Applying Triangle Sum Properties
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Transcript of Applying Triangle Sum Properties
Applying Triangle Sum Properties
Section 4.1
Triangles Triangles are polygons with three sides.
There are several types of triangle: Scalene Isosceles Equilateral Equiangular Obtuse Acute Right
Scalene Triangles Scalene triangles do not have any congruent
sides.
In other words, no side has the same length.
3cm
8cm
6cm
Isosceles Triangle A triangle with 2 congruent sides.
2 sides of the triangle will have the same length.
2 of the angles will also have the same angle measure.
Equilateral Triangles All sides have the same length
Equiangular Triangles All angles have the same angle measure.
Acute Triangle All angles are acute angles.
Right Triangle Will have one right angle.
Obtuse Angle Will have one obtuse angle.
Exterior Angles vs. Interior Angles Exterior Angles are angles that are on the
outside of a figure.
Interior Angles are angles on the inside of a figure.
Interior or Exterior?
Interior or Exterior?
Interior or Exterior?
Triangle Sum Theorem (Postulate Sheet) States that the sum of the interior angles is
180.
We will do algebraic problems using this theorem. The sum of the
angles is 180, so
x + 3x + 56= 1804x + 56= 180
4x = 124x = 31
Find the Value for X
2x + 15
3x
2x + 15 + 3x + 90 = 180
5x + 105 = 180
5x = 75
x = 15
Corollary to the Triangle Sum Theorem (Postulate Sheet) Acute angles of a right triangle are
complementary.
3x + 10
5x +16
Exterior Angle Sum Theorem The measure of the exterior angle of a triangle is equal to
the sum of the non-adjacent interior angles of the triangle
88 + 70 = y
158 = y
2x + 40 = x + 72
2x = x + 32 x = 32
Find x and y
3x + 13
46o
8x - 1
2yo
4.1 Apply Congruence and Triangles4.2 Prove Triangles Congruent by SSS, SAS
Objectives:1. To define congruent triangles2. To write a congruent statement3. To prove triangles congruent by SSS, SAS
Congruent Polygons
Congruent Triangles (CPCTC)
Two triangles are congruent congruent triangles triangles if and only if the ccorresponding pparts of those ccongruent ttriangles are ccongruent.
Congruence Statement
When naming two congruent triangles, order is very important.
Example
Which polygon is congruent to ABCDE?ABCDE -?-
Properties of Congruent Triangles
Example
What is the relationship between C and F?
30
30
75
75
E
F
D
A
C
B
Third Angle Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.
Congruent Triangles
Checking to see if 3 pairs of corresponding sides are congruent and then to see if 3 pairs of corresponding angles are congruent makes a total of SIX pairs of things, which is a lot! Surely there’s a shorter way!
Congruence Shortcuts?
Will one pair of congruent sides be sufficient? One pair of angles?
Congruence Shortcuts?
Will two congruent parts be sufficient?
Congruent Shortcuts?
Will three congruent parts be sufficient?
And if so….what three parts?
Section 4.3Proving Triangles are Congruents by SSS
Draw any triangle using any 3 size lines For me I use lines of 5, 4, and 3 cm’s. Now use the same lengths and see if you can
make a different triangle.
Now measure both triangles angles and see what you get.
3cm
4cm
5cm
3cm
4cm 5cm90
9053 53
37
37
Are the following triangles congruent? Why?
6 6
10
6 6
10
a. YES, all sidesare equal so SSS
108
9
10
6
9
b. No, all sidesare not equal8 ≠ 6, so failsSSS
Use the SSS Congruence Postulate
Decide whether the congruence statement is true. Explain your reasoning.
NLKL
NMKM
SOLUTION
NLMKLM
LMLM
Given
Given
Reflexive Property
So, by the SSS Congruence Postulate,
NLMKLM
4.4:Prove Triangles Congruent by SAS and HL
Goal:Use sides and angles to prove congruence.
