Download Ch 4 Note Sheets Key

Ch 4 Note Sheets L2 Key 3
Name ___________________________
Chapter 4:
Discovering and Proving
Triangle Properties
Note Sheets
S. Stirling
Page 1 of 13
Ch 4 Note Sheets L2 Key 3
4.1 Triangles Sum Conjectures
Name ___________________________
Read the top of page 200. These notes replace pages 200 – 202 in the book.
Rigidity is a property that triangles have. They cannot be shifted, like a quadrilateral can. They retain
their shape.
Triangle Sum Conjecture
The sum of the measures of the angles in every triangle is 180°.
Numeric Example:
Find x.
Find y.
x + x + y = 180
2x + y = 180
y = 180 – 2x
B
101
56
x
A
C
F
x
E
x
y
D
101 + 56 + x = 180
x = 180 – 157 = 23
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.
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.
Easier than 180 – 101 – 23 = 56.
S. Stirling
Page 2 of 13
Ch 4 Note Sheets L2 Key 3
Name ___________________________
Lesson 4.2 Properties of Isosceles Triangles
Remember! With If..then.. statements: If a, then b. The converse of the statement is If b, then a.
Read top of page 206. This vocabulary is review! Know it!!
Label and define the special vocabulary for the isosceles triangle ∆ VBS .
Base angles
B
sides. VB and VS
Base
S
leg
leg
A leg (of an isosceles triangle) is one of the congruent
The base (of an isosceles triangle) is the side that is not a
leg. BS
The vertex angle is the angle between the two legs. ∠ V
V
Vertex angle
The base angles are the pair of angles whose vertices are
the endpoints of the base. ∠B and ∠ S
Do group Investigation 1, Base angles of an Isosceles Triangle on WS page 2.
Isosceles Triangle Conjecture
If a triangle is isosceles, then its base angles are congruent.
V
Given ∆VBS with VB = VS , then ∠ B ≅ ∠ S or
If VB ≅ VS , then ∠ B ≅ ∠ S
S
B
***Note: Since equilateral triangles are a special case of isosceles triangles, any property
that applies to isosceles triangles also applies to equilateral triangles.
Do group Investigation 2, Is the Converse True? on WS page 3. Use a protractor.
Converse of the Isosceles Triangle Conjecture
If a triangle has two congruent angles, then it is an isosceles triangle.
If ∠ B ≅ ∠ S , then ∆VBS is isosceles
B
V
or If ∠ B ≅ ∠ S , then VB ≅ VS .
S. Stirling
Page 3 of 13
S
Ch 4 Note Sheets L2 Key 3
Lesson 4.3 Triangle Inequalities
Name ___________________________
Read of page 215. Do group Investigation 1 on WS page 5.
Triangle Inequality Conjecture
The sum of the lengths of any two sides of a triangle is greater than
the length of the third side.
Is a triangle measuring 5 cm, 4 cm and 2 cm
possible? Sketch it then explain!
Is a triangle measuring 5 cm, 3 cm and 2 cm
possible? Sketch it then explain!
Since 2 + 4 > 5, it makes a triangle.
Since 2 + 3 = 5, it makes segment NOT
a triangle.
Do group Investigation 2, Largest and Smallest Angles in a Triangle on WS page 6.
Side-Angle Inequality Conjecture (This 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. (And visa versa.)
Likewise, if one side is the shortest side, then the angle opposite the shortest side is the
smallest angle. (And visa versa.)
Which is the largest angle? The smallest
angle? Why?
Which is the largest side? Why?
D
K
5.34
2.48
J
6.3
∠K largest, its opposite the longest
side, JI .
∠I smallest, its opposite the shortest
side, JK .
S. Stirling
I
In ∆DAQ , ∠ D largest,
so AQ largest in ∆DAQ .
In ∆QAU ,
A
m∠QAU = 100
Q
36
90
23
54
100
which makes UQ the
largest in ∆QAU
since AQ < UQ .
Overall, UQ is the longest.
Page 4 of 13
57
U
Ch 4 Note Sheets L2 Key 3
Name ___________________________
Label the drawing with the following terms and define the terms:
Exterior angle is an angle that forms a linear pair
with one of the interior angles of a polygon.
Adjacent interior angle is the angle of a polygon
that forms a linear pair with a given exterior angle of
a polygon.
Remote
interior
Exterior
angle
Adjacent
interior
The remote interior angles (of a triangle) are the
interior angles of a triangle that do not share a vertex
with a given exterior angle.
Triangle Exterior Angle Conjecture
The measure of an exterior angle of a triangle is equal to the sum of the measures of the
remote interior angles.
Given: ∆ABC with exterior angle
labeled as shown.
Show:
∠ BCD and angles
x = a+b
x + c = 180 Linear pairs are supplementary
a + b + c = 180 Triangle’s angles add to 180°
Since they both equal 180, x + c = a + b + c
now subtract measure c from both sides
x = a+b
Given ∆ ABC above. If a = 50 and b = 60, what
is the measure of ∠ BCD ? Explain.
