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Triangles and Angles
Standard/Objectives:
Standard 3: Students will learn and apply
geometric concepts.
Objectives:
• Classify triangles by their sides and
angles.
• Find angle measures in triangles
DEFINITION: A triangle is a figure formed
by three segments joining three noncollinear points.
2
Names of triangles
Triangles can be classified by the
sides or by the angle
Equilateral
—3
congruent
sides
Isosceles
Triangle—2
congruent
sides
Scalene—
no
congruent
sides
3
Acute Triangle
3 acute angles
m ABC = 70.26 
m CAB = 41.76 
m BCA = 67.97 
B
C
A
4
Equiangular triangle
• 3 congruent angles. An equiangular
triangle is also acute.
5
Right
Triangle
Obtuse
Triangle
• 1 right angle
6
7
Parts of a triangle
• Each of the three
points joining the
sides of a triangle is a
vertex.(plural:
vertices). A, B and C
are vertices.
• Two sides sharing a
common vertext are
adjacent sides.
• The third is the side
opposite an angle
Side
opposite
A
B
C
adjacent
adjacent
A
8
Right Triangle
hypotenuse
• Red represents the
hypotenuse of a
right triangle. The
sides that form
the right angle are
leg
the legs.
leg
9
Isosceles Triangles
• An isosceles triangle
can have 3 congruent
sides in which case it
is equilateral. When
an isosceles triangle
has only two congruent
sides, then these two base
sides are the legs of
the isosceles triangle.
The third is the base.
leg
leg
10
Identifying the parts of
an isosceles triangle
•
About 7 ft.
5 ft
•
5 ft
Explain why ∆ABC is
an isosceles right
triangle.
In the diagram you
are given that C is
a right angle. By
definition, then
∆ABC is a right
triangle. Because AC
= 5 ft and BC = 5 ft;
AC BC. By
definition, ∆ABC is
also an isosceles
triangle.
11
Identifying the parts of
an isosceles triangle
Hypotenuse & Base
•
About 7 ft.
•
5 ft
leg
5 ft
leg
Identify the legs and
the hypotenuse of
∆ABC. Which side is
the base of the
triangle?
Sides AC and BC are
adjacent to the right
angle, so they are
the legs. Side AB is
opposite the right
angle, so it is t he
hypotenuse. Because
AC BC, side AB is
also the base.
12
13
Using Angle Measures of
Smiley faces are
Triangles
interior angles and
hearts represent the
exterior angles
B
A
C
Each vertex has a pair
of congruent exterior
angles; however it is
common to show only
one exterior angle at
each vertex.
14
Ex. 3 Finding an Angle
Measure.
Exterior Angle theorem: m1 = m A
+m 1
x + 65 = (2x + 10)
65 = x +10
65
x
55 = x
(2x+10)
15
16
Finding angle measures
• Corollary to the
triangle sum
theorem
• The acute angles
of a right triangle
are complementary.
• m A + m B = 90
2x
x
17
Finding angle measures
X + 2x = 90
3x = 90
X = 30
• So m A = 30 and
the m B=60
B
2x
C
x
A
18
19
20
Congruence and
Triangles
Standards/Objectives:
Standard 2: Students will learn and apply
geometric concepts
Objectives:
 Identify congruent figures and
corresponding parts
 Prove that two triangles are congruent
22
23
Identifying congruent figures
 Two geometric figures are congruent if
they have exactly the same size and
NOT CONGRUENT
shape.
CONGRUENT
24
Triangles
Corresponding angles
