Download Ch. 4 Note Sheet L1 Name: A.Simons Page 1 of 18 Rigidity is a

Ch. 4 Note Sheet L1
Name:_____________________________
Triangulation is a procedure
used by surveyors to locate
Read top
of pg 200.
positions by using a network of
triangles with their angles and
distance measurements.
Rigidity
is a property of
triangles, meaning they retain
their shape and can’t be shifted,
unlike
quadrilaterals
.
1
1
3
2
1
1
2
3
2
3
m∠1 + m∠2 + m∠3 = 180
Triangle Sum Conjecture:
The sum of the measures of the angles in every triangle is 180
A.Simons
Page 1 of 18
.
Ch. 4 Note Sheet L1
Name:_____________________________
Numeric Example:
Find x.
Find y.
F
B
101
56
A
C
x
x
x + x + y = 180
2x + y = 180
y = 180 – 2x
E
x
y
D
101 + 56 + x = 180
x = 180 – 157 = 23
Proof: (What definitions and conjectures do we already know?)
Given:
Show:
HJJG
ΔABC with auxiliary line EC & AB .
m∠ 2 + m∠ 4 + m∠5 = 180°
m∠1 + m∠ 2 + m∠3 = 180° Linear pair conjecture
HJJG
HJJG
AC and CB form transversals between parallel lines EC and AB
m∠1 = m∠ 4 and m∠3 = m∠5 because AIA are congruent
Substituting into the first equation above
m∠ 2 + m∠ 4 + m∠5 = 180°
Therefore, the sum of the measures of the angles in every triangle is 180°.
Measure:
∠A , ∠B , ∠E and ∠F , what can
we conclude about
∠C
and
∠D ?
Explain!
F
B
A
C
E
D
Third Angle Conjecture:
If two angles of one triangle are congruent to two angles of another triangle, then the
third angles of the triangles are congruent.
A.Simons
Page 2 of 18
Ch. 4 Note Sheet L1
Name:_____________________________
Numeric Example:
If m∠ K = m∠B and m∠ I = m∠ C , find y. Give reasons for your
answer!
A
K
y
B
101
56
J
23
I
C
y = 56 because the third angles are equal also.
B
Proof: (What definitions and coonjectures do we already know?)
Given:
Show:
m∠A = m∠E and m∠B = m∠F
m∠ C = m∠ D
C
A
F
m∠A + m∠B + m∠C = 180° and
m∠E + m∠F + m∠D = 180° by the triangle sum conjecture.
Since they both equal 180,
m∠A + m∠B + m∠C = m∠ E + m∠F + m∠ D
Now subtract equal measures m∠ A = m∠ E and m∠ B = m∠ F .
m∠C = m∠D Therefore, the third angles are always congruent.
A.Simons
E
Page 3 of 18
D
Ch. 4 Note Sheet L1
Name:_____________________________
Angle between
the two legs.
VERTEX
ANGLE
One of the
congruent sides.
Pair of angles
whose vertices
are the endpoints
of the base.
LEG
LEG
Side that is not
a leg.
BASE
BASE
ANGLE
BASE
ANGLE
Read pg 206. Do investigation 1 pg 207.
Isosceles Triangle Conjecture:
If a triangle is isosceles, then its base angles are congruent.
Given +VBS with VB = VS , then ∠B ≅ ∠S . OR
B
If VB = VS , then ∠B ≅ ∠S
V
S
**What can we say about equilateral triangles? Since they are a special case of isosceles triangle, any
property that applies to isosceles triangles also applies to equilateral triangles.
Do investigation 2 pg 208. Use a protractor.
Converse:
If a triangle has two congruent angles, then it is an isosceles
triangle.
S
B
If <B ≈ <S, then ΔVBS is isosceles OR
If <B ≈ <S, then VB ≈ VS.
V
A.Simons
Page 4 of 18
Ch. 4 Note Sheet L1
Name:_____________________________
QUICK VOCAB REVIEW!
∠A is opposite BC and AC is opposite ∠B
∠A is between BA and AC
(angles between sides)
BC is between ∠B and ∠C
(sides between angles)
B
C
A
INCLUDED ANGLE: angle
formed between two consecutive sides of a polygon.
INCLUDED SIDE: side of a polygon between two consecutive angles.
Read pg 215. Do investigation 1 pg 216.
Triangle Inequality Conjecture:
The sum of the lengths of any two sides of a triangle is greater
than the length of the third side.
Construct a triangle with sides
measuring 5cm, 4cm, and 2cm. Is this
possible?