Given ∆ ABC above. If x = 80 and b = 30, what
is the measure of ∠ A ? Explain.
a+b = x
50 + 60 = 110 = x
80 = a + 50
80 – 50 = 30 = a
The exterior angle equals the sum of the
two remote interior angles.
The exterior angle equals the sum of the
two remote interior angles.
S. Stirling
Page 5 of 13
Ch 4 Note Sheets L2 Key 3
Name ___________________________
Before starting Lesson 4.4:
How do you get equal parts (for congruent triangles)?
How can you use given information to get equal sides and angles in a triangle. You will need to use your
definitions and conjectures that you already know to be true. Complete the following to help you review these
properties. Remember, to mark your diagrams (tic marks and arc marks) to show which parts are equal. Also never
assume things are congruent! You must have a definitions or conjecture to back you up!!
Ways to Get Equal 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 ,
V
∆VBS isosceles
then VB ≅ VS .
B
If
then VB ≅ VS
B
Def. Midpoint
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 .
“Same Segment”
When two triangles share
the exact same segment, you get
a pair of equal segments.
∆BET and ∆WTE share
ET ≅ TE
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
CM bisects AB ,
then AM ≅ MB .
If
M
A
If AM is median in ∆ABC ,
then BM ≅ MC .
Ways to Get Equal Angles and Segments
CD is the perp. bisector of AB ,
then AM ≅ MB
and ∠ CMA = ∠ CMB = 90° .
If
S. Stirling
M
B
If a segment is a median,
then it connects the vertex to the
midpoint of the opposite side.
Def. perpendicular bisector
B
A
Def. Median
A line (or part of a line) that passes through
the midpoint of a segment and
is perpendicular to the segment.
S
C
B
M
A
D
Page 6 of 13
C
Ch 4 Note Sheets L2 Key 3
Name ___________________________
Ways to Get Equal Angles
Isosceles Triangle Conjecture
Vertical Angle Conjecture
If a triangle is isosceles,
then its base angles are congruent.
V
If two angles are vertical,
then they are congruent.
∆VBS isosceles
or if VB ≅ VS ,
then ∠ B ≅ ∠ S .
V
E
R
S
C
If
∠ VET and ∠ CER are vertical,
then ∠ VET ≅ ∠ CER .
If
B
Definition of Angle Bisector
If you have an angle bisector,
then the ray cuts the angle
into two equal angles.
Def. perpendicular lines
A
D
B
B
M
A
C
BD bisects ∠ ABC
then ∠ ABD ≅ ∠ DBC .
C
If two lines are perpendicular,
then they intersect to form equal
90° angles.
CM ⊥ AB , then
m∠ CMA = m∠ CMB = 90° .
If
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
Corresponding Angles Conjecture
T
I
A
If two parallel lines are cut by a transversal,
then corresponding angles are congruent.
A
RA is an altitude of ∆TRI
then m∠ RAT = m∠ RAI = 90° .
E
If
∆TIA and ∆NIR share
∠ TIA ≅ ∠ RIN .
AD EK ,
then ∠ MDA ≅ ∠ K
A
R
K
D
If
I
“Same Angle”
When two triangles share
the exact same angle, you get
a pair of equal angles.
M
.
G
T
N
Alternate Interior Angles Conjecture
If two parallel lines are cut by a transversal,
then alternate interior angles are congruent.
C
A
CB AD ,
then ∠ B ≅ ∠ A .
If
S. Stirling
Page 7 of 13
B
D
Ch 4 Note Sheets L2 Key 3
Name ___________________________
Before starting Lesson 4.4:
Vocab for triangles Triangles have 6 “parts”: 3 angles and 3 sides.
angle
“is opposite” means
A
∠A is opposite BC and AC is opposite ∠B
side
∠A is between BA and AC (angles are between sides)
side
angle B
side
BC is between ∠B and ∠ C (sides are between angles)
angle
C
An included angle is an angle formed between two consecutive sides of a polygon.
A
For sides AC and BC , the included angle is ∠ C
For sides AC and BC , a non-included angle is ∠ A or ∠ B
B
For sides AB and AC , the included angle is ∠ A
C
For sides AB and AC , a non-included angle is ∠ C or ∠ B
An included side is a side of a polygon between two consecutive angles.
For angles ∠ B and ∠ A , the included side is AB
A
For angles ∠ B and ∠ A , a non-included side is AC or CB
B
For angles ∠ C and ∠ A , the included side is AC
C
For angles ∠ C and ∠ A , a non-included side is AB or CB
Lesson 4.4:
Read page 221, then complete the Triangle Congruence Shortcut Investigation, page 1 – 2.
results on page 9 and 10 of these notes.
Summarize your
Whole Class: Complete page 4 of the Triangle Congruence Shortcut Investigation. Remember you must use the
properties you know to be true!! Use your notes pages 6 and 7.
Lesson 4.5:
Complete the Triangle Congruence Shortcut Investigation, page 3.
of these notes.
Summarize your results on page 9 and 10
Whole Class: Complete page 5 of the Triangle Congruence Shortcut Investigation. Remember you must use the
properties you know to be true!! Use your notes pages 6 and 7.