A ≅ P
B ≅ Q
C ≅ R
B
A
Corresponding Sides
AB ≅ PQ
BC ≅ QR
CA ≅ RP
Q
CP
R
25
 If Δ ABC is  to Δ XYZ, which
angle is  to C?
Z
26
27
Thm 4.3
rd
3 angles thm
 If 2 s of one Δ are  to 2
s of another Δ, then the
3rd s are also .
28
Ex: find x
)
22o
87o
(4x+15)o
29
Ex: continued
22+87+4x+15=180
4x+15=71
4x=56
x=14
30
31
Ex: ABCD is  to HGFE, find x
and y.
C
A
9 cm
B
E
D
F
(5y-12)°
91°
B
D
113°
G
4x – 3 cm
A
86°
H
C
4x-3=9
5y-12=113
4x=12
5y=125
x=3
y=25
32
Thm 4.4
Props. of  Δs
A
 Reflexive prop of Δ  Every Δ is  to itself
(ΔABC  ΔABC).
 Symmetric prop of Δ  C
If ΔABC  ΔPQR, then
ΔPQR  ΔABC.
 Transitive prop of Δ  - If
ΔABC  ΔPQR & ΔPQR 
ΔXYZ, then ΔABC 
X
ΔXYZ.
Z
B
P
Q
R
Y
33
34
35
36
37
38
Proving Δs are  :
SSS and SAS
Standards/Benchmarks
Standard 2: Students will learn and apply
geometric concepts
Objectives:
• Prove that triangles are congruent using
the SSS and SAS Congruence Postulates.
• Use congruence postulates in real life
problems such as bracing a structure.
40
Remember?
 As of yesterday, Δs could only be  if
ALL sides AND angles were 
NOT ANY MORE!!!!
There are two short cuts to add.
41
Post. 19
Side-Side-Side (SSS)  post
• If 3 sides of one Δ are  to 3
sides of another Δ, then the
Δs are .
42
A
Meaning:
___
E
___
B
___
C
___
F
___
D
___
If seg AB  seg ED,
seg AC  seg EF &
seg BC  seg DF,
then ΔABC  ΔEDF.
43
Given: seg QR  seg UT, RS  TS,
QS=10, US=10
Prove: ΔQRS  ΔUTS
U
Q
10
R
10
S
T
44
45
46
Proof
Statements
1.
2. QS=US
3. Seg QS  seg US
4. Δ QRS  Δ UTS
Reasons
1. given
2. subst. prop. =
3. Def of  segs.
4. SSS post
47
Post. 20
Side-Angle-Side post. (SAS)
• If 2 sides and the included  of
one Δ are  to 2 sides and the
included  of another Δ, then
the 2 Δs are .
48
• If seg BC  seg YX, seg AC  seg
ZX, and C  X, then ΔABC 
ΔZXY.
B
Y
(
A
C
X
Z
49
50
Given: seg WX  seg. XY, seg VX
 seg ZX,
Prove: Δ VXW  Δ ZXY
W
Z
1
V
X
2
Y
51
Proof
Statements
1. seg WX  seg. XY
seg. VX  seg ZX
2. 1  2
3. Δ VXW  Δ ZXY
Reasons
1. given
2. vert s thm
3. SAS post
52
53
Given: seg RS  seg RQ and seg ST
 seg QT
Prove: Δ QRT  Δ SRT.
S
Q
R
T
54
Proof
Statements
1. Seg RS  seg RQ
seg ST  seg QT
2. Seg RT  seg RT
3. Δ QRT  Δ SRT
Reasons
1. Given
2. Reflex prop 
3. SSS post
55
Given: seg DR  seg AG and
seg AR  seg GR
Prove: Δ DRA  Δ DRG.
D
A
R
G
56
Proof
Statements
1. seg DR  seg AG
Seg AR  seg GR
2. seg DR  Seg DR
3.DRG & DRA are
rt. s
4.DRG   DRA
5. Δ DRG  Δ DRA
Reasons
1. Given
2. reflex. Prop of 
3.  lines form 4 rt. s
4. Rt. s thm
5. SAS post.
57
58
59
60
Proving Triangles are
Congruent: ASA and
AAS
Objectives:
1. Prove that triangles are congruent
using the ASA Congruence Postulate
and the AAS Congruence Theorem
2. Use congruence postulates and
theorems in real-life problems.
62
Postulate 21: Angle-Side-Angle
(ASA) Congruence Postulate
• If two angles and the
B
included side of one
triangle are
congruent to two
angles and the
C
included side of a
second triangle, then
the triangles are
congruent.
A
E
F
D
63
64
Theorem 4.5: Angle-Angle-Side
(AAS) Congruence Theorem
• If two angles and a
B
non-included side of
one triangle are
congruent to two
angles and the
