Download Chapter 4: Discovering and Proving Triangle Properties Note Sheet

Ch 4 Note Sheet L1 Key
Name ___________________________
Chapter 4:
Discovering and Proving
Triangle Properties
Note Sheet
S. Stirling
Page 1 of 15
Ch 4 Note Sheet L1 Key
4.1 Triangles Sum Conjectures
Name ___________________________
These notes replace pages 200 – 202 in the book. See step 4 of the Investigation.
Rigidity is a property that triangles have. They cannot be shifted, like a quadrilateral can. They retain
their shape.
Triangulation is a procedure used by surveyors to locate a position by using triangles.
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
C
x
A
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.
See Ch 4 Worksheet to prove these conjectures.
S. Stirling
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Ch 4 Note Sheet L1 Key
Lesson 4.2 Properties of Isosceles Triangles
Name ___________________________
Read top of page 206. This vocabulary is review! Know it!!
Remember! With If..then.. statements: If a, then b. The converse of the statement is If b, then a.
Label and define the special vocabulary for the isosceles triangle ΔVBS .
A leg (of an isosceles triangle) is one of the congruent
sides. VB and VS
B
S
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
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 page 207.
Isosceles Triangle Conjecture
If a triangle is isosceles, then its base angles are congruent.
Given ΔVBS with VB = VS , then ∠ B ≅ ∠ S
If VB ≅ VS , then ∠B ≅ ∠S .
V
S
or
B
***Note: Since equilateral triangles are a special case of isosceles triangles, any property
that applies to isosceles triangles also apply to equilateral triangles.
Do group Investigation 2, Is the Converse True? on page 208. 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 ΔVBS is isosceles
If ∠ B ≅ ∠ S , then VB ≅ VS .
S. Stirling
or
B
V
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S
Ch 4 Note Sheet L1 Key
Lesson 4.3 Triangle Inequalities
Name ___________________________
Read of page 215. Do group Investigation 1 on page 216.
Triangle Inequality Conjecture
The sum of the lengths of any two sides of a triangle is greater than
third side.
the length of the
Is a triangle measuring 5 cm, 4 cm and 2 cm
possible? Construct it then explain!
Is a triangle measuring 5 cm, 3 cm and 2 cm
possible? Construct 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 page 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, if one side is the shortest side, then the angle opposite the shortest side is the
smallest angle.
Which is the largest angle? The smallest
angle? Why?
Which is the largest side? Why?
D
K
5.34
2.48
J
6.3
I
∠D largest, so AQ
largest in ΔADQ ,
But, m∠QAU = 100
Q
90
23
54
A
which makes UQ the
∠K largest, its opposite the longest
side, JI .
∠I smallest, its opposite the shortest
side, JK .
S. Stirling
57
largest in ΔQAU and
AQ < UQ .
Overall, UQ is the longest.
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Ch 4 Note Sheet L1 Key
Name ___________________________
Do group Investigation 3, Exterior Angle of a Triangle on page 217 – 218 .
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.
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 ΔABC above. If a = 50 and b = 60, what
is the measure of ∠ BCD ? Explain.
Given ΔABC above. If x = 80 and b = 30, what
is the measure of ∠A ? Explain.
50 + 60 = 110
80 – 30 = 50
The exterior angle equals the sum of the
two remote interior angles.
The exterior angle equals the sum of the
two remote interior angles.
Before starting Lesson 4.4:
Vocab for triangles Triangles have 6 “parts”.
A
∠A is opposite BC and AC is opposite ∠B
∠A is between BA and AC (angles are between sides)
BC is between ∠B and ∠C (sides are between angles)
B
C
included angle is an angle formed between two consecutive
sides of a polygon.
included side is a side of a polygon between two consecutive angles.
S. Stirling
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Ch 4 Note Sheet L1 Key
Name ___________________________
Read the top of page 221, then complete the Triangle Congruence Shortcut Investigation, page 1 – 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
A
D
O
C
C
T
G
T
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
E
D
ΔPIZ ≅ Δ_______
ΔZED ≅ Δ_______
By SAS 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.