Construct a triangle with sides
measuring 5cm, 3cm, and 2cm. Is this
possible?
Since 2+4> 5, it makes a triangle!
Since 2+5 =5, it makes a segment not
a triangle!
Do investigation 2 pg 217.
Side-Angle Inequality Conjecture: (Only applies to triangles!)
In a triangle, if one side is the longest side, then the angle opposite
the longest side is the largest angle.
LIKEWISE!
SHORTEST
SIDE…SMALLEST
ANGLE
A.Simons
LONGEST
SIDE…LARGEST
ANGLE
If one side is the shortest side,
then the angle opposite the
shortest side is the smallest
side.
Page 5 of 18
Ch. 4 Note Sheet L1
Name:_____________________________
Which is the largest angle? Which is
the smallest angle? Why?
D
Which is the largest side? Why?
Q
90
23
K
<K largest, its opp.
The longest side JI.
<I smallest, its opp
The shortest side JK.
J
< D largest, so AQ largest in
ΔADQ, But m<QAU = 100
Which makes UQ the largest in
ΔQAU and AQ < UQ
Overall, UQ is the longest.
5.34
2.48
I
6.3
54
A
57
U
Do investigation 3 pg 217-218.
Triangle Exterior Angle Conjecture:
6.3
The measure
of an exterior angle of a triangle is equal to the
sum
57
of the measures of the remote interior angle.
U
B
Remote Interior
Angles:
Interior angles of a
triangle that do not share a
vertex with a given exterior
angle.
Adjacent Interior
Angles:
b
<A and
<B
a
A
<C and
<X
c
Angle of a polygon that
forms a linear pair with a
given exterior angle of a
polygon
x
C
<X and
<C
D
Exterior Angles:
Angle that forms a linear
pair with one of the interior
angles of the polygon.
Given ABC above. If a=50 and b=60,
what is the measurement of <BCD?
Explain!
Given ABC above. If x = 80 and
b = 30, what is the measurement of
<A? Explain!
50+60=110 The exterior angle equals
the sum of the two remote interior
angles.
80-30=50 the exterior angle equals
the sum of the two remote interior
angles.
A.Simons
Page 6 of 18
Ch. 4 Note Sheet L1
Name:_____________________________
Read top of pg 221. Complete Triangle Congruence Shortcut Investigation (pg1-3).
SSS Congruence Conjecture:
If the three sides of one triangle are congruent to the three sides
of another triangle, then the triangles are congruent.
A
D
A
O
C
C
T
T
G
O
ΔCAT ≅ Δ_______
ΔCAT ≅ Δ_______
By SSS Congruence Conjecture.
By SSS Congruence Conjecture.
SAS Congruence Conjecture:
If two sides and the included angle of one triangle are
congruent to two sides and the included angle of another
triangle, then the triangles are congruent.
A
I
I
P
Z
C
T
Z
P
D
E
ΔPIZ ≅ Δ_______
ΔZED ≅ Δ_______
By SAS Congruence Conjecture.
By SAS Congruence Conjecture.
A.Simons
Page 7 of 18
Ch. 4 Note Sheet L1
Name:_____________________________
ASA Congruence Conjecture:
If two angles and the included side of one triangle are
congruent to two angles and the included side of another
triangle, then the triangles are congruent.
D
O
O
I
C
G
Z
P
P
A
ΔDOG ≅ Δ_______
ΔCOP ≅ Δ_______
By ASA Congruence Conjecture.
By ASA Congruence Conjecture.
SAA or AAS Congruence Conjecture:
If two angles and a non-included side of one triangle
are congruent to the corresponding angles and side of
another triangle, then the triangles are congruent.
D
O
P
I
A
G
Z
P
T
C
ΔPIZ ≅ Δ_______
ΔCAP ≅ Δ_______
By SAS Congruence Conjecture.
By SAS Congruence Conjecture.
A.Simons
Page 8 of 18
Ch. 4 Note Sheet L1
Name:_____________________________
Hypotenuse Leg Congruence Conjecture: (SPECIAL CASE)
If the hypotenuse and one leg of a right triangle are congruent to the
hypotenuse and one leg of another right triangle, then the two triangles
are congruent.
O
D
O
I
C
G
Z
P
P
A
ΔDOG ≅ Δ_______
ΔCOP ≅ Δ_______
By HL Congruence Conjecture.
By HL Congruence Conjecture.
SSA or ASS?
If two sides and the no-included angle of one
triangle are congruent to two sides and the non-included
angle of another triangle, then the triangle are NOT
necessarily congruent.
A
C
Draw a counterexample.
