Proofs, Where Your Reasons must go with your Given and Previous Conclusions
DEFINITIONS
Midpoint of a segment: the point that divides the segment into two congruent
segments.
ex.
Given: M is the midpoint of AB
Conclusion: AM = MB
Reason: definition of midpoint
Segment bisector: a segment that intersects the segment at its midpoint.
ex.
Given: CD is a segment bisector of AB passing through M on AB
Conclusion: AM = MB
Reason: definition of segment bisector
Median: a segment that connects the vertex of a triangle to the midpoint of the
side opposite from this angle.
B
D
A
ex.
C
Given: CD is a median of ABC and the diagram above
Conclusion: D is a midpoint of AB
Reason: Definition of a median
Conclusion: AD = DB
Reason: Definition of midpoint
Right triangle: a triangle that has a right angle
F
ex.
Given: diagram at the right
G
E
Conclusion: EGF is a right triangle
Reason: Definition of a right triangle
Angle bisector: a ray that divides an angle into two congruent angles
ex.
Given: YW bisects XYZ
Conclusion: XYZWYZ
Reason: definition of angle bisector
Perpendicular lines: lines that intersect to form right angles
ex.
Given: AD is perpendicular to CD at D
Conclusion: angle ADC is a right angle
Reason: definition of a right angle
Altitude: a perpendicular line segment drawn from the vertex of a triangle to the
line containing the opposite side.
M
ex.
Given: the diagram at the right
N
K
Conclusion: segment MN is perpendicular to line KL
Reason: definition of an altitude
L
Isosceles triangle: a triangle with two congruent sides. Because the two sides are
congruent, the angles across from the two congruent sides are congruent and viceversa. We call these angles the base angles. An equilateral triangle is just a more
specific isosceles triangles b/c it has additional properties.
ex.
B
Given: The diagram to the right
Conclusion: ABC is an isosceles 
A
C
Reason: Definition of an isosceles 
POSTULATES
Reflexive Postulate
ex.
Given: ABC
Conclusion: ABC  ABC
You may think that this is a pointless
postulate, but it is the one students
leave out the most often, so BEWARE
of this one!
Reason: Reflexive Postulate
Symmetric Postulate
ex.
Given: 12
Conclusion: 21
Reason: Symmetric Postulate
Transitive Postulate
ex.
Given: 12, 23
Conclusion: 13
Reason: transitive postulate (it's like a chain)
Substitution Postulate
ex.
Given: 45, 46
Conclusion: : 56
Equals can always be replaced with
equals. This is quite possibly the most
commonly used postulate.
Reason: Substitution Postulate
Addition Postulate
ex.
Given: B is b/w A and C
Conclusion: AB +BC = AC
Reason: Addition Postulate
Given: P is in the interior of angle RST
Conclusion: mRSP + mPST = mRSP
Reason: Addition Postulate
Subtraction Postulate
ex.
Given: B is b/w A and C
Conclusion: AC - BC = AB
Reason: Subtraction Postulate
Given: P is in the interior of angle RST
Conclusion: mRST - mRSP = mPST
Reason: Subtraction Postulate
Multiplication Postulate
ex.
Given: A = 2B and C = 2B
Conclusion: B = B
Reason: Reflexive Postulate
Conclusion: A = C
Reason: Multiplication Postulate or Doubles of equals are equal
Division Postulate
ex.
Given: A = (1/2)B and C = (1/2)B
Conclusion: B = B
Reason: Reflexive Postulate
Conclusion: A = C
Reason: Division Postulate or Halves of equals are equal
Linear Pair Postulate
ex.
Given: 1 and 2 form a linear pair
Conclusion: 1 and 2 are supplementary
Reason: Linear pairs are supplementary
THEOREMS
Vertical angle congruence theorem
1
2
ex.
Given: The diagram above
Conclusion: 1 and 2 are vertical angles
Reason: definition of vertical angles
Conclusion: 1  2
Reason: vertical angles are congruent
Right angles congruence theorem
ex.
Given: the diagram above
Conclusion: 1 and 2 are right angles
Reason: definition of right angles
Conclusion: 1  2
Reason: all right angles are congruent
Congruent supplements theorem
ex.
Given: 1 and 2 are supplementary and 3 and 2 are
supplementary
Conclusion: 2  2
Reason: Reflexive Postulate
Conclusion: 1  3
Reason: supplements to congruent angles are congruent
Congruent complements theorem
ex.
Given: 1 and 2 are complementary and 3 and 2 are
complementary
Conclusion: 2  2
Reason: Reflexive Postulate
Conclusion: 1  3
Reason: complements to congruent angles are congruent
ISOSCELES TRIANGLE THEOREMS
I
J
H
ex.
Given: diagram above
Conclusion: segment IH segment IJ
Reason: If two angles in a triangle are congruent, then the sides
opposite those angles are congruent.
I
H
ex.
J
Given: diagram above
Conclusion: IHJ  IJH
Reason: If two sides in a triangle are congruent, then the angles
opposite those sides are congruent.
PARALLEL LINES THEOREMS
If two parallel lines are cut by a transversal, the alternate interior angles are
congruent.
If two parallel lines are cut by a transversal, the alternate exterior angles
are congruent.
If two parallel lines are cut by a transversal, the corresponding angles are
congruent.
If two parallel lines are cut by a transversal, the same side interior angles
are supplementary.
If two lines are cut by a transversal and the alternate interior angles are
congruent, then the lines are parallel.
If two lines are cut by a transversal and the alternate exterior angles are
congruent, then the lines are parallel.
If two lines are cut by a transversal and the corresponding angles are
congruent, then the lines are parallel.
If two lines are cut by a transversal and the same side interior angles are
supplementary, then the lines are parallel.
TRIANGLE THEOREMS
If two angles of one triangle are congruent to two angles of another
triangle,the third pair of angles must be congruent.
The sum of the two smallest sides in a triangle must be greater than the
third side to form a triangle.
In any triangle, the largest side is across from the largest angle and viceversa.
SSS  SSS
SAS  SAS
ASA  ASA
AAS  AAS
HL  HL
Corresponding Parts of Congruent Triangles are Congruent
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