and a side in common but do not have any interior points in common

Angle

A figure formed by two rays having a common endpoint

Arc

Consists of two points on a circle and all points between them

Base angles

Angles opposite the legs of an isosceles triangle

Betweenness of points

B is between A and C if (1) A, B and C are collinear points, and (2) AB + BC = AC

Betweenness of rays

Ray OB is between ray OA and ray OC if m∠AOB and m∠BOC = m∠AOC

Bisect

Means to separate into two congruent points

Bisector of a line segment

A line, ray or line segment that intersects the line segment at its midpoint, dividing the line segment into two congruent parts

Bisector of a line segment

A ray that originates at the vertex of the angle and divides it into two congruent parts

Center

A fixed point equidistant from all points on a circle

Circle

All points in a plane equidistant from a fixed point

Collinear points

Points which lie on the same line

Compass

An instrument used to draw circles or arcs

Complementary angles

Two angles whose measures have a sum of 90 degrees

Congruent

Having the same length or measure

Congruent angles

Angels that have the same measure

Congruent segments

Segments with the same length

Construction

A geometric drawing done with only a straightedge and a compass

Corresponding angles

Congruent angles of two triangles having a one-to-one relationship with each other

Corresponding sides

Congruent sides of two triangles having a one-to-one correspondence with each other

Defined terms

The words for which specific meanings can be given

Degree

Is 1/360 of a circle and is the unit used to measure the size of the angle

Diameter

A line segment that passes through the center of a circle with endpoints that lie on the circle

Equiangular triangle

An acute triangle having three congruent sides

Equilateral triangle

A triangle having three congruent sides

Exterior of an angle

Consists of all the area lying outside the sides of an angle

Interior of an angle

Consists of all the area lying between the sides of the angle

Isosceles triangle

A triangle with only two congruent sides

Legs

The two congruent sides of an isosceles triangle

Line

Consists of continuous point but lacks width or height, its only dimension is length

Line segment

Consists of two endpoints and all points between those two endpoints and has a definite length

Midpoint

Divides the segment into two congruent parts

Obtuse angle

An angle with a measure greater than 90 degrees and less than 180 degrees

Obtuse triangle

Has one obtuse angle

One-to-one correspondence

Every element in one group or figure is matched with one, and only one, element in another group or figure

Parallel lines

Lines that lie in the same plane but do not intersect

Perpendicular lines

Lines that intersect and form right angles and they form congruent, adjacent angles

Plane

A flat surface

Plane geometry

Deals with figures that lie in a flat surface called a plane; figures that have only the dimensions of length and width

Point

Is the basic building block of all geometric figures

Postulate

Statements that are self-evident, or seem to be true, but cannot be proved

Protractor

An instrument used to measure the size of an angle

Radius

A line segment from the center of a circle to any point on the circle

Ray

Consists of an endpoint and all points along the extension in the direction the ray is drawn

Right angle

An angle measuring 90 degrees

Right triangle

A triangle with one right angle

Scalene triangle

A triangle with none congruent sides

Straight angle

An angle measuring 180 degrees

Straightedge

Unmarked ruler used to draw lines through a given point or to draw lines through two given points

Supplementary angles

Two angles whose measures have a sum of 180 degrees

Triangle

A figure formed by line segments connecting three points that are nog in the same line

Undefined terms

Means they have no precise explanation or formal definition

Vertex

The common endpoint of the two rays that form an angle

Vertex of an isosceles triangle

The angle formed by the legs of an isosceles triangle

Vertical angles

Two pairs of nonadjacent angles formed by two intersecting lines

Vertices

The endpoints of the sides of a triangle

Addition Property

If equal quantities are added to equal quantities, the sums are equal

Subtraction Property

If equal quantities are subtracted from equal quantities

Multiplication Property

If equal quantities are multiplied by equal quantities, the products are equal

Division Property

If equal quantities are divided by equal quantities, the quotients are equal

Partition Property

The whole is equal to the sum of its parts and is greater than any part

Reflexive Property

A quantity is equal (congruent) to itself

Transitive Property

If a first quantity is equal (congruent) to a second quantity and the second quantity is equal (congruent) to a third quantity, then the first and third quantities are equal (congruent)

Substitution Property

A quantity may be substituted for its equal (congruent) quantity

Postulate 1

Only one straight line can be drawn through two points; two points determine a straight line

Postulate 2

A straight line is the shortest line connecting two points in a plane

Postulate 3

A line segment can be bisected at only one point

Postulate 4

Two lines in the same plane either intersect or are parallel. If two lines intersect, four angles are formed at the same point of intersection

Postulate 5

An angle has only one bisector

Postulate 6

All straight angles are congruent

Postulate 7

All right angles are congruent

Postulate 8

In a plane, only one line can be drawn through a point perpendicular to a line

Postulate 9

If one line meets another line, the adjacent angles formed are always supplementary

Postulate 10

Angles that are complements of the same angle, or congruent angles, are congruent

Postulate 11

Angles that are supplements of the same angle, or congruent angles, are congruent

Postulate 12

Vertical angles are congruent

Postulate 13

The sum of the measures of the angles around a common vertex on one side of a line is 180 degrees

Postulate 14

If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent (SSS postulate)

Postulate 15

If two sides and the included angles of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent (SAS postulate)

Postulate 16

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent (ASA postulate)