Proving Proving ΔΔs are s are : : SSS, SAS, HL, ASA, AAS, SSS, SAS, HL, ASA, AAS,
& CPCTC& CPCTC
SSSSSSSide-Side-Side Postulate
• If 3 sides of one Δ are to 3 sides of another Δ, then the Δs are .
More on the SSS PostulateMore on the SSS Postulate
If AB ED, AC EF, & BC DF, then ΔABC ΔEDF.
E
D
F
A
B
C
Use the SSS Congruence Postulate
Write a proof.
GIVEN KL NL, KM NM
Proof It is given that KL NL and KM NM
By the Reflexive Property, LM LN.
So, by the SSS Congruence Postulate, KLM NLM
PROVE KLM NLM
EXAMPLE 1:EXAMPLE 1:
GUIDED PRACTICE
Decide whether the congruence statement is true. Explain your reasoning.
SOLUTION
1. ACB CAD
BC ADGIVEN :
PROVE : ACB CAD
PROOF: It is given that BC AD By Reflexive propertyAC AC, But AB is not congruent CD.
YOUR TURN:YOUR TURN:
GUIDED PRACTICE
Therefore the given statement is false and ABC is not Congruent to CAD because corresponding sides are not congruent
YOUR TURN (continued):YOUR TURN (continued):
1.
GUIDED PRACTICE
Decide whether the congruence statement is true. Explain your reasoning.
SOLUTION
QT TR , PQ SR, PT TSGIVEN :
PROVE : QPT RST
PROOF: It is given that QT TR, PQ SR, PT TS. So bySSS congruence postulate, QPT RST. Yes, the statement is true.
QPT RST 2.
YOUR TURN:YOUR TURN:
SASSASSide-Angle-Side Postulate
• If 2 sides and the included of one Δ are to 2 sides and the included of another Δ, then the 2 Δs are .
More on the SAS Postulate• If BC YX, AC ZX, & C X,
then ΔABC ΔZXY.
B
A C X
Y
Z)(
EXAMPLE 2 Use the SAS Congruence Postulate
Write a proof.
GIVEN
PROVE
STATEMENTS REASONS
BC DA, BC AD
ABC CDA
1. Given1. BC DAS
Given2. 2. BC AD
3. BCA DAC 3. Alternate Interior Angles Theorem
A
4. 4. AC CA Reflexive Property of Congruence
S
Example 2:Example 2:
EXAMPLE 2
STATEMENTS REASONS
5. ABC CDA SAS Congruence Postulate
5.
Example 2 (continued):Example 2 (continued):
Given: DR AG and AR GR
Prove: Δ DRA Δ DRG.
D
AR
G
Example 4:Example 4:
Statements_______ 1. DR AG; AR GR2. DR DR3.DRG & DRA are
rt. s4.DRG DRA5. Δ DRG Δ DRA
Reasons____________1. Given 2. Reflexive Property3. lines form right s
4. Right s Theorem
5. SAS PostulateD
A GR
Example 4 (continued):Example 4 (continued):
HLHLHypotenuse - Leg Theorem
• If the hypotenuse and a leg of a right Δ are to the hypotenuse and a leg of a second Δ, then the 2 Δs are .
ASAASAAngle-Side-Angle Congruence
Postulate
• 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.
AAS AAS Angle-Angle-Side Congruence
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.
Proof of the Angle-Angle-Side (AAS) Proof of the Angle-Angle-Side (AAS) Congruence TheoremCongruence Theorem
Given: A D, C F, BC EF
Prove: ∆ABC ∆DEF
Paragraph Proof
You are given that two angles of ∆ABC are congruent to two angles of ∆DEF. By the Third Angles Theorem, the third angles are also congruent. That is, B E. Notice that BC is the side included between B and C, and EF is the side included between E and F. You can apply the ASA Congruence Postulate to conclude that ∆ABC ∆DEF.
A B
C
D
E
F
Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.
Example 5:Example 5:
In addition to the angles and segments that are marked, EGF JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. Thus, you can use the AAS Congruence Theorem to prove that ∆EFG ∆JHG.
Example 5 (continued):Example 5 (continued):
Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.
Example 6:Example 6:
In addition to the congruent segments that are marked, NP NP. Two pairs of corresponding sides are congruent. This is not enough information to prove the triangles are congruent.
Example 6 (continued):Example 6 (continued):
Given: AD║EC, BD BC
Prove: ∆ABD ∆EBC
Plan for proof: Notice that ABD and EBC are congruent. You are given that BD BC. Use the fact that AD ║EC to identify a pair of congruent angles.
Example 7:Example 7:
Statements:1. BD BC2. AD ║ EC3. D C
4. ABD EBC
5. ∆ABD ∆EBC
Reasons:
1. Given
2. Given
3. If || lines, then alt. int. s are
4. Vertical Angles Theorem
5. ASA Congruence Postulate
Example 7 (continued):Example 7 (continued):
CPCTC
CPCTC
STATEMENTS REASONS
Given:
Prove: D B