corresponding non- C
included side of a
second triangle, then
the triangles are
congruent.
A
E
F
D
65
Theorem 4.5: Angle-Angle-Side
(AAS) Congruence Theorem
Given: A  D, C
 F, BC  EF
Prove: ∆ABC  ∆DEF
B
A
E
C
F
D
66
Theorem 4.5: Angle-Angle-Side
(AAS) Congruence Theorem
You are given that two angles of
∆ABC are congruent to two
angles of ∆DEF. By the Third
Angles Theorem, the third
angles are also congruent.
That is, B  E. Notice that
BC is the side included
between B and C, and EF C
is the side included between
E and F. You can apply
the ASA Congruence
Postulate to conclude that
∆ABC  ∆DEF.
B
A
E
F
D
67
68
Ex. 1 Developing Proof
Is it possible to prove
the triangles are
congruent? If so,
state the postulate or
theorem you would
use. Explain your
reasoning.
H
E
G
F
J
69
Ex. 1 Developing Proof
A. In addition to the angles
and segments that are
marked, EGF JGH
by the Vertical Angles
Theorem. Two pairs of
corresponding angles
and one pair of
corresponding sides are
congruent. You can use
the AAS Congruence
Theorem to prove that
∆EFG  ∆JHG.
H
E
G
F
J
70
Ex. 1 Developing Proof
Is it possible to prove
the triangles are
congruent? If so,
state the postulate or
theorem you would
use. Explain your
reasoning.
N
M
Q
P
71
Ex. 1 Developing Proof
B. In addition to the
congruent segments
that are marked, NP
 NP. Two pairs of
corresponding sides
are congruent. This
is not enough
information to prove
the triangles are
congruent.
N
M
Q
P
72
Ex. 1 Developing Proof
Is it possible to prove
the triangles are
congruent? If so,
state the postulate or
theorem you would
use. Explain your
reasoning.
UZ ║WX AND UW
║WX.
U
1
2
W
Z
3
4
X
73
Ex. 1 Developing Proof
The two pairs of
parallel sides can be
used to show 1 
3 and 2  4.
Because the included
side WZ is congruent
to itself, ∆WUZ 
∆ZXW by the ASA
Congruence
Postulate.
U
1
2
W
Z
3
4
X
74
75
Ex. 2 Proving Triangles are
Congruent
Given: AD ║EC, BD  BC
Prove: ∆ABD  ∆EBC
Plan for proof: Notice that
ABD and EBC are
congruent. You are
given that BD  BC
. Use the fact that AD ║EC
to identify a pair of
congruent angles.
C
A
B
D
E
76
C
A
Proof:
B
D
Statements:
1. BD  BC
2. AD ║ EC
3. D  C
4. ABD  EBC
5. ∆ABD  ∆EBC
E
Reasons:
1.
77
C
A
Proof:
B
D
Statements:
1. BD  BC
2. AD ║ EC
3. D  C
4. ABD  EBC
5. ∆ABD  ∆EBC
E
Reasons:
1. Given
78
C
A
Proof:
B
D
Statements:
1. BD  BC
2. AD ║ EC
3. D  C
4. ABD  EBC
5. ∆ABD  ∆EBC
E
Reasons:
1. Given
2. Given
79
C
A
Proof:
B
D
Statements:
1. BD  BC
2. AD ║ EC
3. D  C
4. ABD  EBC
5. ∆ABD  ∆EBC
E
Reasons:
1. Given
2. Given
3. Alternate Interior
Angles
80
C
A
Proof:
B
D
Statements:
1. BD  BC
2. AD ║ EC
3. D  C
4. ABD  EBC
5. ∆ABD  ∆EBC
E
Reasons:
1. Given
2. Given
3. Alternate Interior
Angles
4. Vertical Angles
Theorem
81
C
A
Proof:
B
D
Statements:
1. BD  BC
2. AD ║ EC
3. D  C
4. ABD  EBC
5. ∆ABD  ∆EBC
E
Reasons:
1. Given
2. Given
3. Alternate Interior
Angles
4. Vertical Angles
Theorem
5. ASA Congruence
Theorem
82
Note:
• You can often use more than one method
to prove a statement. In Example 2, you
can use the parallel segments to show
that D  C and A  E. Then you
can use the AAS Congruence Theorem to
