PROPERTIES and SPECIAL PAIRS of ANGLES
Properties of Equality:
Addition Property of Equality:
Algebra Ex. If x  2  10 then x  12
Geometry Ex. If AB  CD then AB  BC  BC  CD .
Subtraction Property of Equality:
Algebra Ex. If x  4  15 then x  11
Geometry Ex. If AB  BC  BC  CD then AB  CD .
Multiplication Property of Equality:
x
Algebra Ex. If  3 then x  6
2
Geometry Ex. If m A  45 then 2(m A)  90
Division Property of Equality:
Algebra Ex. If 4x  24 then x  6
Geometry Ex. If 2(m A)  180 then m A  90
Substitution Property:
Algebra Ex, If x  2 then 3x  1  3(2)  1
Geometry ex. If m A  m B and m A  m B  90 then
m A  m A  90
Reflexive, Symmetric, and Transitive Properties
Reflexive Property of Equality: m A  m A
Reflexive Property of Congruence: AB  AB
Symmetric Property of Equality: If AB  CD then CD  AB
Symmetric Property of Congruence: If R  P then P  R
Transitive Property of Equality: If AB  CD and CD  JK then AB  JK.
Transitive Property of Congruence: If A  B and B  C ,
then A  C.
Remember them : R,S,T—1,2,3
(1 object, 1 equation; 2 objects, 2 equations; 3 objects, 3 equations )
1. Two Types of reasoning:
a. Inductive Reasoning: Looking at specific examples to draw a
general conclusion; (Looking for a trend or pattern in specific
cases.) Conclusion may not necessarily follow.
b. Deductive Reasoning: Using proven (general) facts (such as
theorems, definitions, postulates) to form a logical argument (or
proof) from which the conclusion must follow.
2. Parts of a 2-column deductive proof
a. Statement: This is the theorem which is to be proven. It should be
written in If/Then form.
b. The Given: This information comes from the hypothesis of the
statement. Items should be named and the information about them
should be specific. This information is assumed to be true.
c. A Picture: A diagram should be drawn to illustrate the given
information. It should be labeled so objects can be referred to by
name.
d. The Prove: This information comes from the conclusion of the
statement. It should be specific. Items should be named according
to the given and the diagram.
e. The Proof: The proof consists of two columns.
1.
2.
3.
4.
Statements
The first Statement is
(almost) always the
information in the Given. It is
what starts the whole ball
rolling.
Each new statement (usually
an equation) that shows
some relationship about the
items which are given, other
objects in the picture, or the
items you just made your
previous statement about.
Use properties, theorems
etc,. to try to reach the final
conclusion, the Prove.
The last statement is always
what you are trying to prove.
Reasons
Each new statement you make must be
backed up by a property, Postulate,
theorem or definition to defend how you
know what you have just said is true. The
only exception is the reason for the first
statement. The only reason that you know
it is true, is that the information was GIVEN
to you. So for reason #1 you write
“GIVEN”
Note: The last statement must be
supported with a definition, theorem, etc,
just like all of the rest.
3. Perpendicular Lines: Two lines that meet to form right angles.
(Segments, rays, and even planes can be perpendicular if they meet to
form right angles.) We write: l  m to say that lines l and m are
perpendicular. In a diagram a “box” at the vertex in the interior of an angle
also means it is a right angle.
4. Congruent angles: Two angles that have the same measure.
We write: ABC  DEF .
If ABC  DEF then m ABC  m DEF (or vice versa)
5. Complementary angles: Two angles the sum of whose measures is
90 . If A and B are complementary then m A  m b  90 .
Example: If m A  35 and m B  55 then A and B are
complements of each other.
6. Supplementary Angles: Two angles the sum of whose measures is
180 . If A and B are supplementary then m A  m B  180 .
Example: If m A  109 and m B  71 then A and B are
supplements of each other.
7. Angle Addition Postulate: If C is a point in the interior of DAB then
m DAC  m CAB  m DAB . In other words: The sum of the measures
of two adjacent angles is equal to the measure of larger angle in which
they lie.
m DAC = 86°
m CAB = 38°
C
D
B
A
m DAB = 124°
m DAC + m CAB = 124°
8. Linear Pair: Two adjacent angles whose sides form opposite rays.
(Note: Linear pairs can be identified from a picture)
9. The Supplement Theorem: The angles in a linear pair are
supplementary,
m ACD + m DCB = 180°
D
m ACD = 44°
m DCB = 136°
A
C
TSU form a linear pair. m RST  11x  5 and
Example: Suppose RST and
m TSU  9x  25 . Find x.
Solution:
B
11x  5
 9 x  25  180
20 x  20  180
20 x  200
x  10
10. Vertical Angles: Two non-adjacent angles formed when 2 lines intersect.
Vertical angles are always congruent. (Note: Vertical angles can be
identified from a pictured.)
The pairs of vertical angles pictured below are:
AED and
AEC and
m AED = 131°
A
m AEC = 49°
C
D
E
m DEB = 49°
m CEB = 131°
B
(Notice the linear pairs are still supplementary as well.)
CEB
DEB
8. The Congruent Supplements Theorem: If two angles are supplementary
to the same angle then the angles are congruent to each other.
Given:
A is supplementary to G
F is supplementary to G
Prove:
A F
A
F
Statements
1.
A is supplementary to G
F is supplementary to G
G
Reasons
1.
2.
3.
3. Transitive property of Equality
4.
5.
6.
m Am F
6.
9.. The Congruent Complements Theorem: If two angles are complementary
to the same angle then the angles are congruent to each other.
The proof of this theorem is left for the student. (Hint: The proof is almost the
same as the one above but with a few minor changes.)
10. The Vertical Angles Theorem: If two angles are vertical angle, then they
are congruent.
Given:
1, 3 are vertical angles
1
2
3
Prove:
1 3
Statements
Reasons
1.
2.
3.
11. Theorem: All right angles are congruent. (To be proven by the student)
12. Theorem: If two lines intersect to form congruent adjacent angles, then
the lines are perpendicular.
1 2
Given: 1  2
l
Prove: l  t
t
Statements
1.
2.
3.
4.
5.
6.
7.
8.
9.
1 2
1 and 2 are Supp.
m 1  m 2  180
m 1 m 2
m 1  m 1  180
2(m 1)  180
m 1  90
1 is a right angle
l t
Reasons