Transcript Chapter_4.4_Proving_Congruence_web

Chapter 4.4 Proving triangle
congruence SSS,SAS
Objective: Use the SSS and SAS
postulates for proving angle congruence.
Check.4.35 Prove that two triangles are congruent by applying the
SSS, SAS, ASA, AAS, and HL congruence statements.
CLE 3108.4.4 Develop geometric intuition and visualization through
performing geometric constructions with straightedge/compass and
with technology.
CLE 3108.4.8 Establish processes for determining congruence and
similarity of figures, especially as related to scale factor, contextual
applications, and transformations.
Forget past mistakes. Forget failures. Forget everything except what you're going to do now
and do it." William Durant
Side-Angle-Side (SAS)
Congruence
Side-Side-Side (SSS)
Congruence
• If the sides of one triangle are
congruent to the sides of a second
triangle then the triangles are
congruent.
• 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
• ABC FDE
• ABC FDE
F
D
F
A
A
D
E
B
C
B
C
E
Constructions – Congruent Triangles Using Sides
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•
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•
•
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Draw a triangle, label the vertices X, Y,
and Z
Elsewhere on the paper, use a straight
edge to construct segment RS Such that
RS  XZ
Using R as the center, draw and arc with
radius equal to XY
Using S as the center draw and arc with a
radius equal to YZ.
Let T be the point of intersection of the two
arcs.
T
Draw RT and ST to form RST
R
Y
Z
X
S
Constructions – Congruent Triangles using 2 sides
and an included angle
•
•
•
•
•
Draw a triangle, label the vertices A, B,
and C
Elsewhere on the paper, use a straight
edge to construct segment KL Such that
KL  BC
Construct and angle congruent to B using
KL as a side of the angle and K as the
vertex.
Construct JK such that JK  BA.
Draw JL to complete KJL
C
A
B
J
K
L
Copy the two Triangles using the requested methodology
B
Using 2 sides and an included angle create
XYZ  DEF
Using Sides create
STV  ABC
E
D
A
C
F
Can you prove Congruence?
Side, Side, Side - SSS
Can you prove Congruence?
Side, Angle, Side - SAS
Can you prove Congruence?
Not enough Information
Can you prove Congruence?
Not enough Information
Can you prove Congruence?
Either SSS or SAS
Page 197 34.
B
4
C
3
$
Given: ABCD, ADCB, ADDC,
2
ABBC, AD|| BC, AB|| CD
1
A
D
Prove: ACD CAB
Statement
Reason
1. ABCD, ADCB
1. Given
2. AC  CA
2. Transitive Property
3. ADDC, ABBC
3. Given
4. AD|| BC, AB|| CD
4. Given
5. D and B is a right Angle 5. Definition of  lines
6. 1  4, 2  3
6. Alternate Interior ’s are 
7. D B
7. Definition of Right Angles
8. ACD CAB
8. Def of congruent triangles
Given: ABAC,BYCY
Prove: BYA CYA
B
Y
A
Statement
1. ABAC,BYCY
2. AY  AY
3. BYA CYA
Reason
1. Given
2. Reflexive Property
3. SSS
C
Given: X is the midpoint of BD,
X is the midpoint of AC
Prove: DXC BXA
Statement
1. X is the midpoint of BD,
X is the midpoint of AC
2. DX  XB, CX  XA
3. DXC  BXA
4. DXC BXA
Reason
1. Given
A
D
X
C
2. Definition of Midpoint
3. Vertical Angles are 
4. SAS
B
Practice Assignment
• Block Page 267, 10 – 18 even
• Honors; Page 267, 10 – 28 even