S. Stirling
Page 8 of 13
Ch 4 Note Sheets L2 Key 3
Name ___________________________
The following properties to guarantee congruent triangles:
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
C
D
T
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.
O
A
G
C
I
T
Z
P
∆CAT ≅ ∆DGO
∆PIZ ≅ ∆TAC
By SSS Congruence Conjecture.
By SAS Congruence Conjecture.
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.
SAA or AAS Congruence Conjecture
If two angles and a non-included side of
one triangle are congruent to the two
angles and the same non-included side of
another triangle, then the triangles are
congruent.
I
D
O
G
D
O
I
D
O
I
G
Z
P
G
Z
P
Z
P
∆DOG ≅ ∆ZPI
∆DOG ≅ ∆ZPI
By ASA Congruence Conjecture.
By AAS Congruence Conjecture.
S. Stirling
Page 9 of 13
Ch 4 Note Sheets L2 Key 3
Name ___________________________
The following properties DO NOT guarantee congruent triangles:
SSA or ASS Congruence?
If two sides and the non-included angle of one triangle are congruent to two sides and the
non-included angle of another triangle, then the triangles are NOT necessarily congruent.
A
Draw a counterexample.
I
I
I
C
T
Z
P
∆ZIP ≅ ∆_______
Z
Z
P
P
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
∆ZIP ≅ ∆_______
By NOT necessarily congruent.
S. Stirling
Z
P
I
Z
P
Page 10 of 13
Ch 4 Note Sheets L2 Key 3
Name ___________________________
4.6 Corresponding Parts of Congruent Triangles
Note: Once you can prove that two triangles are congruent, by using the conjectures above, then any of
the corresponding parts will be equal.
Definition of Congruent Triangles:
If two triangles are congruent, then all of their corresponding parts (sides and angles) are
congruent.
Can also be stated “Corresponding Parts of Congruent Triangles are Congruent” or
CPCTC.
When you write congruent triangles you must match the corresponding vertices. (To show how the match up.)
So if you know that ∆ CAT ≅ ∆ DOG then you can
say any of the following:
∠C ≅ ∠D
∠A ≅ ∠O
∠T ≅ ∠G
CA ≅ DO
AT ≅ OG
CT ≅ DG
O
A
T D
C
G
by Def. of Congruent Triangles or by CPCTC.
How to prove (show deductively) that parts are equal.
P
Example 1:
∠ PAC ≅ ∠ TAC and PA ≅ AT
∠ PCA ≅ ∠ TCA ?
Know:
Is
∠ PAC ≅ ∠ TAC
given
PA ≅ AT
given
A
T
.
C
CA ≅ AC
Shared side
∆PAC ≅ ∆TAC
SAS Congruence
∠ PCA ≅ ∠ TCA
CPCTC or Def. Congruence
S. Stirling
Page 11 of 13
Ch 4 Note Sheets L2 Key 3
Name ___________________________
Example 2:
Given: Z is midpoint of
I
P
DI and EP .
PI ≅ DE
Prove:
Z
IZ = ZD
Z is midpoint of
DI and EP
D
Def. of Midpoint
∠PZI ≅ ∠EZD
given
PZ = ZE
Vertical angles =
Def. of Midpoint
∆ZIP ≅ ∆ZDE
SAS Congruence
PI ≅ DE
CPCTC or Def. Congruence
Example 3:
Know:
Is
R
E
RE TC and TR CE .
RE ≅ EC ?
What is congruent?
RE TC
given
∠ RET ≅ ∠ ETC
||, so alt. int. ∠ s =
T
C
TR CE
given
TE = TE
∠ RTE ≅ ∠ CET
||, so alt. int. ∠ s =
Shared side
∆TRE ≅ ∆ECT
and EC are
NOT corresponding
sides. So, probably not
equal!
RE
ASA Congruence
RE ≅ TC , RT ≅ EC or
∠R ≅ ∠C
Try Example A and Example B on Ch 4 Worksheet page 13. CPCTC or Def. Congruence
S. Stirling
E
Page 12 of 13
Ch 4 Note Sheets L2 Key 3
4.8 Proving Special Triangle Conjectures
Name ___________________________
Equilateral/Equiangular Triangle Conjecture
Every equilateral triangle is equiangular.
Conversely, every equiangular triangle is equilateral.
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.
Isosceles Triangle with vertex angle A.
A
Bisect vertex angle A, label the intersection X.
Measure all parts of the resulting figure.
List all possible congruent parts:
m∠ C = m ∠ B
base angles congruent
m∠ CAX = m∠ BAX
C
angle bisector
m∠ CXA = m∠ BXA = 90
AX is an altitude of ∆CAB
B
X
CA = BA
legs of isosceles triangle
CX = XB
AX is a median of ∆CAB
AX is a perpendicular bisector of
∆CAB
Isosceles Triangle with vertex angle A, and base angle C.
Bisect base angle C, label the intersection X. Z
Draw median CM .
Draw perpendicular bisector MY .
Draw altitude CZ .
A
Y
X
M
B
C
Are any of these the same segment? NO
S. Stirling
Page 13 of 13