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Justifications Review
Justifications come in three forms: properties, definitions and theorems. They are commonly used in proofs in
geometry. You know other geometric definitions and properties, but they are less commonly used in geometric
proofs.
Properties that are frequently used in geometric proofs:
Reflexive Property (of equality or congruence)
(a = a)
Symmetric Property (of equality or congruence)
(if a = b then b = a)
Transitive Property (of equality, inequality, or congruence)
(if a = b and b = c then a = c)
Angle addition property.
(If ABC and CBD are adjacent then mABC + mCBD = mABD)
Definitions that are frequently used in geometric proofs:
Definition of midpoint
If M is the midpoint of AB then AM = MB (or AM MB)
If a point cuts a segment into 2 equal parts, then it is the midpoint of the
segment.
Definition of right angle
If an angle is a right angle then it has measure of 90°.
If an angle has measure 90°, then it is a right angle.
Definition of congruence
If two figures are isometries of one another, then they are congruent.
If two figures are congruent, then they are isometries of one another.
Definition of circle
A circle is the set of all points in a plane at a certain distance (its radius)
from a certain point (its center.)
Definition of angle bisector
(see middle of p. 127) – you can use the name for this one
Definition of reflection
(see top of p. 185) – you can use the name for this one
Definition of supplementary angles
The measures of two angles add to 180° iff they are supplementary.
Definition of complementary angles
The measures of two angles add to 90° iff they are complementary.
You must also know the definitions of scalene, isosceles, and equilateral triangles.
Theorems that are frequently used in geometric proofs:
|| lines alternate interior angles 
and
alternate interior angles || lines
|| lines alternate exterior angles 
and
alternate exterior angles || lines
|| lines corresponding angles 
and
corresponding angles || lines

Vertical angles are congruent
Reflections preserve angle measure, betweenness, collinearity, & distance (ABCD Theorem)
The sum of the measures of the angles of a triangle is 180°
The sum of the measures of the angles of a polygon with n sides is 180(n-2)
If a point lies on the line of reflection, then it is its own reflection.
A. Prove that this construction creates an equilateral triangle.
C
a. Draw a segment (AB)
b. Draw two circles with segment AB as a radius.
c. Draw a dot at the intersection point of the two circles (C)
A
d. Connect the endpoints of the original segment to the dot
B
you just drew.
Prove: ∆ABC is equilateral
Statements
Justifications
1. AB = AC
1. ________________________________________________________
2. AB = BC
2. ________________________________________________________
3. AC = BC
3. ________________________________________________________
4. ∆ABC is equilateral
4. ________________________________________________________
B. Given the figure, prove that 4 24. (All lines that look parallel are parallel.)
Statements
Justifications
0. All lines that look
parallel are parallel
0.Given
1. 4 12
1.
2. 12 14
2.
__
.
3. 4 10
3.
__
.
4. 10 22
4.
__
.
5. 22 24
5.
__
.
6. 4 24
6.
__ .
.
1
L
2
8
__ .
3
7
4
6
10
9
16
15
18
17
Given: LK is the  bisector of JH.
Prove: ∆LJK  ∆LHK
J
__
24
11
12
14
13
20
19
23
5
22
21
Conclusions
Justifications
0. LK is the 
bisector of JH.
_________________________________
1. ___________
_________________________________
2. ___________
_________________________________
3. ___________
_________________________________
4. ___________
_________________________________
5. ___________
_________________________________
H
K