& congruent triangles ncscos: 2.02; 2.03 u.e.q: how do we prove the congruence of triangles, and...
TRANSCRIPT
& CONGRUENT TRIANGLES
NCSCOS: 2.02; 2.03
U.E.Q: How do we prove the congruence of
triangles, and how do we use the congruence of triangles solving real-life
problems?
The triangle is the first geometric shape you will study. The use of this shape has a long history. The
triangle played a practical role in the lives of ancient Egyptians and Chinese as an aid to surveying land.
The shape of a triangle also played an important role in triangles to represent art forms. Native
Americans often used inverted triangles to represent the torso of human beings in paintings or
carvings. Many Native Americans rock carving called petroglyphs. Today, triangles are frequently
used in architecture.
Temple of Diana at EphesusTemple of Diana at Ephesus
Pyramids of GizaPyramids of GizaStatue of ZeusStatue of Zeus
On a cable stayed bridge the cables attached to each tower transfer the weight of the roadway to the tower.
You can see from the smaller diagram that the cables balance the weight of the roadway on both sides of each tower.
In the diagrams what type of angles are formed by each individual cable with the tower and roadway?
What do you notice about the triangles on opposite sides of the towers?
Why is that so important?
We can find triangles everywhere:
In nature In man-made structures
ReplayReplaySlideSlide
Classifying Triangles
Equilateral Equilateral
3 congruent sides3 congruent sides
Isosceles Isosceles
At least 2 congruent sidesAt least 2 congruent sides
Scalene Scalene
No congruent sidesNo congruent sides
Equilangular Equilangular
3 congruent angles3 congruent angles
Acute Acute
3 acute angles3 acute angles
Obtuse Obtuse
1 obtuse angle1 obtuse angle
Right Right
1 right angle1 right angle
Vertex: the point where two sides of a triangle meet
Adjacent Sides: two sides of a triangle sharing Adjacent Sides: two sides of a triangle sharing a common vertexa common vertexHypotenuse: side of the triangle across from Hypotenuse: side of the triangle across from the right anglethe right angleLegs: sides of the right triangle that form Legs: sides of the right triangle that form the right anglethe right angleBase: the non-congruent sides of an Base: the non-congruent sides of an isosceles triangleisosceles triangle
Label the following on the right
triangle:VerticesHypotenuseLegs
VertexVertex
VertexVertexVertexVertex
HypotenuseHypotenuse
LegLeg
LegLeg
Label the following on the isosceles triangle:
BaseCongruent adjacent
sidesLegs
m<1 = m<A + m<B
Adjacent Adjacent sideside
BaseBase
Adjacent Adjacent SideSide
LegLeg LegLeg
Interior Angles: angles inside the triangle
(angles A, B, and C)
AA
BB
CC
11
22
33
Exterior Angles: Exterior Angles: angles adjacent to the angles adjacent to the interior anglesinterior angles
(angles 1, 2, and 3)(angles 1, 2, and 3)
The sum of the measures of the interior angles of a triangle is 180o.
AA
BB
CC
<A + <B + <C = 180<A + <B + <C = 180oo
The measure of an exterior angle of a triangle is equal to the sum of the measures of two nonadjacent interior angles.
AA
BB
11
m<1 = m <A + m <B m<1 = m <A + m <B
The acute angles of a right triangle are complementary. BB
AA
m<A + m<B = 90m<A + m<B = 90oo
NCSCOS: 2.02; 2.03NCSCOS: 2.02; 2.03
2 figures are 2 figures are congruent if they congruent if they have the exact same have the exact same size and shape.size and shape.
When 2 figures are When 2 figures are congruent the congruent the corresponding corresponding parts are parts are congruent. (angles congruent. (angles and sides)and sides)
Quad ABDC is Quad ABDC is congruent to Quad congruent to Quad EFHGEFHG
AABB
CC DD
EEFF
GGHH
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
)))))))))))))) ((
((((
(())))))))
))))))
((((
If If ΔΔ ABC is ABC is to to ΔΔ XYZ, which XYZ, which angle is angle is to to C?C?
If 2 If 2 s of one s of one ΔΔ are are to 2 to 2 s of another s of another ΔΔ, , then the 3rd then the 3rd s are s are also also ..
))
2222oo
)))) 8
787oo
))
))))(4x+15)(4x+15)oo
22+87+4x+15=18022+87+4x+15=1804x+15=714x+15=71
4x=564x=56x=14x=14
AABB
DD CC
FFEE
GGHH
9191oo
8686oo
9cm9cm
(5y-12)(5y-12)oo
4x-3cm4x-3cm
113113oo
4x-3=9 5y-12=1134x-3=9 5y-12=113
4x=12 5y=1254x=12 5y=125
x=3 y=25x=3 y=25
Reflexive prop of Δ - Every Δ is to itself (ΔABC ΔABC).
Symmetric prop of Δ - If ΔABC ΔPQR, then ΔPQR ΔABC.
