How to Interpret a Diagram
You are allowed to assume:
1. lines and segments are straight
2. straight angles
3. collinearity of points
4. betweenness of points
5. relative position of points
You should not assume:
1. right angles
2. congruent angles
3. congruent segments
4. relative size of segments and angles
Given: AB
BC
DC
BC
B
C
Prove:
A
D
B
C
A postulate is an unproved assumption that is taken to be true.
A theorem is a proved statement.
A definition states the meaning of a term. A definition can not
be proved.
Definitions are reversible.
Postulates and theorems are not necessarily reversible.
Prove perpendicular lines form congruent angles.
The measure of the supplement of an angle is 60 less than 3 times the
complement of the angle. Find the measure of all three angles.
K
Given: Diagram as shown.
Prove: GHK is supp. to KHJ.
G
H
J
If angles are supplementary to the same angle, then they are congruent.
If angles are supplementary to congruent angels, then they are congruent.
If angles are complementary to the same angle, then they are congruent.
If angels are complementary to congruent angles, then they are congruent.
If segments are congruent, their like multiples are congruent
(Multiplication Property for Segments)
Given: AB
CD
EF = 2(AB)
GH = 2(CD)
Prove: EF
GH
Statements
Reasons
Substitution - If two items are equal (or congruent) then one can be
used in place of the other.
If a = b and b + c = d, then a + c = d.
Transitivity - A special type of substitution, in which the ordering of the
items is important. Transitivity swaps one side for
another.
If segments are congruent to the same segment, then they are
congruent to each other (Transitive Property for Segment Congruence).
If AB
CD and CD
EF, then AB
EF
If segments are congruent to congruent segments, then they are
congruent to each other (Transitive Property for Segment Congruence).
If AB
CD, CD
EF and EF
GH, then AB
GH
If segments are equal to the same segment, then they are equal to
each other (Transitive Property for Segment Equivalence).
If AB = CD and CD = EF, then AB = EF
If segments are equal to congruent segments, then they are equal to
each other (Transitive Property for Segment Equivalence).
If AB = CD, CD = EF and EF = GH, then AB = GH
If angles are congruent to the same angle, then they are congruent to
each other (Transitive Property for Angle Congruence).
If ABC
DEF and DEF
GHJ, then ABC
GHJ
If angles are congruent to congruent angles, then they are congruent to
each other (Transitive Property for Angles Congruence).
If ABC
ABC
DEF, DEF
GHJ, and GHJ
KLM, then
KLM
If angles are equal to the same angles, then they are equal to each other
(Transitive Property for Angles Equivalence).
If m ABC = m DEF and m DEF = m GHJ, then m ABC = m GHJ
If angles are equal to congruent angles, then they are equal to each
other (Transitive Property for Angles Equivalence).
If m ABC = m DEF, m DEF = m GHJ, and m GHJ = m KLM, then
m ABC = m KLM
Vertical angles are congruent.
Given: Diagram as shown
Prove: 1
2
X
R
Statements
3
1
2
S
T
Y
Reasons