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Chapter 1:
Section 1.8
Definitions:
1. Betweenness: A number is between two others if it’s greater than one of them
and less than the other.
Postulates:
1. Distance Postulate: There are 3 parts to the distance postulate
a. Uniqueness Property: On a line there is a unique distance between two
points.
b. Distance Formula: If two points on a line have coordinates x and y.
c. Additive Property: If B is on line segment AC ( -- ), then AB + BC = AC
Chapter 2:
Section 2.7
Definitions:
1. The Triangle Inequality: Any side of a triangle is shorter than the other two sides
combined.
a<b+c
b<a+c
c<a+b
Chapter 3:
Section 3.1
Definitions:
1. Angle: The union of two rays that have the same endpoint.
2. Sides: The sides of an angle are the two rays that form it.
3. Vertex: the vertex of the angle is the common endpoint of the two rays.
4. Interior: The convex set is inside of the angle.
5. Exterior: The convex set is outside of the angle.
6. Measure: Indicates the amount of openness of the interior of the angle.
7. Degree: A unit of measure used for the measure of an angle, arc, or rotation
8. Bisector: A ray in the interior of an angle that divides angle into two angles of
equal measures.
Postulates:
1. Unique Measure Assumption: Every angle has a unique measure from 0 degrees
to 180 degrees.
2. Unique Angle Assumption: Given any ray VA (→) and any real number r between
0 and 180, there is a unique angle BVA in each half-plane of line VA (<-->) such
that m<BVA = r
3. Zero Angle Assumption:If ray VA (→) and ray VB (→) are the same ray then
m<AVB = 0
4. Straight Angle Assumption: If ray VA (→) and ray VB (→) are opposite rays, then
m<AVB = 180
5. Angle Addition Property: If ray VC (→) (except for point V) is in the interior of
<AVB, then m<AVC + m<CVB = m<AVB.
Section 3.2
Definitions:
1. Central angle of a circle: An angle whose vertex is the center of the circle.
2. Minor arc: The points of circle O that are on or in the interior of angle AOB
3. Major arc: The points of circle O that are on or in the exterior of angle AOB.
4. Semicircle: An arc of a circle whose endpoints are the endpoints of a diameter of
the circle.
5. Degree measure of a minor arc or semicircle: The measure of the central angle
AOB
6. Degree measure of a major arc (ACB): 360 - (measure of minor arc AB)
7. Concentric: Two or more circles that lie in the same plane and have the same
center.
8. Images: The result of applying a transformation to an original figure or preimage.
9. Preimages: The original figure in a transformation.
10. Clockwise direction: The direction in which the hands move on the nondigital
clock, designated by negative magnitude.
11. Counterclockwise direction: The direction opposite that which the antecedent is
true but the consequent is false. An example which shows conjecture to be false.
12. Magnitude: In a rotation, the amount that the preimage is turned about the
center of rotation, measured in degrees from -180 degrees (clockwise) to 180
degrees (counterclockwise), positive/negative measure angle POP, where P is
the image of P under the rotation and O is its center
Section 3.3
Definitions:
1. If m is the measure of an angle, then the angle is:
a.
b.
c.
d.
e.
zero if and only if m = o
acute if and only if 0 < m < 90
right if and only if m = 90
obtuse if and only if 90 < m < 180
straight if and only if m = 180
2. Complementary Angles: When two measures add up to 90°
3. Supplementary Angles: When two measures add up to 180°
4. Adjacent Angles: Two non straight and nonzero angles with a common side
interior to the angle formed by the noncommon sides.
5. Linear Pair: Two adjacent angles if and only if their noncommon sides are
opposite rays.
6. Vertical Angles: Two non-straight angles if and only if the union of their side is
two lines.
Theorem:
1. Linear Pair Theorem: If two angles for a linear pair, then they are supplementary.
2. Vertical Angles Theorem: If two angles are vertical angles, then the have equal
measures.
Postulates:
1. Angle Addition Postulate: <ABD (Angle ABD) + <DBC (Angle DBC) = <ABC (Angle
ABC)
Section 3.4
Postulates:
1. Postulates of Equality:
a. Reflexive Property: a = a
b. Symmetric Property: If a = b; b = a
c. Transitive Property of Equality: If a = b and b = c, then a = c
2. Postulates of Equality and Operations:
a. Addition Property: You can add/subtract a number from both sides of an
equation.
b. Multiplication Property: You can multiply/divide an equation by the same
number on both sides (except 0).
c. Substitution Property: If a = b, you can replace a with b anywhere.
3. Postulates of Inequality and Operations:
a. Transitive Property: If a < b and b < c, then a < c.
b. Addition Property: If a < b, then a + c < b + c.
c. Multiplication Property: If a < b and c > 0, then ac < bc.
If a < b and c < 0 , then ac > bc.
4. Postulates of Equality and Inequality:
a. Equation of Inequality Property: If a and b are positive numbers and a + b
= c, then c > a and c > b.
b. Substitution Property: If a = b, then a may be substituted for b in any
expression.
Section 3.5
Definition:
1. Proof argument: Contains a conclusion and a justification.
2. Justification: a reason why a statement is true.
● There are 3 types of justifications: postulates/properties, definitions, and theorems.
Section 3.6:
Definitions:
1. Transversal: A line that intersects 2 others.
2. Corresponding angles: A pair of angles in similar locations when two lines are
intersected by a transversal.
3. Slope: In the coordinate plane, the change in y-value divided y the corresponding
changes in x-values.
Postulates
1. Corresponding Angle Postulate: Two (coplanar) lines are parallel if and only if
corresponding angles created by a transversal are equal.
Theorem:
1. Parallel Lines and Slopes: Two nonvertical lines are parallel if and only if they
have the same slope.
2. Transactivity of Parallelism: In a plane if l // m and m // n, the l // n.
Section 3.7
Definitions:
1. Perpendicular: If 2 lines are perpendicular, they form a 90° angle.
2. Parallel: 2 lines are parallel if they never intersect
Theorem
1. Two perpendiculars: If 2 coplanar line l and m are each perpendicular to the
same line, then they are parallel to each other.
2. Perpendicular to Parallels: In a plane, if a line is perpendicular to one of 2 parallel
lines, then it’s also perpendicular to the other.
3. Perpendicular Lines and Slopes: 2 nonvertical lines are perpendicular if and only
if the product of their slopes is -1.
4. Parallel Lines and Slopes: The slope of parallel lines are equal.
Section 3.8
Definitions:
1. Bisector of a segment: Has 2 equal pieces even though it does not create a 90°
angle.
2. perpendicular bisector: Splits a segment into two segments and creates a 90°
angle.
3. Construction: A drawing which is made using only an unmarked straightedge and
a copass following certain prescribed rules.
4. Unmarking straightedge: An instrument with a straight edge (straight line such as
a ruler etc.)
5. compass: An instrument for drawing circles.
6. Algorithm: A finite sequence of steps leading to a desired end.
Chapter 4:
Section 4.1
Definitions:
1. preimage: preimage of an object is the object we are reflecting.
2. reflecting line: The line over which a preimage is reflected. Also called line of
reflection or mirror.
3. reflection image of P overline m: If p is not on m, the reflection image of p is the
point Q such that m is the perpendicular bisector of line segment PQ. If P is on
m, the reflection image is point P itself.
4. transformation: a correspondence between 2 sets of points such that:
a. each point in the preimage set has a unique image.
b. each point in the image has exactly one preimage.
5. mapping: a transformation.
Section 4.2
Definition:
1. Reflection image of a figure: The set of all reflection images of the points of the
original figure.
Postulates:
1. Reflection: Reflections preserve:
a. angle measure
b. betweenness
c. collinearity
d. distance
e. orientation
Theorem:
1. Figure Reflection: If a figure is determined by certain points, then its reflection
image is the corresponding figure determined by the reflection images of those
points.
Section 4.4
Definition:
1. Composite: The composite of a first transformation S and a second
transformation T; denoted T°S is the transformation that maps each point P onto
T (S(P))
2. translation or slide: reflecting over 2 parallel lines
3. direction: the direction of a translation is which way it travels
a. Perpendicular to the lines and towards the second line.
4. Magnitude: the magnitude of a translation is the distance it travels.
a. TWICE the distance between parallel lines
Theorem:
1. Two-Reflection Theorem for Translation: If m // l, the translator r (lowercase m)
composite (°) r (lowercase l) has magnitude two times the distance between l
and m, in the direction from l perpendicular to m
Section 4.5
Definition:
1. Rotation: The composite of two reflections over intersecting lines; the
transformation “turns” the preimage onto the final image about a fixed point (its
center). Also called turn.
Theorem:
1. Two-Reflection Theorem for Rotations: If m intersects l, the rotation r (lowercase
m) composite ( ° ) r (lowercase l) has center at the point of intersection of m and
l and has magnitude twice the measure of non-obtuse angle formed by these
lines, in the direction from l to m.
Section 4.7
Definition:
1. isometry: A transformation that is a reflection or a composite of reflections. Also
called congruence transformation or distance-preserving transformation.
a. 3 types of isometries: 1 Reflection = Reflection; 2 Reflections =
Translation & Rotation; Reflection + Translation = glide reflection.
2. concurrent: Two or more lines that have a point in common
Chapter 5:
Section 5.1
Definition:
1. corresponding parts: Angles or sides that are images of each other under a
transformation.
Theorem:
1. Corresponding parts of congruent figures: If two figures are congruent, then any
pair of corresponding parts is congruent.
2. A-B-C-D: Every isometry preserves Angle measure, Betweenness, Collinearity
(lines), and Distance (length of segments).
Section 5.2
Theorem:
1. Equivalence Properties of congruent: For any figures F, G, H:
a. Reflexive: F congruent to F
b. Symmetric: If F congruent G, then G congruent F
c. Transitive: If F congruent G and G congruent H, the F congruent H
2. Segment Congruence: Two segments are congruent if and only if they have the
same length.
3. Angle Congruence: Two angles are congruent if and only if they have the same
measure.
Section 5.4
Definition:
1. two-column form: A form of written proof in which the conclusion are written in
one column, the justifications beside them in a second column.
2. interior angles: Angles formed by two lines and a transversal whose interiors are
partially between the lines; the angles of a polygon.
3. exterior angles: An angle formed by two lines and a transversal whose interior
contains no points between the two lines. An angle which forms a linear pair
with an angle of a given polygon.
4. alternate interior angles: Angles formed by two lines and a transversal whose
interiors are partially between the two lines an on different sides of transversal.
5. alternate exterior angles: Angles formed by two lines and a transversal whose
interior are not between the two lines and are on different sides of the
transversal.
Theorem:
1. Lines ⇒AIA congruent (Alternate Interior Angles Theorem): If two parallel lines
are cut by a transversal , then alternate interior angle are congruent.
2. AIA congruent ⇒// lines: If two lines are cut by a transversal and form congruent
alternate interior angles, then the lines are parallel.
Section 5.5
Definition:
1. equidistant: At the same distance
Theorem:
1. Perpendicular bisector: If a point is on the perpendicular bisector of a segment,
then it is equidistant from the endpoints of the segment.
Section 5.6
Definition:
1. uniquely determined: One and exactly one. Example: The line between 2 points,
the perpendicular bisector of a segment, the measure of an angle, and the
midpoint of a segment.
2. auxiliary figure: gives help or assisting. Auxiliary figures are objects we add to a
diagram to help us prove something.
Theorem:
1. Uniqueness of Parallels: Through a point not on a line, there is exactly one line
parallel to the given line.
Section 5,7
Theorem:
1. Triangle-sum: The sum of the measures of tje amg;es pf a triangle is 180 degree
2. quadrilateral-sum: The sum of the measures of the angles of a convex
quadrilateral is 360 degrees
3. polygon-sum: The sum of the measures of the angles of a convex n-gon is (n - 2)
x 180 degrees.