prove that the triangles are congruent.
83
84
85
86
Using Congruent Triangles
Objectives:
Use congruent triangles to plan and write
proofs.
Use congruent triangles to prove
constructions are valid.
88
Planning a proof
Knowing that all pairs of corresponding
parts of congruent triangles are congruent
can help you reach conclusions about
congruent figures.
89
Planning a proof
For example, suppose you want
to prove that PQS ≅ RQS
in the diagram shown at the
right. One way to do this is
to show that ∆PQS ≅ ∆RQS
by the SSS Congruence
Postulate. Then you can
use the fact that
corresponding parts of
congruent triangles are
congruent to conclude that
PQS ≅ RQS.
Q
R
P
S
90
Ex. 1: Planning & Writing a Proof
Given: AB ║ CD, BC ║
DA
Prove: AB≅CD
Plan for proof: Show
that ∆ABD ≅ ∆CDB.
Then use the fact that
corresponding parts of
congruent triangles are
congruent.
B
A
C
D
91
Ex. 1: Planning & Writing a Proof
Solution: First copy the
diagram and mark it with the
given information. Then mark
any additional information you
can deduce. Because AB and
CD are parallel segments
intersected by a transversal, and
BC and DA are parallel
segments intersected by a
transversal, you can deduce that
two pairs of alternate interior
angles are congruent.
B
A
C
D
92
Ex. 1: Paragraph Proof
Because AD ║CD, it follows
from the Alternate Interior
Angles Theorem that ABD
≅CDB. For the same
reason, ADB ≅CBD
because BC║DA. By the
Reflexive property of
Congruence, BD ≅ BD. You
can use the ASA
Congruence Postulate to
conclude that ∆ABD ≅
∆CDB. Finally because
corresponding parts of
congruent triangles are
congruent, it follows that AB
≅ CD.
B
A
C
D
93
Ex. 2: Planning & Writing a Proof
Given: A is the midpoint of
MT, A is the midpoint of SR.
Prove: MS ║TR.
Plan for proof: Prove that
∆MAS ≅ ∆TAR. Then use
the fact that corresponding
parts of congruent triangles
are congruent to show that
M ≅ T. Because these
angles are formed by two
segments intersected by a
transversal, you can
conclude that MS ║ TR.
M
R
A
S
T
94
M
Given: A is the midpoint of MT, A is the
midpoint of SR.
Prove: MS ║TR.
A
Statements:
Reasons:
1.
1.
2.
3.
4.
5.
6.
A is the midpoint of MT, A
is the midpoint of SR.
MA ≅ TA, SA ≅ RA
MAS ≅ TAR
∆MAS ≅ ∆TAR
M ≅ T
MS ║ TR
R
S
T
Given
95
M
Given: A is the midpoint of MT, A is the
midpoint of SR.
Prove: MS ║TR.
R
A
S
Statements:
Reasons:
1.
1.
Given
2.
Definition of a midpoint
2.
3.
4.
5.
6.
A is the midpoint of MT, A
is the midpoint of SR.
MA ≅ TA, SA ≅ RA
MAS ≅ TAR
∆MAS ≅ ∆TAR
M ≅ T
MS ║ TR
T
96
M
Given: A is the midpoint of MT, A is the
midpoint of SR.
Prove: MS ║TR.
R
A
S
Statements:
Reasons:
1.
1.
Given
2.
3.
Definition of a midpoint
Vertical Angles Theorem
2.
3.
4.
5.
6.
A is the midpoint of MT, A
is the midpoint of SR.
MA ≅ TA, SA ≅ RA
MAS ≅ TAR
∆MAS ≅ ∆TAR
M ≅ T
MS ║ TR
T
97
M
Given: A is the midpoint of MT, A is the
midpoint of SR.
Prove: MS ║TR.
R
A
S
T
Statements:
Reasons:
1.
1.
Given
2.
3.
4.
Definition of a midpoint
Vertical Angles Theorem
SAS Congruence Postulate
2.
3.
4.
5.
6.
A is the midpoint of MT, A
is the midpoint of SR.
MA ≅ TA, SA ≅ RA
MAS ≅ TAR
∆MAS ≅ ∆TAR