I
T
Z
I
P
Z
P
ΔZIP ≅ Δ_______
By NOT necessarily congruent.
AAA Congruence Conjecture:
If three angles of one triangle are congruent to the corresponding
angles of another triangle, then the triangles are NOT necessarily
congruent.
A
C
Draw a counterexample.
I
T
Z
I
P
Z
P
ΔZIP ≅ Δ_______
By NOT necessarily congruent.
A.Simons
Page 9 of 18
Ch. 4 Note Sheet L1
Name:_____________________________
To see if you have congruent triangles, you will be checking for a SSS, SAS, ASA or AAS
marked on the matching pair of triangles that you be given (as shown above). Sometimes the parts will
be marked equal in the diagram. That’s the easy stuff. Other times, you will be given information that
you will “translate” into equal sides and angles in order to get your congruence. You will need to
deduce this information from definitions or conjectures that you already know to be true. Complete the
following to help you review these statements. Remember, to mark your diagrams with the equal parts.
Also never assume things are congruent! You must have a definitions or conjecture to back you up!!
SEGMENTS!
Converse of the Isosceles Triangle Conj.
Def. Isosceles Triangle
If a triangle has two congruent angles,
then it is an isosceles triangle.
V
If a triangle is isosceles,
then its legs are congruent.
S
If ∠ B ≅ ∠ S ,
then VB ≅ VS
ΔVBS isosceles
then VB ≅ VS .
B
Def. segment bisector
S
If a point is a midpoint,
then it divides the segment into
two equal segments.
G
If M is the midpoint of SG ,
then SM ≅ MG .
When two triangles share
the exact same segment, you get
a pair of equal segments.
ΔBET and ΔWTE share
ET ≅ TE
A.Simons
B
W
O
E
C
If a line (or part of a line) is a bisector,
then it passes through the
midpoint of the segment.
M
T
S
B
If
Def. Midpoint
“Same Segment”
V
CM bisects AB ,
then AM ≅ MB .
If
B
M
A
B
Def. Median
If a segment is a median,
then it connects the vertex to the
midpoint of the opposite side.
If AM is median in ΔABC ,
then BM ≅ MC .
Page 10 of 18
M
A
C
Ch. 4 Note Sheet L1
Name:_____________________________
ANGLES!
Isosceles Triangle Conjecture
If a triangle is isosceles,
then its base angles are congruent.
V
ΔVBS isosceles
or VB ≅ VS ,
then ∠B ≅ ∠S .
If two angles are vertical,
then they are congruent.
S
If
V
E
R
C
If ∠ VET and ∠ CER are vertical,
then ∠ VET ≅ ∠ CER .
B
Definition of Angle Bisector
If you have an angle bisector,
then the ray cuts the angle
into two equal angles.
A
Def. perpendicular lines
B
JJJG
BD bisects ∠ABC
then ∠ ABD ≅ ∠ DBC .
C
If two lines are perpendicular,
then they intersect to form equal
90° angles.
D
C
If
B
M
D
A
CD ⊥ AB , then
m∠CMA = m∠CMB = 90° .
If
R
Def. Altitude
If a segment is an altitude,
then it goes from a vertex
perpendicular to the line that
contains the opposite side.
T
Vertical Angle Conjecture
Corresponding Angles Conjecture
E
If two parallel lines are cut by a transversal,
then corresponding angles are congruent.
T
I
A
RA is an altitude of ΔTRI
then m∠ RAT = m∠ RAI = 90° .
If
B
A
HJJG HJJG
If AB & DC ,
then ∠ EFA ≅ ∠ FGD .
F
C
G
D
Alternate Interior Angles Conjecture
“Same Angle”
When two triangles share
the exact same angle, you get
a pair of equal angles.
If two parallel lines are cut by a transversal,
then alternate interior angles are congruent.
I
A
R
G
T
N
ΔTIA and ΔNIR share
∠TIA ≅ ∠RIN .
HJJG HJJG
If LK & NJ ,
then ∠KHM ≅ ∠HMN
Def. perpendicular bisector
ANGLES AND
SIDES!
Complete the
Triangle
Congruence
Shortcut
A.Simons
J
.
N
M
B
M
CD is the perp. bisector of AB ,
then AM ≅ MB
and ∠ CMA = ∠ CMB = 90° .
If
K
L
C
A line (or part of a line) that passes through
the midpoint of a segment and
is perpendicular to the segment.
H
A
Page 11 of 18
D
Ch. 4 Note Sheet L1
Name:_____________________________
Once you can PROVE that
two triangles are congruent, by
using the conjectures above,
then ANY of the
corresponding parts will be
equal!