D
O
O
I
C
G
Z
P
P
A
ΔDOG ≅ Δ_______
ΔCOP ≅ Δ_______
By ASA Congruence Conjecture.
By ASA Congruence Conjecture.
S. Stirling
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Ch 4 Note Sheet L1 Key
Name ___________________________
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.
Special Case: Hypotenuse Leg Congruence Conjecture
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.
D
O
O
I
C
G
Z
P
P
A
ΔDOG ≅ Δ_______
ΔCOP ≅ Δ_______
By HL Congruence Conjecture.
By HL Congruence Conjecture.
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
C
Draw a counterexample.
I
T
Z
I
P
Z
P
ΔZIP ≅ Δ_______
By NOT necessarily congruent.
S. Stirling
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Ch 4 Note Sheet L1 Key
Name ___________________________
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
I
Z
P
Z
P
ΔZIP ≅ Δ_______
By NOT necessarily congruent.
How do you get equal parts in order to get congruent triangles?
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!!
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 ,
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
S. Stirling
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.
M
A
If AM is median in ΔABC ,
then BM ≅ MC .
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C
Ch 4 Note Sheet L1 Key
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.
ΔVBS isosceles
or VB ≅ VS ,
then ∠ B ≅ ∠ S .
S
V
E
R
C
∠VET and ∠CER are vertical,
then ∠ VET ≅ ∠ CER .
If
If
B
Def. perpendicular lines
Definition of Angle Bisector
If you have an angle bisector,
then the ray cuts the angle
into two equal angles.
D
B
C
If two lines are perpendicular,
then they intersect to form equal
90° angles.
A
B
M
D
A
JJJG
If BD bisects ∠ ABC
then ∠ ABD ≅ ∠ DBC .
CD ⊥ AB , then
m∠CMA = m∠CMB = 90° .
C
If
Corresponding Angles Conjecture
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
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
If two parallel lines are cut by a transversal,
then alternate interior angles are congruent.
I
“Same Angle”
When two triangles share
the exact same angle, you get
a pair of equal angles.
A
R
G
T
N
ΔTIA and ΔNIR share
∠TIA ≅ ∠RIN .
HJJG HJJG
If LK & NJ ,
then ∠KHM ≅ ∠HMN
K
H
L
J
.
M
N
Ways to Get equal Angles and Sides
Def. perpendicular bisector
A line (or part of a line) that passes through
the midpoint of a segment and
is perpendicular to the segment.
Complete the Triangle
Congruence Shortcut
Investigation, page 4 – 5.
S. Stirling
CD is the perp. bisector of AB ,
then AM ≅ MB
and ∠ CMA = ∠ CMB = 90° .
C
B
M
A
If
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Ch 4 Note Sheet L1 Key
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 ΔCAT ≅ ΔDOG then you can say any of the following:
∠C ≅ ∠D
∠A ≅ ∠O
∠T ≅ ∠G
CA ≅ DO
AT ≅ OG
CT ≅ DG
Examples: How to prove (show deductively) that parts are equal.
P
First example, informal paragraph :
∠PAC ≅ ∠TAC and PA ≅ AT
∠PCA ≅ ∠TCA ?
Know:
Is
The triangles share a side, so
So…
CA = CA
A
.
of course.
T
C
ΔPAC ≅ ΔTAC by SAS Congruence
and ∠ PCA ≅ ∠ TCA because CPCTC or Corresponding Parts of Congruent Triangles are
Congruent.
I
P
A bit more formal paragraph:
Given: Z is midpoint of
Prove:
Z
DI and EP .
PI ≅ DE ?
D
E
Z is midpoint of DI and EP was given, so
IZ = ZD and PZ = ZE by Definition of Midpoint
∠PZI ≅ ∠EZD by the Vertical Angles Conjecture (or vertical angles are congruent.)
Therefore, ΔZIP ≅ ΔZDE by SAS Congruence and
PI ≅ DE because CPCTC or Corresponding Parts of Congruent Triangles are Congruent.