Vocabulary Leg of a right triangle: In a right triangle, a In a right triangle, a
side adjacent to the right angle is called a leg.side adjacent to the right angle is called a leg. Hypotenuse:In a right triangle, the side In a right triangle, the side
opposite the right angle is called the opposite the right angle is called the hypotenuse.hypotenuse.
LegLeg
HypotenuseHypotenuse
Before we start…let’s get a few things straight
INCLUDED SIDE
A B
C
X Z
Y
Angle-Side-Angle (ASA) Congruence Postulate
Two angles and the INCLUDED side
Angle-Angle-Side (AAS) Congruence Postulate
Two Angles and One Side that is NOT included
} Your Only Ways To Prove Triangles Are
Congruent
NO BAD WORDS
Overlapping sides are congruent in
each triangle by the REFLEXIVE property
Vertical Angles
are congruen
t
Alt Int Angles are congruent
given parallel
lines
Things you can mark on a triangle when they aren’t marked.
Ex 1
statement. congruence a Write.
and ,, and In
LE
NLDENDΔLMNΔDEF
DEF NLM
Ex 2
What other pair of angles needs to be marked so that the two triangles are congruent by AAS?
F
D
E
M
L
N
NE
Ex 3
What other pair of angles needs to be marked so that the two triangles are congruent by ASA?
F
D
E
M
L
N
LD
Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.
ΔGIH ΔJIK by AAS
G
I
H J
KEx 4
ΔABC ΔEDC by ASA
B A
C
ED
Ex 5
Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.
ΔACB ΔECD by SASB
A
C
E
D
Ex 6
Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.
ΔJMK ΔLKM by SAS or ASA
J K
LM
Ex 7
Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.
Not possible
K
J
L
T
U
Ex 8
Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.
V
Postulate 20:Side-Angle-Side (SAS) Congruence Postulate
If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.
If Side ,
Angle , and
Side ,
then .
UV
U
RS
R
R UW
U
T
T VS WR
Example 1:Use the SAS Congruence Postulate Write a proof.
Given ,
Prove
JN LN KN MN
JKN LMN
21N
J
MK
L
Statements Reasons
1. , 1. Given
JN
K
LN
MN N
2. 1 2 Vertical Angles T2. h em eor
3. 3. SAS Congruence Postul ateJKN LMN
Example 2:Use SAS and properties of shapes
In the diagram, is a rectangle.
What can you conclude about
and ?
ABCD
ABC CDA By the ,
. Opposite sides of a rectangl
Right Angles Congruence Theorem
e are congruent,
so and .AB CD BC
B D
DA
and are congruent by SAS Congruen the ce
Postulate
.
ABC CDA
Checkpoint
In the diagram, , , and pass
through the center of the circle.
Also, 1 2 3 4.
AB CD EF
M
Statements Reaso
Prove
n
t
s
tha .DMY BMY
1. 3 4 1. Given 2. 2. Definition of a DM BM
3. 3. Reflexive Property of
Congruence
MY MY
4. 4. SAS Congruence
Postulate
DMY BMY
Checkpoint
In the diagram, , , and pass
through the center of the circle.
Also, 1 2 3 4.
AB CD EF
M
What can you conclude about and ?AC BD
Because they are vertical angles,
. All points on a circle are the
same distance from the center, so
. By the SAS Congruence
Postulate, . Corresponding parts
of congruent trian
AMC BMD
AM BM CM DM
AMC BMD
gles are congruent, so you
know .AC BD
Theorem 4.5:Hypotenuse-Leg Congruence Theorem If the hypotenuse and a leg of a right
triangle are congruent to the hypotenuse and a leg of a second triangle, then the two triangles are congruentcongruent.
Example 3:Use the Hypotenuse-Leg Theorem Write a proof.
Given ,
,
,
is a bisector of .
Prove
AC EC
AB BD
ED BD
AC BD
ABC EDC
Example 3:Use the Hypotenuse-Leg Theorem
Statements Reasons
H 1. 1. GivenAC EC
2. , 2.
