THEOREMS WE KNOW PROJECT
1
Name_________________ Date___________
Period_____
This is a list of all of the theorems that you know and that will be helpful when
working on proofs for the rest of the unit. In the Notes section I would like you to
write anything that will help you remember the theorem, such as an example
problem, writing the theorem in your own word, a picture of what the
theorem represents, etc. In the Proof section I would like you write the proof of
the theorem. These are all theorems that you have seen and/or written the proof of
before in math 3 or in previous classes. If you do not remember the proof use your
book, the internet (remember to cite your source), your classmates, and as always
Mr. G and I as a resource.
The two proofs that you are responsible for are due Monday 2-3-14.
Notes and 3 questions on proofs are due Wednesday 2-5-14
The final project will be due Friday 2-7-14.
Theorems of Geometry
Angles:
If two angles are supplements of the same angle, then they are equal in
measure.
Notes:
Proof:
Statements
Reasons
<DAB≅<HEF
Given
<DAB+<BAC=180° Definition of
B
supplementary
angles
<HEF+<FEG=180° Definition of
A
D
C
supplementary
F
angles
180-<DAB=<BAC
Property of
subtraction
180-<HEF=<FEG
Property of
H
E
G
subtraction
180-<HEF=<BAC
Substitution
property
<BAC and <FEG
Properties of
are equal to 180equality
<DAG therefore
they are equal
<BAC=<FEG
Equality
If two angles are complements of the same angle, then they are equal in
measure
Statements
Reasons
<DAB≅<HEF
Given
<DAB+<BAC=90° Definition of
complementary
angles
2
<HEF+<FEG=90°
90-<DAB=<BAC
90-<HEF=<FEG
180-<HEF=<BAC
<BAC and <FEG
are equal to 180<DAG therefore
they are equal
Notes:
Definition of
complementary
angles
Property of
subtraction
Property of
subtraction
Substitution
property
Properties of
equality
Proof:
Prove: <BOD = <DBA
Know:
 <BOD+<COB=180° by
supplementary angles
 <COA+<COB=180° by
supplementary angles
 If <COA+<COB=180 then
<COA=180-<COB
 If <BOD+<COB=180 then
<BOD=180-<COB
 Therefore <BOD=<COA
The sum of the measures of the angles of a triangle is 180.
Notes:
Proof:
An exterior angle of a triangle is equal in measure to the sum of the
measures of its two remote interior angles.
Notes:
Proof:
THEOREMS WE KNOW PROJECT
Name_________________ Date___________
Period_____
If two sides of a triangle are equal in measure, then the angles opposite
those sides are equal in measure.
Notes:
Proof:
Statement
Reasons
Given
Each angle has
one unique
angle bisector
An angle
bisector is an
ray whose
endpoint is the
vertex of the
angle and
which divides
the angles into
two congruent
angles
Reflexive
property a
quantity is
congruent to
itself.
SAS- if two
sides and the
included angle
of one triangle
are congruent
to the
corresponding
parts of
another
triangle, the
triangles are
congruent.
C.P.C.T.C.
If two angles of a triangle are equal in measure, then the sides opposite
3
4
those angles are equal in measure
Notes:
Proof:
If a triangle is equilateral, then it is also equiangular, with three 60
angles
Notes:
Proof:
Statement
Reason
ΔABC is equilateral Given
AC @ BC; AB @ AC Def. of
equilateral
triangle
<A≅<B;<B≅<C
Isosceles
triangle
theorem
<A≅<C
Transitive
property
ΔABC is equal
All angles are
angular
equal
360°÷3=60°
Property of
division
All 3 angle
Division
measures are 60°
above.
If a triangle is equiangular, then it is also equilateral.
Notes:
Proof:
A
Statement
Reason
<A≅<B
Given
<B≅<C
Given
Two angles
AB @ BC
congruent,
opposite sides
B
C
are congruent
Two angles are
BC @ AC
congruent
opposite sides
are congruent
Transitive
AB = AC
property
THEOREMS WE KNOW PROJECT
5
Name_________________ Date___________
Period_____
The sum of the angle measures of an n-gon is given by the formula
S(n)=(n-2)180
Notes:
Proof:
The sum of the exterior angle measures of an n-gon, one angle at each
vertex is 360.