M ≅ T
MS ║ TR
98
M
Given: A is the midpoint of MT, A is the
midpoint of SR.
Prove: MS ║TR.
R
A
S
T
Statements:
Reasons:
1.
1.
Given
2.
3.
4.
5.
Definition of a midpoint
Vertical Angles Theorem
SAS Congruence Postulate
Corres. parts of ≅ ∆’s are ≅
2.
3.
4.
5.
6.
A is the midpoint of MT, A
is the midpoint of SR.
MA ≅ TA, SA ≅ RA
MAS ≅ TAR
∆MAS ≅ ∆TAR
M ≅ T
MS ║ TR
99
M
Given: A is the midpoint of MT, A is the
midpoint of SR.
Prove: MS ║TR.
R
A
S
T
Statements:
Reasons:
1.
1.
Given
2.
3.
4.
5.
6.
Definition of a midpoint
Vertical Angles Theorem
SAS Congruence Postulate
Corres. parts of ≅ ∆’s are ≅
Alternate Interior Angles
Converse.
2.
3.
4.
5.
6.
A is the midpoint of MT, A
is the midpoint of SR.
MA ≅ TA, SA ≅ RA
MAS ≅ TAR
∆MAS ≅ ∆TAR
M ≅ T
MS ║ TR
100
101
Ex. 3: Using more than one pair of
triangles.
Given: 1≅2, 3≅4.
Prove ∆BCE≅∆DCE
Plan for proof: The only
information you have about
∆BCE and ∆DCE is that 1≅2
and that CE ≅CE. Notice,
however, that sides BC and DC
are also sides of ∆ABC and
∆ADC. If you can prove that
∆ABC≅∆ADC, you can use the
fact that corresponding parts of
congruent triangles are
congruent to get a third piece of
information about ∆BCE and
∆DCE.
D
C
2
1
4
E
3
A
B
102
Given: 1≅2, 3≅4.
Prove ∆BCE≅∆DCE
C
2
1
E
A
B
Statements:
Reasons:
1.
2.
3.
4.
5.
6.
1. Given
1≅2, 3≅4
AC ≅ AC
∆ABC ≅ ∆ADC
BC ≅ DC
CE ≅ CE
∆BCE≅∆DCE
4
3
103
Given: 1≅2, 3≅4.
Prove ∆BCE≅∆DCE
C
2
1
E
4
3
A
B
Statements:
Reasons:
1. 1≅2, 3≅4
2. AC ≅ AC
1. Given
2. Reflexive property of
Congruence
3.
4.
5.
6.
∆ABC ≅ ∆ADC
BC ≅ DC
CE ≅ CE
∆BCE≅∆DCE
104
Given: 1≅2, 3≅4.
Prove ∆BCE≅∆DCE
C
2
1
E
4
3
A
B
Statements:
Reasons:
1. 1≅2, 3≅4
2. AC ≅ AC
1. Given
2. Reflexive property of
Congruence
3. ASA Congruence
Postulate
3. ∆ABC ≅ ∆ADC
4. BC ≅ DC
5. CE ≅ CE
6. ∆BCE≅∆DCE
105
Given: 1≅2, 3≅4.
Prove ∆BCE≅∆DCE
2
1
C
E
4
3
A
B
Statements:
Reasons:
1. 1≅2, 3≅4
2. AC ≅ AC
1. Given
2. Reflexive property of
Congruence
3. ASA Congruence
Postulate
3. ∆ABC ≅ ∆ADC
4. BC ≅ DC
5. CE ≅ CE
6. ∆BCE≅∆DCE
4.
Corres. parts of ≅ ∆’s are ≅
106
Given: 1≅2, 3≅4.
Prove ∆BCE≅∆DCE
2
1
C
E
4
3
A
B
Statements:
Reasons:
1. 1≅2, 3≅4
2. AC ≅ AC
1. Given
2. Reflexive property of
Congruence
3. ASA Congruence
Postulate
3. ∆ABC ≅ ∆ADC
4. BC ≅ DC
5. CE ≅ CE
6. ∆BCE≅∆DCE
4.
5.
Corres. parts of ≅ ∆’s are ≅
Reflexive Property of
Congruence
107
Given: 1≅2, 3≅4.
Prove ∆BCE≅∆DCE
2
1
C
E
4
3
A
B
Statements:
Reasons:
1. 1≅2, 3≅4
2. AC ≅ AC
1. Given
2. Reflexive property of
Congruence
3. ASA Congruence
Postulate
3. ∆ABC ≅ ∆ADC
4. BC ≅ DC
5. CE ≅ CE
6. ∆BCE≅∆DCE
4.
5.
6.
Corres. parts of ≅ ∆’s are ≅
Reflexive Property of
Congruence
SAS Congruence Postulate
108
Ex. 4: Proving constructions are
valid
In Lesson 3.5 – you learned to copy an
angle using a compass and a straight edge.
The construction is summarized on pg. 159
and on pg. 231.
Using the construction summarized above,
you can copy CAB to form FDE. Write
a proof to verify the construction is valid.
109
Plan for proof
Show that ∆CAB ≅ ∆FDE.
Then use the fact that
corresponding parts of
congruent triangles are
congruent to conclude that
CAB ≅ FDE. By
construction, you can