G
iv
en
AB BD
ED BD
3. and are 3. Definition of lines
righ .t angles
B D
4. and 4. Definition of a
right trian are . gles right triang le
ABC EDC
5. is a bisector 5.
Give
n
of .
AC
BD
Example 3:Use the Hypotenuse-Leg Theorem
Statements Reasons
L 6. 6. Definition of segment
bisector
BC DC
7. 7.
HL Congruence
The or em
ABC EDC
Example 4:Choose a postulate or theorem
Gate The entrance to a ranch
has a rectangular gate as shown
in the diagram. You know that
. What postulate
or theorem can you use to
conclude that ?
AFC EFC
ABC EDC
Example 4:Choose a postulate or theorem
You are given that is a rectangle, so and
are . Because opposite sides of a rectangle
are , . You are also given
right angles
con that
, so . The hypotenuse and a
gruent
leg
of
ABDE B D
AB
AFC EFC AC
DE
EC
each triangle is congruent.
HL Congruence TheYou can use the to conclude
th
orem
at .ABC EDC
Using Congruent Triangles: CPCTC
Academic Geometry
Proving Parts of Triangles Congruent
You know how to use SSS, SAS, ASA, and AAS to show that the triangles are congruent.
Once you have triangles congruent, you can make conclusions about their other parts because, by definition, corresponding parts of congruent triangles are congruent. Abbreviated CPCTC
Proving Parts of Triangles CongruentIn an umbrella frame, the stretchers are congruent and they
open to angles of equal measure.
Given SL congruent to SR
<1 congruent <2
Prove that the angles formed by the shaft
and the ribs are congruent
shaft
stretcher
ribl r3 4
1 2
c
s
Proving Parts of Triangles Congruent
Prove <3 congruent <4
Statement Reason
shaft
stretcher
ribl r3 4
1 2
c
s
Proving Parts of Triangles CongruentGiven <Q congruent <R
<QPS congruent <RSP
Prove SQ congruent PR
Statements Reasons
r
p q
s
Proving Parts of Triangles CongruentGiven <DEG and < DEF are right angles.
<EDG congruent <EDF
Prove EF congruent EG
Statements Reasons
d
ef
g
4.7 Isosceles and Equilateral Triangles
Chapter 4Congruent Triangles
4.5 Isosceles and Equilateral Triangles
Isosceles Triangle:
Base
Leg Leg
Vertex Angle
Base Angles
*The Base Angles are Congruent*
Isosceles Triangles Theorem 4-3 Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent
A
B
C
<A = <C
Theorem 4-4 Converse of the Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the sides opposite those angles are congruent
Isosceles Triangles
A
B
C
Given: <A = <CConclude: AB = CB
Theorem 4-5 The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base
Isosceles Triangles
A
B
C
Given: <ABD = <CBDConclude: AD = DC and
BD is ┴ to AC
D
Equilateral Triangles Corollary: Statement that immediately follows
a theorem
Corollary to Theorem 4-3:If a triangle is equilateral, then the triangleis equiangular
Corollary to Theorem 4-4:If a triangle is equiangular, then the triangle is equilateral
Using Isosceles Triangle TheoremsExplain why ΔRST is isosceles.
R
V
S
W
T
UGiven: <R = <WVS,
VW = SWProve: ΔRST is isosceles
Statement Reason
3. m<R = m<WVS
1. Given1. VW = SW
2. m<WVS = m<S 2. Isosceles Triangle Thm.
3. Given
4. m<S = m<R 4. Transitive Property
5. ΔRST is isosceles 5. Def Isosceles Triangle
Using AlgebraFind the values of x and y:
) )
y°
x°
63°
L
O
N
M
ΔLMN is isosceles
m<L = m< N = 6363°m<LM0 = y = m<NMO
63 + 63 + y + y = 180
y°
126 + 2y = 180- 126 -126
2y = 542 2
y = 27
27 + 63 + x = 180
90 + x = 180-90 -90
x = 90
27°
LandscapingA landscaper uses rectangles and equilateral triangles
for the path around the hexagonal garden. Find the value of x.
x°