Congruent Triangles Triangles are congruent when all corresponding sides and interior angles are congruent. The triangles will have the same shape and size, but one may be a mirror image of the other.
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Congruent Triangles
Triangles are congruent when all corresponding sides and interior angles are congruent. The triangles will have the same shape and size, but one may be a mirror image of the other.
The Triangle Congruence Postulates &Theorems
LAHALLHL
FOR RIGHT TRIANGLES ONLY
AASASASASSSS
FOR ALL TRIANGLES
SSS postulateSSS (side, side, side) postulate
If three sides of a triangle are congruent to its three corresponding sides of another triangle, then the two triangles are congruent.
AB ED ,≅BC EF and≅CA FD≅
∆ABC ∆DEF≅
Look at these two triangles
SAS postulateSAS Postulate (Side-Angle-Side)
If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
Look at these triangles.
AC ≅XZ C ≅ Z
CB ZY≅
∆ABC ∆XYZ≅
EXAMPLE 1
Write a proof.
GIVEN
PROVE
STATEMENTS REASONS
BC DA≅ , BC AD
∆ABC ∆ ≅ CDA
1. Given1. BC DA≅S
Given2. 2. BC AD3. BCA ≅ DAC 3. Alternate Interior
Angles TheoremA
4. 4. AC ≅ CA Reflexive propertyS
EXAMPLE 1
STATEMENTS REASONS
5. ABC ≅ CDA SAS Postulate5.
Given: RS RQ and ST QT Prove: Δ QRT Δ SRT.
Q
RS
T
EXAMPLE 2
STATEMENT REASON ________ 1. RS RQ; ST QT 1. Given 2. RT RT 2. Reflexive 3. Δ QRT Δ SRT 3. SSS Postulate
RQ S
T
EXAMPLE 2
ASA PostulateASA Postulate (Angle-Side-Angle)
If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
Look at these triangles.
B ≅ E
BC ≅ EF
C ≅ F
∆ABC ∆≅ DEF
AAS TheoremAAS (Angle-Angle-Side) Theorem
If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the triangles are congruent.
Look at these triangles.
B ≅ E
C ≅ F
AC ≅ DF
∆ABC ∆≅ DEF
Given: AD║EC, BD BCProve: ∆ABD ∆EBC
EXAMPLE 4
Statements:1. BD BC2. AD ║ EC3. D C
4. ABD EBC
5. ∆ABD ∆EBC
Reasons:1. Given2. Given3. If || lines, then alt. int.
s are 4. Vertical Angles Theorem5. ASA Congruence
Postulate
EXAMPLE 5
GIVEN - EGF JGH, EF HJPROVE - ∆EFG ∆JHG
EXAMPLE 5
STATEMENTS REASONS
1. EFG JHG 1. Given
2. EF HJ 2. Given3. EGF JGH 3. Vertical angles
theorem4. ∆EFG ∆JHG 4. AAS Theorem
Given: YR MA and AR RMProve: Δ MYR Δ AYR
Y A
R
M
Try to solve this.
CPCTC Theorem• CPCTC states that if
two or more triangles are proven congruent by any method, then all of their corresponding angles and sides are congruent as well.
Given: YR MA and AR RMProve: AY MY
Y A
R
M
Try to solve this.
To prove that triangles are congruent we are going to use these theorems and postulates.1.The (SSS) Side-Side-Side postulate2.The (SAS) Side-Angle-Side postulate3.The (ASA) Angle-Side-Angle postulate4.The (AAS) Angle-Angle-Side theorem
2. GIVEN; DE CE, EA EBPROVE; ∆DAB ∆CBA
1. GIVEN; circle with center H AHB FHBPROVE; A F
H
A FB
D
E
C
B
A
Prove the following. ( 20 pts. )
Assignment
1.What is the HL theorem?
2.What is the LL theorem?
• Reference; Plane Geometry for Secondary Schools
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