assume the following
statements:
– AB ≅ DE Same compass
setting is used
– AC ≅ DF Same compass
setting is used
– BC ≅ EF Same compass
setting is used
C
A
B
F
D
E
110
C
Given: AB ≅ DE, AC ≅ DF, BC ≅ EF
Prove CAB≅FDE
2
1
A
B
4
3
F
D
E
Statements:
Reasons:
1.
2.
3.
4.
5.
1.
AB ≅ DE
AC ≅ DF
BC ≅ EF
∆CAB ≅ ∆FDE
CAB ≅ FDE
Given
111
C
Given: AB ≅ DE, AC ≅ DF, BC ≅ EF
Prove CAB≅FDE
2
1
A
B
4
3
F
D
E
Statements:
Reasons:
1.
2.
3.
4.
5.
1.
2.
AB ≅ DE
AC ≅ DF
BC ≅ EF
∆CAB ≅ ∆FDE
CAB ≅ FDE
Given
Given
112
C
Given: AB ≅ DE, AC ≅ DF, BC ≅ EF
Prove CAB≅FDE
2
1
A
B
4
3
F
D
E
Statements:
Reasons:
1.
2.
3.
4.
5.
1.
2.
3.
AB ≅ DE
AC ≅ DF
BC ≅ EF
∆CAB ≅ ∆FDE
CAB ≅ FDE
Given
Given
Given
113
C
Given: AB ≅ DE, AC ≅ DF, BC ≅ EF
Prove CAB≅FDE
2
1
A
B
4
3
F
D
E
Statements:
Reasons:
1.
2.
3.
4.
5.
1.
2.
3.
4.
AB ≅ DE
AC ≅ DF
BC ≅ EF
∆CAB ≅ ∆FDE
CAB ≅ FDE
Given
Given
Given
SSS Congruence Post
114
C
Given: AB ≅ DE, AC ≅ DF, BC ≅ EF
Prove CAB≅FDE
2
1
A
B
4
3
F
D
E
Statements:
Reasons:
1.
2.
3.
4.
5.
1.
2.
3.
4.
5.
AB ≅ DE
AC ≅ DF
BC ≅ EF
∆CAB ≅ ∆FDE
CAB ≅ FDE
Given
Given
Given
SSS Congruence Post
Corres. parts of ≅ ∆’s
are ≅.
115
Q
Given: QSRP, PT≅RT
Prove PS≅ RS
2
1
4
3
P
T
Statements:
Reasons:
1. QS  RP
2. PT ≅ RT
1. Given
2. Given
R
S
116
117
118
119
120
121
Isosceles,
Equilateral and
Right s
Pg 236
Standards/Objectives:
Standard 2: Students will learn and apply geometric concepts
Objectives:
• Use properties of Isosceles and equilateral triangles.
• Use properties of right triangles.
123
Isosceles triangle’s special parts
A is the vertex angle (opposite
A
the base)
 B and C are base angles
(adjacent to the base)
B
C
Base
124
Thm 4.6
Base s thm
• If 2 sides of a  are , the the s opposite them are .( the base s of an
isosceles  are )
A
If seg AB  seg AC,
then  B   C
B
125
C
126
Thm 4.7
Converse of Base s thm
• If 2 s of a  are , the sides opposite them are .
A
If  B   C,
then seg AB 
seg AC
B
127
C
Corollary to the base s thm
• If a triangle is equilateral, then it is equiangular.
If seg AB  seg
BC  seg CA,
then A  B 
C
B
A
C
128
Corollary to converse of the base angles thm
• If a triangle is equiangular, then it is also equilateral.
)
A
If A  B  C, then seg AB  seg BC 
seg CA
B
(
129
C
Example:
find x and y
• X=60
• Y=30
X
120
Y
130
131
132
Thm 4.8
Hypotenuse-Leg (HL)  thm
• If the hypotenuse and a leg of
one right  are  to the
hypotenuse and leg of another
right , then the s are .
A
_
If seg AC  seg XZ
and seg BC  seg YZ,
then  ABC   XYZ
C
_ Y
_
X
_
B
Z
133
Given: D is the midpt of seg CE, BCD and FED are
rt s and seg BD  seg FD.
Prove:  BCD   FED
B
C
F
D
E
134
Proof
Statements
1.
2.
3.
D is the midpt of seg CE, 
BCD and <FED are rt  s and
seg BD  to seg FD
Seg CD  seg ED
 BCD   FED
Reasons
1. Given
2.
3.
Def of a midpt
HL thm
135
Are the 2 triangles  ?
Yes, ASA
or AAS
)
(
136
Find x and y.
x
75
y
60
90
y
x
2x + 75=180
x
x=60
y=30
2x=105
x=52.5
y=75
137
Find x.
56ft
(
8xft
))
56=8x
7=x
((
138
139
140
141
Triangles and
Coordinate Proof
Objectives:
1.
2.
Place geometric figures in a coordinate
plane.
Write a coordinate proof.
143
Placing Figures in a
Coordinate Plane