Notes:
Proof:
Lines
If two parallel lines are intersected by a transversal, then alternate interior angles
are equal in measure.
Notes:
Proof:
Given: a||b
Prove: <1≅<3
Angle 1 is equal to angle 4 because
corresponding
angels are equal. Angle 3 is
6
4
equal to Angle 4 because of vertical angles
3
5
theorem. Angle 1 is equal to angle 3
1
because of transitive property. Therefor if
2
a transversal intersects 2 parallel lines
alternate interior angles are equal.
If two parallel lines are intersected by a transversal, the co-interior angels are
6
supplementary.
Notes:
Proof:
If two lines are intersected by a transversal and corresponding angles are equal in
measure, then the lines are parallel.
Notes:
Proof:
Statements
Reasons
<ACL≅<MAR
Given
S
(Corresponding
Angles)
L
P
C
<PCS≅<ACL
Vertical Angles
<MAR≅<QAC
Vertical Angles
<MAR+<QAM=180° Supplementary
Q
A
angles
R
<MAR+<CAR=180°
Supplementary
M
angles
<QAM≅<CAR
If two angles are
supplementary to
the same angle
they are
congruent
<CAR≅<SCL
Corresponding
angles
<PCA≅<SCL
Vertical angles
The transversal
PL || QR
intersects the
two lines with
the same angles.
If two lines are intersected by a transversal and alternate interior angles are equal
in measure, then the lines are parallel.
Notes:
Proof:
7
THEOREMS WE KNOW PROJECT
Name_________________ Date___________
Period_____
If two lines are intersected by a transversal and co-interior angles are
supplementary, then the lines are parallel.
Notes:
Proof:
If two lines are perpendicular to the same transversal, then they are parallel.
Notes:
Proof: Lines k and l are cut by t, the
transversal. <1(top right of line k) and
<5(top of line “l” left) are corresponding
angles, along with <3(bottom right of line
k) to <7(bottom right of line l), <2(top left
of line k) to <6(bottom left of line l), and
<4(bottom left of line k) to <8(bottom left
of line l). The definition of corresponding
angles is, “if two parallel lines are cut by a
transversal, then the corresponding angles
are congruent”. The converse of that
statement is, “if the corresponding angles
are congruent, the lines are parallel.” Since
all angles equal 90 degrees, all
corresponding angles are congruent. Thus,
two line perpendicular to a transversal are
parallel.
If a transversal is perpendicular to one of two parallel lines, then it is perpendicular
to the other one also.
Notes:
Proof: What is given? r || l,t ^ r
What do you need to prove? r ^ l
Statements
r || l,t ^ r
<l is a right angle
M<l=90°
m<1≅m<2
Reason
Given
Def. of
perpendicular lines
Def. of right angles
Corresponding
8
angles
Def. of congruent
angles
m<2=90°
Substitution
property
<2 is a right angle Def. of right angle
t ^l
Def. of
perpendicular lines
If a point is the same distance from both endpoints of a segment, then it lies on the
perpendicular bisector of the segment.
Notes:
Proof:
Statement
Reason
m<1=m<2
Triangles:
If a line is drawn from a point on one side of a triangle parallel to another side, the it
forms a triangle similar to the original triangle
Notes:
Proof:
In a triangle, a segment that connects the midpoints of two sides is parallel to the
third side and half as long.
Notes:
Proof:
9
THEOREMS WE KNOW PROJECT
Name_________________ Date___________
Period_____
If two angles and the included side of one triangle are equal in measure to the
corresponding angles and side of another triangle, then the triangles are congruent.
(ASA)
Notes:
Proof:
<ABC=<ADC
Given
Line DC=Line BC
Given
Angle DCA= Angle
ACB
Line AC= Line AC
Given
Triangle ADCTriangle ABC
Reflexive property of
equality
SAS
If two angles and a non-included side of one triangle are equal in measure to the
corresponding angles and sides of another triangle, then the two triangles are
congruent. (AAS)
Notes:
Proof:
If two sides and the included angle of one triangle are equal in measure to the
corresponding sides and angle of another triangle, then the triangles are congruent.
(SAS)
Notes:
Proof:
10
If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles
formed are similar to the original triangle and to each other.