Congruent Triangles:
If two triangles are congruent, then
all of their corresponding parts (sides
and angles) are congruent.
Also called
Corresponding Parts
of Congruent
Triangles are
Congruent
Corresponding triangles must match corresponding vertices!
(Shows how they match-up!)
So if you know that
following:
ΔCAT ≅ ΔDOG then you can say any of the
∠C ≅ ∠D
∠A ≅ ∠O
∠T ≅ ∠G
CA ≅ DO
AT ≅ OG
CT ≅ DG
CPCTC
How to prove that parts are equal! (Show Deductively)
P
1
∠PAC ≅ ∠TAC and PA ≅ AT .
∠PCA ≅ ∠TCA ?
Know:
Is
The triangles share a side, so
So…
CA = CA
A
T
of course.
C
ΔPAC ≅ ΔTAC by SAS Congruence
and ∠ PCA ≅ ∠ TCA because CPCTC or Corresponding Parts of Congruent Triangles are Congruent.
2
Given: Z is midpoint of
Prove:
PI ≅ DE ?
DI and EP .
I
P
Z
DI and EP was given, so
IZ = ZD and PZ = ZE by Definition of Midpoint
D
∠PZI ≅ ∠EZD by the Vertical Angles Conjecture (or vertical angles are congruent.)
Therefore, Δ ZIP ≅ Δ ZDE by SAS Congruence and
PI ≅ DE because CPCTC or Corresponding Parts of Congruent Triangles are Congruent.
Z is midpoint of
A.Simons
E
Page 12 of 18
Ch. 4 Note Sheet L1
Name:_____________________________
TRICKY!
Know:
Is
R
RE & TC and TR & CE .
RE ≅ EC ?
E
What is congruent?
T
C
RE & TC given so ∠RET ≅ ∠ETC and
TR & CE given so ∠RTE ≅ ∠CET because parallel lines make equal alternate interior angles.
TE = TE the triangles share a side,
ΔTRE ≅ ΔECT by ASA Congruence. So any of the corresponding sides are congruent.
RE
and
EC are NOT corresponding sides.
Note: you could say any of the following though:
So, probably not equal.
RE ≅ TC , RT ≅ EC or ∠R ≅ ∠C .
TRICKIER!
Start by un-overlapping possible pairs of congruent triangles:
LE = EP and LV = OP
Prove: ∠ ALP ≅ ∠ APL
A
Given:
V
O
E
L
P
LE = EP so ∠ELP ≅ ∠EPL If isosceles, then base angles are equal.
LP = PL Same segment
LV = OP Given information
∠LOP ≅ ∠PVL by SAS Congruence
∠ALP ≅ ∠APL CPCTC
Try Proofs in Worksheet Packet!
A.Simons
Page 13 of 18
Ch. 4 Note Sheet L1
Name:_____________________________
4.7 Page 238 Example B Flowchart Proof
Given:
Prove:
EC ≅ AC and ER ≅ AR
∠A ≅ ∠E
E
R
C
EC ≅ AC
Given
ER ≅ AR
Given
A
ΔREC ≅ ΔRAC
∠A ≅ ∠E
SSS Cong. Conj.
CPCTC
RC ≅ RC
Same segment
4.7 Page 239 Top. Explain why the angle bisector construction works. Flowchart Proof
Given:
Prove:
∠ABC with BA ≅ BC and CD ≅ AD
BD is the angle bisector of ∠ABC .
BA ≅ BC
Given
CD ≅ AD
Given
ΔBAD ≅ ΔBCD
∠1 ≅ ∠2
SSS Cong. Conj.
CPCTC
BD ≅ BD
Same segment
BD
is the angle bisector of
∠ABC .
Def. of angle bisector
A.Simons
Page 14 of 18
Ch. 4 Note Sheet L1
Name:_____________________________
NEED EQUAL
SEGMENTS TO
GET…
B
Def. Isosceles Triangle
Def. Median
If a triangle’s legs are congruent,
then it is isosceles.
If a segment connects the vertex to the
midpoint of the opposite side,
A
then it is a median.
V
VB ≅ VS
then ΔVBS isosceles.
S
then AM is median in ΔABC .
B
Def. Midpoint
Def. segment bisector
C
If a line (or part of a line) passes through the
midpoint of the segment,
then it is a bisector.
M
S
If a point divides the segment into
two equal segments,
then it is a midpoint.
M
If SM ≅ MG , then M is
G
the midpoint of SG .
NEED EQUAL
ANGLES TO
GET….
