Download Guided Notes - Proving Triangle Congruence with SSS and SAS

Geometry
Guided Notes
Proving Triangle Congruence Using SSS and SAS
Name: __________________________________
Date: ______________________ Period: ______
Proving Triangle Congruence Using SSS and SAS
To prove that two triangles are congruent you need to show that ________________________
______________________________________________________________________________
(SSS)__________________________________________________- If three sides of one
triangle are congruent to three sides of a second triangle, then the two triangles are congruent.
̅̅̅̅̅ ≅ _____________,
If side 𝑴𝑵
side ̅̅̅̅̅
𝑵𝑷 ≅ _____________, and
side ̅̅̅̅̅
𝑷𝑴 ≅ _____________, then
ΔMNP ≅ ΔQRS
(SAS) _________________________________________________- 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 ̅̅̅̅
𝑷𝑸 ≅ _____________,
side ∡𝑸 ≅ _____________, and
side ̅̅̅̅
𝑸𝑺 ≅ _____________, then
ΔPQS ≅ ΔWXY
SAS______________________________________Not SAS_________________________
Geometry
Guided Notes
Proving Triangle Congruence Using SSS and SAS
Name: __________________________________
Date: ______________________ Period: ______
Example #1: State the included angle of the following sides of the given triangle.
ΔAEB: ̅̅̅̅
𝐴𝐸 𝑎𝑛𝑑 ̅̅̅̅
𝐸𝐵
_____________________
̅̅̅̅ 𝑎𝑛𝑑 ̅̅̅̅
𝐴𝐵
𝐸𝐵
_____________________
ΔMNO: ̅̅̅̅̅
𝑀𝑁 𝑎𝑛𝑑 ̅̅̅̅
𝑂𝑁
_____________________
̅̅̅̅̅ 𝑎𝑛𝑑 ̅̅̅̅
𝑀𝑂
𝑂𝑁
_____________________
Example #2: Decide whether enough information is given to prove that the triangles are congruent.
ΔSTU, ΔPUT
S
P
ΔABC, ΔEBD
A
U
T
D
B
C
E
Proving Triangle Congruence
1. Examine the drawing or make a sketch.
2. Mark any congruent sides and/or angles.
3. Determine what congruence postulate could be used to prove that the triangles are
congruent (SSS, SAS, AAS, ASA or HL).
4. List the statements: givens, congruent parts, etc.
5. Fill in the justifications.
Geometry
Guided Notes
Proving Triangle Congruence Using SSS and SAS
Name: __________________________________
Date: ______________________ Period: ______
Properties of Congruent Triangles
1. Every Triangle is congruent to itself (reflexive).
2. If ΔABC ≅ ΔPQR, then ΔPQR ≅ ΔABC (symmetric).
3. If ΔABC ≅ ΔPQR and ΔPQR ≅ ΔTUV, then ΔABC ≅ ΔTUV (transitive).
A
Example #3:
Given: ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐷𝐶 ; ̅̅̅̅
𝐴𝐶 ≅ ̅̅̅̅
𝐷𝐵
B
C
Prove: ∆ABC ≅ ∆DCB
D
1. What congruence postulate could we use to prove ∆ABC ≅ ∆DCB?
2. What side(s) must we show congruent?
3. What property states that something (side, angle, etc.) is equal to itself?
4. Now that you have answered these questions, complete a 2-column proof.
Statements
Justifications
Geometry
Guided Notes
Proving Triangle Congruence Using SSS and SAS
Name: __________________________________
Date: ______________________ Period: ______
N
Example #4:
Given: C is the midpoint of ̅̅̅̅
𝑁𝐵 ;
̅̅̅̅̅
C is the midpoint of 𝑀𝐴
M
A
C
Prove: ∆NMC ≅ ∆BAC
B
1. What congruence postulate could we use to prove ∆NMC ≅ ∆BAC?
2. What does the definition of midpoint tell us?
3. What sides are congruent by the definition of midpoint?
4. Can we show that all sides are congruent? If not can we show that at least one angle is
congruent?
5. Now that you have answered these questions, complete a 2-column proof.
Statements
Justifications
Geometry
Guided Notes
Proving Triangle Congruence Using SSS and SAS
Name: __________________________________
Date: ______________________ Period: ______
Example #5:
Y
X
̅̅̅̅ ≅ ̅̅̅̅̅
̅̅̅̅ || ̅̅̅̅̅
Given: 𝑋𝑌
𝑊𝑍 ; 𝑋𝑌
𝑊𝑍
Prove: ∆WXY ≅ ∆YZW
W
Z
1. What congruence postulate could we use to prove ∆WXY ≅ ∆YZW?
2. If ̅̅̅̅
𝑋𝑌 || ̅̅̅̅̅
𝑊𝑍, then ̅̅̅̅̅
𝑊𝑌 can be considered a ______________________?
3. If you have answered number 2, then you can determine what parts are congruent. What
postulates/Theorems can be used to justify that these parts are congruent?
4. Now that you have answered these questions, complete a 2-column proof.
Statements
Justifications
Geometry
Guided Notes
Proving Triangle Congruence Using SSS and SAS
Name: __________________________________
Date: ______________________ Period: ______
Example #6:


D
Perpendicular lines form 4 right angles.
All right angles are congruent.
Given: ̅̅̅̅
𝐷𝐸 ⊥ ̅̅̅̅
𝐴𝐶 ; ̅̅̅̅
𝐷𝐸 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ̅̅̅̅
𝐴𝐶
A
Prove: ΔAED ≅ ΔCED
Statements
Justifications
E
C