Transitive prop of Δ - If ΔABC ΔPQR & ΔPQR ΔXYZ, then ΔABC ΔXYZ.
AA
BB
CCPP
RR
XX
YY
ZZ
RR
PP
NN
MM
9292oo
9292o o
Statements ReasonsStatements Reasons1. 1. given1. 1. given2. m2. mP=mP=mN 2. subst. N 2. subst. prop =prop =
3. 3. P P N N 3. def of 3. def of s s
4. 4. RQP RQP MQN 4. vert MQN 4. vert s s thmthm
5. 5. R R M 5. 3M 5. 3rdrd s s thmthm
6. 6. ΔΔRQP RQP ΔΔ MQN 6. def of MQN 6. def of ΔΔss
In Lesson 4.2, you learned that if all six In Lesson 4.2, you learned that if all six pairs of corresponding parts (sides and pairs of corresponding parts (sides and angles) are congruent, then the angles) are congruent, then the triangles are congruent.triangles are congruent.
Corresponding PartsCorresponding Parts
ABC ABC DEFDEF
BB
AA
CC
EE
DD
FF
1.1. AB AB DEDE
2.2. BC BC EFEF
3.3. AC AC DFDF
4.4. A A D D
5.5. B B E E
6.6. C C F F
If all the sides of one triangle are congruent to all of the sides of a second triangle, then the triangles are congruent. (SSS)
Use the SSS Postulate to show the two triangles are congruent. Find the length of each side.
AC =
BC =
AB =
MO =
NO =
MN =
5
7 2 25 7 74
5
7 2 25 7 74
ABC MNOV V
K
J
L
K
J
L
K is the angle between JK and KL. It is called the included angle of sides JK and KL.
What is the included angle for sides KL and JL?
L
J
L
K
QP
R
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 triangles are congruent. (SAS)
S
A SS
A
S
JKL PQRV V by SAS
S
N
L
W
K
Given: N is the midpoint of LW N is the midpoint of SK
Prove: LNS WNKV V
N is the midpoint of LWN is the midpoint of SK
Given
,LN NW SN NK Definition of Midpoint
LNS WNK Vertical Angles are congruent
LNS WNKV V SAS Postulate
K
J
L
K
J
L
JK is the side between J and K. It is called the included side of angles J and K.
What is the included side for angles K and L?
KL
K
J
L
Z
XY
If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent. (ASA)
JKL ZXYV V by ASA
W
HA
KS
AW WK
Given: HA || KS
Prove: HAW SKWV V
HA || KS, Given
HAW SKW Alt. Int. Angles are congruent
HWA SWK Vertical Angles are congruent
HAW SKWV V ASA Postulate
AW WK
METEORITESMETEORITESWhen a meteoroid (a piece of rocky or When a meteoroid (a piece of rocky or metallic matter from space) enters Earth’s metallic matter from space) enters Earth’s atmosphere, it heatsup, leaving a trail of atmosphere, it heatsup, leaving a trail of burning gases called a meteor. Meteoroid burning gases called a meteor. Meteoroid fragments that reach Earth without fragments that reach Earth without burningup are called meteorites.burningup are called meteorites.
On December 9, 1997, an extremely bright meteor lit up the sky
above Greenland. Scientists attempted to find meteorite fragments by collecting data from eyewitnesses who had seen the meteor pass through the sky. As shown, the scientists were able to describe sightlines from observers in different towns. One sightline was from observers in Paamiut (Town P) and another was from observers in Narsarsuaq (Town N). Assuming the sightlines were accurate, did the scientists have enough information to locate any meteorite fragments? Explain. ( this example is taken from your text book pg. 222
Note: is not Note: is not SSS, SAS, or ASA.SSS, SAS, or ASA.
TSC
B
A
R
H I
J
K
M L P N
O
V W
U
Identify the congruent triangles (if any). State the postulate by which the triangles are congruent.
ABC STRV V by SSSby SSS
PNO VUWV V by SASby SAS
JHIV
H
M
T
A
Given:
Prove: MH HT
MATV is isosceles with vertex bisected by AH.MAT
• Sides MA and AT are congruent by the definition of an isosceles triangle.
• Angle MAH is congruent to angle TAH by the definition of an angle bisector.
• Side AH is congruent to side AH by the reflexive property.
• Triangle MAH is congruent to triangle TAH by SAS.
• Side MH is congruent to side HT by CPCTC.
A line to one of two || lines is to the other line.A line to one of two || lines is to the other line.
NM
Q
O
P
|| ,QM PO QM MO
QM PO
Given:
Prove: QN PN
|| ,
,
QM PO QM MO
QM PO MO
GivenGiven
PO MO
has midpoint Nhas midpoint N
90
90
om QMNom PON
Perpendicular lines intersect at 4 right Perpendicular lines intersect at 4 right angles.angles.
QMN PON Substitution, Def of Congruent AnglesSubstitution, Def of Congruent Angles
Definition of MidpointDefinition of Midpoint
QMN PONV V SASSAS
QN PN CPCTCCPCTC
Triangles may be proved congruent by Side – Side – Side (SSS) PostulateSide – Angle – Side (SAS) Postulate, and Angle – Side – Angle (ASA) Postulate.