Converse of the Isosceles Triangle Conj.
If a triangle has two congruent angles,
then it is an isosceles triangle.
V
S
B
A
D
B
∠ABD ≅ ∠DBC ,
JJJG
then BD bisects ∠ ABC .
C
Def. Altitude
C
If two lines intersect to form equal
adjacent angles,
then they are perpendicular.
B
M
D
A
Corresponding Angles Conjecture
E
If two lines are cut by a transversal and
the corresponding angles are congruent,
then the lines are parallel.
A
∠EFA ≅ ∠FGD ,
HJJG HJJG
then AB & DC .
B
F
C
If
G
D
Alternate Interior Angles Conjecture
If two lines are cut by a transversal and
the alternate interior angles are congruent,
then the lines are parallel.
R
m∠RAT = m∠RAI = 90
A °,
then RA is an altitude of ΔTRI
A.Simons
A
m∠CMA = m∠CMB ,
then CD ⊥ AB .
If
If a segment goes from a vertex
perpendicular to the line that
contains the opposite side,
then it is an altitude.
B
If
Definition of Angle Bisector
If
AM ≅ MB ,
then CM bisects AB .
If
Def. perpendicular lines
If a ray cuts the angle into two
equal angles, then you have
an angle bisector.
C
If BM ≅ MC ,
If
If ∠ B ≅ ∠ S ,
then ΔVBS isosceles.
M
T
I
.
If ∠ KHM ≅ ∠ HMN
HJJG HJJG
then LK & NJ .
H
K
L
,
J
N
M
Page 15 of 18
Ch. 4 Note Sheet L1
Need Equal
Segments and Angles
to get….
Name:_____________________________
Def. perpendicular bisector
If a line (or part of a line) that passes through the midpoint of
a segment and is perpendicular to the segment,
C
then it is the perpendicular bisector.
AM ≅ MB and ∠CMA = 90° ,
then CD is the perp. bisector of AB .
If
B
M
A
D
SUMMARY
Basic Procedure for Proofs
“parts” refers to sides and/or angles.
1. Get equal parts by using given info. and known definitions and conjectures.
2. State the triangles are congruent by SSS, SAS, ASA or AAS.
3. Use CPCTC to get more equal parts.
4. Connect that info. to what you were trying to prove .
Hints [if you get stuck]:
Mark the diagram with what you have stated as congruent in your proof.
( If given M is the midpoint of
midpoint before marking the diagram!)
AB , convert it to AM = MB by def. of
Look at the diagram to find equal parts.
Brainstorm and then apply previous conjectures and definitions.
Work (or think) backwards!
Draw overlapping triangles separately.
Re-draw figures without all of the “extra segments” in there.
Draw an auxillary line.
Break a problem into smaller parts.
A.Simons
Page 16 of 18
Ch. 4 Note Sheet L1
Name:_____________________________
Investigate. What can we conclude about Isocceles Triangles and the measurement of it’s altitude, median
and angle bisector from one base angle?
Isosceles Triangle with vertex angle A,
Isosceles Triangle with vertex angle B.
Median BM
AM = 1.64 cm
MC = 1.64 cm
A
Altitude BE
m∠ABE = 90 .
D
Median BM
Altitude BL
Angle Bisector BS
AB = 5.43 cm
AC = 5.43 cm
CB = 6.96 cm
A
VB = VS
B
C
C
Angle Bisector BD
m∠ABD = 27D
m∠DBC = 27D
B
A
L
A
E,M,D
C
S
M
B
B
C
Vertex Angle Bisector Conjecture:
In an isosceles triangle, the bisector of the vertex
angle is also the altitude and the median to the base and the
perpendicular bisector of the base.
Equilateral/Equiangular Triangle Conjecture:
Every equilateral triangle is equiangular.
Conversely, every equiangular triangle is
equilateral.
A.Simons
Page 17 of 18
Ch. 4 Note Sheet L1
Name:_____________________________
Look at the following isosceles triangles, what can we conclude about segments from the base angles?
A
A
N
M
P
A
S
T
L
C
C
C
B
B
B
Medians MC = BN
Angle Bisectors PC = BS
Altitudes BT = LC
Medians to the Congruent Sides Theorem:
In an isosceles triangle, the medians to the congruent
sides are congruent.
Angle Bisectors to the Congruent Sides Theorem:
In an isosceles triangle, the angle bisectors to the
congruent sides are congruent.
Altitudes to the Congruent Sides Theorem:
In an isosceles triangle, the altitudes to the
congruent sides are congruent.
A.Simons
Page 18 of 18