Notes:
Proof:
In any right triangle, the square of the length of the hypotenuse is equal to the sum
of the squares of the lengths of the legs
Notes:
Proof:
If the altitude is drawn to the hypotenuse of a right triangle, then the measure of the
altitude is the geometric mean between the measures of the parts of the hypotenuse.
Notes:
B
Statements
Reasons
h=altitude
By definition
ΔABD is similar to
Altitude creates
h
ΔBCD
similar triangles
A
Properties of similar
BD DC
D
C
=
triangles
AD BD
Properties of similar
h
DC
=
triangles
AD
h
Properties of ratio
Definition of
h = AD· DC
geometric mean.
The sum of the lengths of any two sides of a triangle is greater than the length of the
third side.
Notes:
Proof:
A
Statements
Reasons
h2 = AB· DC
BE is the shortest
distance from vertex
B to AE
BA>BE.
BA^2=AE^2+BE^2AB>BC
short distance
theorem
Pythagorean
theorem
1
1
THEOREMS WE KNOW PROJECT
Name_________________ Date___________
Period_____
B
C
Triangle AEC=
Pythagorean
AC^2=AE^2+EC^2= theorem.
AC>EC
AC^2=
EC^AE^2
AB^2-BE^2= EC^2AE^2
AB+AC>BE+BC
AB+AC>BC
substitution
property
addition property
segment addition
postulate
In an isosceles triangle, the medians drawn to the legs are equal in measure.
Notes:
Proof:
a
e
Statement
ΔABC is isosceles
Draw medians
BD and CE
d
Reasons
Given
Through any 2
points there is 1 line
Properties of an
isosceles triangle
Definition of
AB = AC
congruence
Multiplication
1
1
AB = AC
property
2
2
A median bisects the
1
1
BE = AB; DC = AC line it passes
2
2
through
Substitution
BE = DC
property
Definition of
BE @ DC
congruence
<B≅<C
Property of an
isosceles triangle
Reflexive property
BC @ BC
ΔEBC≅ΔDCB
SAS theorem
C.P.C.T.C.
CE @ DC
AB @ AC
b
c
Quadrilaterals:
In a parallelogram, the diagonals have the same midpoint.
Notes:
Proof:
Statement:
A quadrilateral ABCD is
Proof:
Given
12
a parallelogram if AB is
parallel to CD and BC is
parallel to DA.
AB ll CD
Definition of a
parallelogram
L BAE is congruent to L
DCE
Alternate interior
angles postulate
AB is congruent to CD
Opposite sides in
a parallelogram
L ABE is congruent to L
CDE
Alternate interior
angles postulate
Triangle AEB is congruent
to triangle DEC
ASA
AE is congruent to EC
CPCTC
BE is congruent to ED
CPCTC
In a kite, the diagonals are perpendicular to each other.
Notes:
Proof:
In a rectangle, the diagonals are equal in measure.
Notes:
Proof:
In a parallelogram, opposite sides are equal in measure.
Notes:
Proof:
Statement
B
<ABD≅<BDC
Reason
Alternate interior
angles
THEOREMS WE KNOW PROJECT
A
D
C
1
3
Name_________________ Date___________
Period_____
<DBC≅<ADB
Alternate interior
angles
Reflexive property
DB @ DB
ΔADB≅ΔCBD
ASA
C.P.C.T.C.
AB @ DC; AD @ BC
If a quadrilateral is a parallelogram, then consecutive angles are supplementary.
Notes
Proof: Lets consider two consecutive angles DAB
and ABC. Draw the straight line AE as the
continuation of the side AB of the parallelogram
ABCD. Then the angle CBE is congruent to the
angle DAB as these angles are the corresponding
angles at the parallel lines AC and BC and the
transverse AE. The angles ABC and CBE are
adjacent supplementary angles and make in sum
the straight angle ABE of 180°. Therefore, two
consecutive angles DAB and ABC are non-adjacent
supplementary angles and make in sum the
straight angle of 180°. Similarly, consider two
other consecutive angles ABC and BCD. Draw the
straight line BF as the continuation of the side BC
of the parallelogram ABCD. Then the angle DCF is
congruent to the angle ABC as these angles are the
corresponding angles at the parallel lines DC and
AB and the transverse BF. The angles BCD and DCF
are adjacent supplementary angles and make in
sum the straight angle BCF of 180°. Therefore, two
consecutive angles ABC and BCD are non-adjacent
supplementary angles and make in sum the
straight angle of 180°. You can repeat these steps
for the other two sets of consecutive angles.
