Need Help?

Get in touch with us

bannerAd

ASA and AAS Congruence

Sep 10, 2022
link

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. 

Angle-Side-Angle (ASA) Congruence Postulate: 
Angle-Side-Angle (ASA) Congruence Postulate: 

If Angle ∠A≅∠D,

Side AC-≅DF, and 

Angle ∠C≅∠F,

then ∆ABC≅∆DEF

parallel

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. 

Angle-Angle-Side (AAS) Congruence Theorem: ASA and AAS Congruence

Given: ∠B=∠E, ∠C=∠F and AC=DF

To prove: ΔABC ≅ ΔDEF

Proof:  

In ∆ABC

parallel

∠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. 

What is a flow proof? 

Solution: 

Given: ∠A≅∠D, ∠C≅∠F and BC≅EF

To prove: ΔABC≅ΔDEF

ΔABC≅ΔDEF

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

Write a flow proof to show that ΔABE≅ΔADE.

ΔABE

Solution: 

Given: CE−⊥BD− and ∠CAB≅∠CAD

To prove: ΔABE≅ΔADE

Proof: 

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. 

Example  4

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. 

Example 5

Solution: 

Given: AD∥EC, BD≅BC

To prove: △ABD ≅ △EBC 

Proof:  

Proof

Exercise

  1. In the diagram, AB ⊥ AD, DE ⊥ AD, and AC ≅ DC. Prove that △ABC ≅ △DEC.
In the diagram, AB ⊥ AD, DE ⊥ AD, and AC ≅ DC. Prove that △ABC ≅ △DEC.
  1. Use the AAS congruence theorem, prove that △HFG ≅ △GKH.
Use the AAS congruence theorem, prove that △HFG ≅ △GKH.
  1. In the diagram, ∠S ≅ ∠U and  ≅ . Prove that △RST ≅ △VUT.
In the diagram, ∠S ≅ ∠U and  ≅ . Prove that △RST ≅ △VUT.
  1. Use the ASA congruence theorem to prove that △NQM ≅ △MPL.
Use the ASA congruence theorem to prove that △NQM ≅ △MPL.
  1. Use the ASA congruence theorem to prove that △ABK ