Review of Concepts – Chapter 3
Definition of Angles – an angle is the union of two rays that have the same endpoint.
Angle Measure Postulate –
a. Unique measure assumption: Every angle has a unique measure from 0 to 180 degrees.
b. Unique angle assumption: Given any ray VB and a real number r between 0 and 180, there is a unique
angle BVA on each
side of ray VB such that m∠BVA = r º.
c. Straight angle assumption: If ray VA and ray VB are opposite rays, then m ∠BVA = 180 º.
d. Zero angle assumption: If ray VA and ray VB are the same ray, then m ∠BVA = 0 º.
Definition of Types of Angles: If m is the measure of an angle, then the angle is:
a zero angle if and only if m = 0;
an acute angle if and only if 0 < m < 90;
a right angle if and only if m = 90;
an obtuse angle if and only if 90 < m < 180;
a straight angle if and only if m = 180.
Definition of Adjacent Angles - Two nonstraight and nonzero angles are adjacent angles if and only if a commom
side is in the interior of the angle formed by the noncommon sides.
Definition of Angle Bisector – Ray VR is an angle bisector of ∠PVQ if and only if ray VR (except point V) is in
the interior of ∠PVQ, and m∠PVQ = m∠RVQ.
Review of Concepts – Chapter 3
Angle Measure Postulate –
e. Angle Addition assumption: If angles AVC and CVB are adjacent angles, then m∠AVC + m∠CVB =
m∠AVB
Definition of Complementary Angles and Supplementary Angles – If the measures of two angles are r and s, then
the angles are:
complementary angles if and only if r + s = 90;
supplementary angles if and only if r + s = 180.
Equal Angle Measures Theorem –
If two angles have the same measure, their complements have the same measure.
If two angles have the same measure, their supplements have the same measure.
Definition of a Linear Pair – Two adjacent angles are a linear pair if and only if their noncommon sides are
opposite rays.
Linear Pair Theorem – If two angles form a linear pair, then they are supplementary.
Definition of Vertical Angles – Two nonstraight angles are vertical angles if and only if the union of their sides is
two lines.
Vertical Angles Theorem – If two angles form a vertical angles, then their measure are equal.
Review of Concepts – Chapter 3
Postulates of Equality – For any real numbers a, b, and c:
Reflexive Property of Equality: a = a
Symmetric Property of Equality: If a = b, then b = a.
Transitive Property of Equality: If a = b and b = c, then a = c.
Postulates of Equality and Operations – For any real numbers a, b, and c:
Addition Property of Equality: If a = b, then a + c = b + c.
Multiplication Property of Equality: If a = b, then a * c = b * c.
Postulates of Inequality and Operations – For any real numbers a, b, and c:
Transitive Property of Inequality: If a < b and b < c, then a < c.
Addition Property of Equality: If a < b, then a + c < b + c.
Multiplication Property of Equality:
If a < b and c > 0, then a * c < b * c.
If a < b and c < 0, then a * c > b * c.
Postulates of Equality and Inequality – For any real numbers a, b, and c:
Equation to Inequality:
If a and b are positive numbers and a + b = c, then c > a and c > b.
Substitution Property:
If a = b, then a may be substituted fo b in any expression.
Review of Concepts – Chapter 3
Definition of Proof – A proof of a conditional is a sequence of justified conclusions starting with the antecedent
and ending with the consequent.
Corresponding Angles Postulate – Suppose two coplanar lines are cut by a transversal (a line that intersects two
other lines).
a. If two corresponding angles have the same measure, then the lines are parallel.
b. If the lines are parallel, then the corresponding angles have the same measure.
Parallel Lines and Slopes Theorem – Two nonvertical lines are parallel if and only if they have the same slope.
Transitivity of Parallelism Theorem – If line l is parallel to line m and to line m is parallel to line n, then line l is
parallel line n.
Definition of Transformation – A transformation is a correspondence between two sets of points A and B such
that:
1. Each point in set A corresponds to exactly one point in set B.
2. Each point in set B corresponds to exactly one point in set A.
Definition of Size change (Size transformation) –
When k ≠ 0, the transformation under which the image of (x, y) is the point (kx, ky) is the size change (size
transformation) of magnitude k and center (0, 0).
Review of Concepts – Chapter 3
Sk Theorem 1: Parallel Property –
Under a size change Sk, the line through any two preimage points is parallel to the line through their images.
Sk Theorem 2: Collinearity is Preserved –
Under a size change Sk, the images of collinear points are collinear.
Sk Theorem 3: Angle Measure if Preserved –
Under a size change Sk, an angle and its image have the same measure.
Two Perpendiculars Theorem –
If two coplanar lines l and m are each perpendicular to the same line, then they are parallel to each other.
Perpendicular to Parallels Theorem –
In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other.
Perpendicular Lines and Slopes Theorem –
Two nonvertical lines are perpendicular if and only if the product of their slopes is -1.
Review of Concepts – Chapter 3
From a figure you can assume
From a figure you cannot assume
collinearity and betweeness of points
drawn on the same line
intersections of lines at a given point
collinearity of three or more points that
are not drawn on lines
parallel lines
points in the interior of an angle, on an
angle, or in the exterior of an angle
exact measures of angles or segments