ASA and AAS Congruence

Key Concepts

• Identify congruent triangles

Introduction

Angle-Side-Angle (ASA) 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 two triangles are congruent. 

If Angle ∠A≅∠D,

Side AC-≅DF, and 

Angle ∠C≅∠F,

then ∆ABC≅∆DEF

Prove Triangles Congruent by ASA and AAS 

Angle-Angle-Side (AAS) 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 two triangles are congruent. 

Given: ∠B=∠E, ∠C=∠F and AC=DF

To prove: ΔABC ≅ ΔDEF

Proof:  

In ∆ABC

∠A+∠B+∠C=180° _____________ (1) (By angle sum property) 

In ∆DEF ∠D+∠E+∠F=180°_____________ (2) (By angle sum property) 

From (1) and (2), 

∠A+∠B+∠C= ∠D+∠E+∠F

∠A+∠E+∠F=∠D+∠E+∠F (Given ∠B=∠E and ∠C=∠F) 

⇒∠A=∠D _____________________ (3) 

Now, in ΔABC and ΔDEF

∠A=∠D (from (3)) 

AC = DF (Given) 

∠C=∠F (Given) 

∴ΔABC≅ΔDEF (AAS congruency) 

Hence proved. 

What is a flow proof? 

A flow proof is a step of proof to be written for a theorem. A flow proof uses arrows to show the flow of a logical argument. 

Example 1: Prove the Angle-Angle-Side congruence theorem for the given figures. 

Solution: 

Given: ∠A≅∠D, ∠C≅∠F and BC≅EF

To prove: ΔABC≅ΔDEF

Example 2: In the diagram,CE ⊥ BD and ∠CAB≅∠CAD.

Write a flow proof to show that ΔABE≅ΔADE.

Solution: 

Given: CE−⊥BD− and ∠CAB≅∠CAD

To prove: ΔABE≅ΔADE

Proof: 

Example 3: OB is the bisector of ∠AOC, PM

⊥ OA and PN ⊥ OC. Show that ∆MPO ≅ ∆NPO. 

Solution:  

In ∆MPO and ∆NPO 

PM ⊥ OM and PN ⊥ ON 

⇒∠PMO = ∠PNO = 90° 

Also, OB is the bisector of ∠AOC 

Then ∠MOP = ∠NOP 

OP = OP common 

∴ ∆MPO ≅ ∆NPO (By AAS congruence postulate) 

Example 4: Prove that △CBD ≅ △ABD from the given figure. 

Solution: 

Given: BD- is an angle bisector of ∠CDA, ∠C ≅ ∠A 

To prove: △CBD ≅ △ABD  

Proof:  

BD- is an angle bisector of ∠CDA, ∠C ≅ ∠A (Given) 

∠CDB ≅ ∠ADB (By angle bisector) 

DB ≅ DB (Reflexive property) 

△CBD ≅ △ABD (Definition of AAS) 

Hence proved. 

Example 5: Prove that △ABD ≅ △EBC in the given figure. 

Solution: 

Given: AD∥EC, BD≅BC

To prove: △ABD ≅ △EBC 

Proof:  

Exercise

1. In the diagram, AB ⊥ AD, DE ⊥ AD, and AC ≅ DC. Prove that △ABC ≅ △DEC.

2. Use the AAS congruence theorem, prove that △HFG ≅ △GKH.

3. In the diagram, ∠S ≅ ∠U and  ≅ . Prove that △RST ≅ △VUT.

4. Use the ASA congruence theorem to prove that △NQM ≅ △MPL.

5. Use the ASA congruence theorem to prove that △ABK ≅ △CBJ.

6. Use the AAS congruence theorem to prove that △XWV ≅ △ZWU.

7. Use the AAS congruence theorem to prove that △NMK ≅ △LKM.

8. Prove that △FDE ≅ △BCD ≅ △ABF from the given figure.

9. If m∥n, find the value of x.

10. Prove that △HJK ≅ △LKJ from the given figure.

Concept Summary

We have learned five methods for proving that the triangles are congruent. 

What have we learned

• Understand and apply Angle-Side-Angle (ASA) congruence postulate.
• Understand and apply Angle-Angle-Side (AAS) congruence postulate.
• Understand the definition of a flow proof.
• Prove theorems on Angle-Side-Angle (ASA) and Angle-Angle-Side (AAS).
• Solve problems on Angle-Side-Angle (ASA) and Angle-Angle-Side (AAS) postulates.