Download Triangle Congruence Proofs 4

Triangle Congruence Proofs 4
Objectives:
G.CO.8: Explain how the criteria for triangle congruence (ASA,SAS, SSS, and AAS) follow
from the definition of congruence in terms of rigid motions.
G.CO.7: Use the definition of congruence in terms of rigid motions to show that two
triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent.
G.SRT.5: Use congruence and similarity criteria for triangles to solve problems and
prove relationships in geometric figures.
For the Board: You will be able to use congruent triangles to prove segments and angles congruent.
Bell Work:
State whether or not you can use the following information to prove two triangles are congruent
1. ASA
2. SSA
3. SAS
4. SSS
5. AAS
6. AAA
Anticipatory Set:
Definition of Congruent Triangles (CPCTC)
Two triangles are congruent if and only if all three angle and all three sides of one triangle are
congruent to the corresponding three angles and three sides of another triangle.
Instruction:
Given: AB||CD, BC||DA
Prove: AB  CD
B
C
A
Proof:
1.
2.
3.
4.
5.
6.
Statements
AB||CD, BC||DA
<CBD  <ADB
<ABD  <CDB
BD  BD
ΔBAC  ΔDCB
AB  CD
D
1.
2.
3.
4.
5.
6.
Reasons
Given
Alternate Interior Angles Theorem
Alternate Interior Angles Theorem
Reflexive Property of Congruence
ASA
CPCTC
M
Given: A midpoint of MT and SR
Prove: <M  <T
R
A
Proof:
1.
2.
3.
4.
5.
Statements
A midpoint MT and SR
MA  TA, SA  RA
<MAS  <TAR
ΔMAS  ΔTAR
<SMA  <RTA
1.
2.
3.
4.
5.
Reasons
Given
Definition of Midpoint
Vertical Angle Theorem
SAS
CPCTC
S
T
Base Angles Theorem
A
If two sides of a triangle are congruent then the angles opposite them are congruent.
Given: AB  AC
Prove: <B  <C.
Proof:
1.
2.
3.
4.
4.
6.
6.
Statements
Let X be the midpoint of BC
Draw AX
AB  AC
BX  CX
AD  AD
ΔABD  ΔACD
<B  <C
Reasons
B
1. Every segment has a unique midpoint.
2. Through two points there is exactly one segment.
3. Given
4. Definition of Midpoint
5. Reflexive Property of Congruence
6. SSS
7. CPCTC
Base Angles Converse Theorem
If two angles of a triangle are congruent then the sides opposite them are congruent.
Given: <B  <C
Prove: AB  AC.
Proof:
Statements
1. Draw the bisector of <A
2. The bisector intersects BC at X
3. <B  <C
4. <BAX  <CAX
5. AD  AD
5. ΔABX  ΔACX
6. AB  AC
Assessment:
Question student pairs.
Independent Practice:
Handout 4
1.
2.
3.
4.
5.
5.
6.
Reasons
Every angle has a unique bisector.
Two lines intersect in at most 1 point.
Given
Definition of Angle Bisector
Reflexive Property of Congruence
AAS
CPCTC
C