So far, you have studied two-column proofs,
paragraph proofs, and flow proofs. A
COORDINATE PROOF involves placing
geometric figures in a coordinate plane.
Then you can use the Distance Formula (no,
you never get away from using this) and the
Midpoint Formula, as well as postulate and
theorems to prove statements about figures.
144
Ex. 1: Placing a Rectangle in a
Coordinate Plane


Place a 2-unit by 6-unit rectangle in a
coordinate plane.
SOLUTION: Choose a placement that
makes finding distance easy (along the
origin) as seen to the right.
145
Ex. 1: Placing a Rectangle in a
Coordinate Plane

One vertex is at the
origin, and three of the
vertices have at least
one coordinate that is
0.
6
4
2
-5
5
-2
-4
146
Ex. 1: Placing a Rectangle in a
Coordinate Plane

One side is centered at
the origin, and the xcoordinates are
opposites.
4
2
-5
5
-2
-4
-6
147
Note:

Once a figure has been placed in a
coordinate plane, you can use the Distance
Formula or the Midpoint Formula to measure
distances or locate points
148
149
150
Ex. 2: Using the Distance
Formula

A right triangle has legs
of 5 units and 12 units.
Place the triangle in a
coordinate plane.
Label the coordinates
of the vertices and find
the length of the
hypotenuse.
6
4
2
5
10
-2
-4
-6
-8
151
Ex. 2: Using the Distance
Formula
One possible placement is
shown. Notice that one leg
is vertical and the other leg
is horizontal, which assures
that the legs meet as right
angles. Points on the same
vertical segment have the
same x-coordinate, and
points on the same
horizontal segment have
the same y-coordinate.
6
4
2
5
10
-2
-4
-6
-8
152
Ex. 2: Using the Distance
Formula
You can use the Distance
Formula to find the
length of the
hypotenuse.
d = √(x2 – x1)2 + (y2 – y1)2
= √(12-0)2 + (5-0)2
= √169
= 13
6
4
2
5
10
-2
-4
-6
-8
153
Ex. 3 Using the Midpoint
Formula