S. Stirling
Page 10 of 15
Ch 4 Note Sheet L1 Key
This one is tricky:
Name ___________________________
R
Know:
Is
E
RE & TC and TR & CE .
RE ≅ EC ?
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.
So, probably not equal.
Note: you could say any of the following though:
RE ≅ TC , RT ≅ EC or ∠R ≅ ∠C .
This one is trickier! Start by un-overlapping possible pairs of congruent triangles:
Given:
Prove:
LE = EP and LV = OP
∠ALP ≅ ∠APL
A
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 Example A and Example B on Ch 4 Worksheet.
S. Stirling
Page 11 of 15
Ch 4 Note Sheet L1 Key
4.7 Flowchart Thinking
4.7 Page 238 Example B Flowchart Proof
Given:
Prove:
Name ___________________________
E
EC ≅ AC and ER ≅ AR
∠A ≅ ∠E
R
C
EC ≅ AC
A
Given
ER ≅ AR
Given
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
ΔBAD ≅ ΔBCD
∠1 ≅ ∠2
SSS Cong. Conj.
CPCTC
Given
BD ≅ BD
Same segment
BD
is the angle bisector of
∠ABC .
Def. of angle bisector
S. Stirling
Page 12 of 15
Ch 4 Note Sheet L1 Key
Name ___________________________
What do you need to know to get…
You’ll need Equal Segments to get…
Def. Isosceles Triangle
B
Def. Median
If a triangle’s legs are congruent,
then it is isosceles.
V
VB ≅ VS
then ΔVBS isosceles.
If
If a segment connects the vertex to the
midpoint of the opposite side,
A
then it is a median.
S
M
C
If BM ≅ MC ,
B
then AM is median in ΔABC .
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 ,
AM ≅ MB ,
then CM bisects AB .
G
If
then M is the midpoint of SG .
B
A
You’ll need Equal Angles to get…
Converse of the Isosceles Triangle Conj.
Def. perpendicular lines
If a triangle has two congruent angles,
then it is an isosceles triangle.
V
If two lines intersect to form equal
adjacent angles,
then they are perpendicular.
If ∠ B ≅ ∠ S ,
then ΔVBS isosceles.
S
A
Corresponding Angles Conjecture
C
∠EFA ≅ ∠ FGD ,
HJJG HJJG
then AB & DC .
B
F
C
If
Def. Altitude
G
D
Alternate Interior Angles Conjecture
R
If two lines are cut by a transversal and
the alternate interior angles are congruent,
then the lines are parallel.
T
I
A
m∠RAT = m∠RAI = 90° ,
then RA is an altitude of ΔTRI
E
If two lines are cut by a transversal and
the corresponding angles are congruent,
then the lines are parallel.
A
D
B
∠ABD ≅ ∠DBC ,
JJJG
then BD bisects ∠ ABC .
∠KHM ≅ ∠HMN
HJJG HJJG
then LK & NJ .
If
If
S. Stirling
D
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
M
If
B
Definition of Angle Bisector
If a ray cuts the angle into two
equal angles, then you have
an angle bisector.
C
.
H
K
L
J
,
N
Page 13 of 15
M
Ch 4 Note Sheet L1 Key
Name ___________________________
You’ll need Equal Angles and Sides to get…
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.
S. Stirling
Page 14 of 15
Ch 4 Note Sheet L1 Key
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 B
Isosceles Triangle with vertex angle A, AB = AC.
Median BM
A
Altitude BL
L
Angle Bisector BS
A
Median BM
AM = 1.64 in.
MC = 1.64 in.
D
M
E
C
M
S
AB = 3.36 in.
AC = 3.36 in.
CB = 4.34 in.
Angle Bistector BD
m∠ABD = 27°
m∠DBC = 27°
B
C
Altitude BE
m∠AEB = 90°
B
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
M
A
A
N
P
S
L
C
B
Medians MC = BN
S. Stirling
T
C
C
B
Angle Bisectors PC = BS
B
Altitudes BT = LC
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