Therefore if a quadrilateral is a parallelogram
then all the of the consecutive angles are
supplementary.
If a quadrilateral is a parallelogram, then opposite angles are equal in measure.
Notes:
Proof:
Statements
Reasons
E
Given
AD || BC
14
D
CD || AB
<BCD≅<CDE
<CDE≅<BAD
<BCD≅<BAD
<FAB≅<ABC
A
C
F
<FAB≅<ADC
<ABC≅<ADC
Given
Alternate int. Angles
Corresponding angles
Transitive property
Alternate interior
angles
Corresponding angles
Transitive property
B
The sum of the measures of the angles of a quadrilateral is 360.
Notes:
Proof:
Quadrilaterals can be
Definition of a
divided into two
quadrilateral
triangle
The angles of triangles Triangle Angle Sum
are equal to 180
Theorem
degrees
Two triangles angles
Additive property of
add up to 360 degrees
addition
Quadrilaterals angles
Substitution property
add up to 360 degrees
of addition
Help from: http://www.mathwords.com/
a/additive_property_of_equality.htm
If both pairs of opposite angles of a quadrilateral are equal in measure, then the quadrilateral is
a parallelogram.
1
5
THEOREMS WE KNOW PROJECT
Name_________________ Date___________
Period_____
Notes:
Proof: We need to prove the opposite angles are
congruent. So, we need to prove that L A = L C
and L B = L D.
Statement:
Reason:
LCBE + LCBA =
180degrees, LFCB +
LDCB = 180 degrees.
Supplementary angles
theorem
LCBE is congruent to
LDAB
LBCF is congruent to
LADC
Corresponding angles
postulate
LCBE is congruent to
LBCD
LBCF is congruent to
LABC
Alternate interior angles
postulate
Hence, LDAB is
congruent to LDCB
Steps 1,2, and 3
If the two diagonals of a quadrilateral bisect each other, then the quadrilateral is a
parallelogram.
Notes:
Proof:
B
Statements
Reasons
Quadrilateral ABCD given
A
P
C
D
Line AP is
congruent to PD.
Line BP is congruent
to PC
Angle APB is
congruent to angle
CPD
Triangle ABP is
congruent to triangle
CPD
Angle BCD is
congruent to angle
CBA
Angle BCD is
congruent to angle
CBA
AB||DC
Diagonals bisect so
diagonals bisect each
other
vertical angle
theorem.
SAS
CPCTC
alternate interior
theorem
converse of parallel
transversal theorem
16
line AP is congruent
to PD and line CP is
congruent to BD
Angle CAP is
congruent to angle
BDC
Angle APC is
congruent to angle
DBP
Triangle APC is
congruent to
Triangle BDP
Angle ABC is
congruent to angle
BCD
Line AC||BD
alternate interior
theorem
vertical angles
theorem
SAS
alternate interior
angle theorem
converse of parallel
transversal theorem
Quadrilateral ABCD definition of
parallelogram
In an isosceles trapezoid, (1) the legs are equal in measure, (2) the diagonals are equal in
measure, and (3) the two angles at each base are equal in measure.
Notes:
Proof:
a
b
Statement
Reasons
c
d
Trapezoid ABCD is
isosceles
Given
<D and <C are base
angles
Definition of base
angles
<D≅<C
Properties of an
isosceles trapezoid
AD @ BC
Given
Draw diagonal
segments AC and BC
Through any two
points, there is exactly
one line
DC @ DC
Reflexive property of
congruence
ΔADC ≅ΔBDC
SAS theorem
1
7
THEOREMS WE KNOW PROJECT
Name_________________ Date___________
Period_____
C.P.C.T.C
AC @ BD
Rubric:
Theorems We Know Project ____/50
40 Points
35 Points
30 Points
20 Points
10 Points
All
39-32
31-24
23-16
15-8
theorems
theorems
theorems
theorems
theorems
have notes. have notes. have notes
have notes. have notes
Proofs (points taken off for each missing proof out of 10):
0 Points
8 or less
theorems
have notes.