In the diagram, ∆MLN ≅
∆KLN). Find the
coordinates of point L.
Solution: Because the
triangles are congruent, it
follows that ML ≅ KL. So,
point L must be the
midpoint of MK. This
means you can use the
Midpoint Formula to find the
coordinates of point L.
160
140
120
100
80
60
40
20
-50
50
100
150
200
-20
-40
-60
-80
-100
-120
154
Ex. 3 Using the Midpoint
Formula

160
L (x, y) = x1 + x2, y1 +y2
2
2
140
120
100
Midpoint Formula
80
=160+0 , 0+160
2
2
60
40
20
Substitute values
= (80, 80)
Simplify.
-50
50
100
150
200
-20
-40
-60
-80
-100
-120
155
156
157
Writing Coordinate Proofs

Once a figure is placed in a coordinate plane,
you may be able to prove statements about
the figure.
158
Ex. 4: Writing a Plan for a
Coordinate Proof




Write a plan to prove that SQ bisects PSR.
Given: Coordinates of vertices of ∆PQS and ∆RQS.
Prove SQ bisects PSR.
Plan for proof: Use the Distance Formula to find the
side lengths of ∆PQS and ∆RQS. Then use the
SSS Congruence Postulate to show that ∆PQS ≅
∆RQS. Finally, use the fact that corresponding parts
of congruent triangles are congruent (CPCTC) to
conclude that PSQ ≅RSQ, which implies that SQ
bisects PSR.
159
Ex. 4: Writing a Plan for a
Coordinate Proof
7


Given: Coordinates of
vertices of ∆PQS and
∆RQS.
Prove SQ bisects
PSR.
6
5
4
S
3
2
1
-5
P
Q
R
5
-1
-2
-3
-4
160
-5
NOTE:


The coordinate proof in Example 4 applies to
a specific triangle. When you want to prove a
statement about a more general set of
figures, it is helpful to use variables as
coordinates.
For instance, you can use variable
coordinates to duplicate the proof in Example
4. Once this is done, you can conclude that
SQ bisects PSR for any triangle whose
coordinates fit the given pattern.
161
162
No coordinates – just variables
y
S
x
P
(-h, 0)
(0, k)
(0, 0)
R
(h, 0)
163
Ex. 5: Using Variables as
Coordinates


Right ∆QBC has leg lengths of
h units and k units. You can
find the coordinates of points B
and C by considering how the
triangle is placed in a
coordinate plane.
Point B is h units horizontally
from the origin (0, 0), so its
coordinates are (h, 0). Point C
is h units horizontally from the
origin and k units vertically
from the origin, so its
coordinates are (h, k). You
can use the Distance Formula
to find the length of the
hypotenuse QC.
C (h, k)
hypotenuse
k units
Q (0, 0)
h units
B (h, 0)
164
Ex. 5: Using Variables as
Coordinates
OC = √(x2 – x1)2 + (y2 – y1)2
= √(h-0)2 + (k - 0)2
= √h2 + k2
C (h, k)
hypotenuse
k units
Q (0, 0)
h units
B (h, 0)
165
Ex. 5 Writing a Coordinate
Proof

Given: Coordinates of
figure OTUV
Prove ∆OUT  ∆UVO
Coordinate proof:
Segments OV and UT
have the same length.
OV = √(h-0)2 + (0 - 0)2=h

UT = √(m+h-m)2 + (k - k)2=h



6
4
T (m , k)
U (m +h, k)
2
O (0, 0)
-5
V (h,5 0)
-2
-4
-6
166
Ex. 5 Writing a Coordinate
Proof

Horizontal segments UT
and OV each have a slope
of 0, which implies they are
parallel. Segment OU
intersects UT and OV to
form congruent alternate
interior angles TUO and
VOU. Because OU  OU,
you can apply the SAS
Congruence Postulate to
conclude that ∆OUT 
∆UVO.
6
4
T (m , k)
U (m +h, k)
2
O (0, 0)
-5
V (h,5 0)
-2
-4
-6
167
168