Parts of triangles may be shown to be congruent by Congruent Parts of Congruent Triangles are Congruent (CPCTC).
If two angles and a non included side of one triangle are congruent to two angles and non included side of a second triangle, then the two triangles are congruent.
Do you need Do you need all six ?all six ?
NO !NO !
SSSSSSSASSASASAASAAASAAS
Solve a real-world problemSolve a real-world problem
Structural SupportStructural Support
Explain why the bench with the diagonal support is Explain why the bench with the diagonal support is stable, while the one without the support can collapse.stable, while the one without the support can collapse.
Solve a real-world problemSolve a real-world problem
The bench with a diagonal support forms triangles with The bench with a diagonal support forms triangles with fixed side lengths. By the fixed side lengths. By the SSSSSS Congruence Postulate, Congruence Postulate, these triangles cannot change shape, so the bench is these triangles cannot change shape, so the bench is stable. The bench without a diagonal support is not stable. The bench without a diagonal support is not stable because there are many possible quadrilaterals stable because there are many possible quadrilaterals with the given side lengths.with the given side lengths.
SOLUTIONSOLUTION
Warning:Warning: No SSA Postulate No SSA Postulate
AA CC
BB
DD
EE
FF
NOT CONGRUENTNOT CONGRUENT
There is no such There is no such thing as an SSA thing as an SSA
postulate! postulate!
Warning:Warning: No AAA Postulate No AAA Postulate
AA CC
BB
DD
EE
FF
There is no such There is no such thing as an AAA thing as an AAA
postulate! postulate!
NOT CONGRUENTNOT CONGRUENT
Tell whether you can use the Tell whether you can use the given information at determine given information at determine whether whether ABC ABC DEF DEF
A A D, ABD, ABDE, ACDE, ACDF DF
AB AB EF, BC EF, BC FD, AC FD, AC DE DE
The Congruence Postulates & The Congruence Postulates & TheoremTheorem
SSS SSS correspondencecorrespondence
ASA ASA correspondencecorrespondence
SAS SAS correspondencecorrespondence
AAS AAS correspondencecorrespondence
SSA SSA correspondencecorrespondence
AAA AAA correspondencecorrespondence
Name That PostulateName That Postulate
SASSASASAASA
SSSSSSSSASSA
(when possible)(when possible)
Name That PostulateName That Postulate(when possible)(when possible)
ASAASA
SASSAS
AAAAAA
SSASSA
Name That PostulateName That Postulate(when possible)(when possible)
SASSAS
SASSAS
SASSASReflexive Reflexive PropertyProperty
Vertical Vertical AnglesAngles
Vertical Vertical AnglesAngles
Reflexive Reflexive PropertyProperty SSASSA
HW: Name That PostulateHW: Name That Postulate(when possible)(when possible)
ClosureClosureIndicate the additional information needed to Indicate the additional information needed to enable us to apply the specified congruence enable us to apply the specified congruence postulate.postulate.
For ASA: For ASA:
For SAS:For SAS:
For AAS:For AAS:
Let’s PracticeLet’s PracticeIndicate the additional information needed to Indicate the additional information needed to enable us to apply the specified congruence enable us to apply the specified congruence postulate.postulate.
For ASA: For ASA:
For SAS:For SAS:
B B DD
For AAS:For AAS: A A FF
AC AC FEFE
Now For The Fun Part…
J
S H0
Write a two column ProofGiven: BC bisects AD and A D
Prove: AB DC
A A C C
EE
BB D D
The two angles in an isosceles triangle adjacent to the base of the triangle are called base angles.
The angle opposite the base is called the vertex angle.
Base AngleBase Angle Base AngleBase Angle
Vertex AngleVertex Angle
If two sides of a triangle are congruent, then the angles opposite them are congruent.
CBthenACABIf ,
AA
CC BB
If two angles of a triangle are congruent, then the sides opposite them are congruent.
CAABthenCBIf ,
If a triangle is equilateral, then it is equiangular.
If a triangle is equiangular, then it is equilateral.
?BAIs
AA
CC
BB
AABB
CC
AA
CC
BB
YesYesYesYes
NoNo
If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent.
DEFABCthenDFACandEFBCIf ,
AA
BB CC
DD
EE FF
Find the measure of the missing angles and tell which theorems you used.
50°50°AA
BB
CC
m B=80° m B=80°
(Base Angle Theorem)(Base Angle Theorem)
m C=50°m C=50°
(Triangle Sum (Triangle Sum Theorem)Theorem)
AA
BB
CC
m A=60°m A=60°
m B=60°m B=60°
m C=60°m C=60°
(Corollary to the Base (Corollary to the Base Angles Theorem)Angles Theorem)
Is there enough information to prove the triangles are congruent?
SS
RR
TT
UUVV
WW
YesYesNoNo NoNo
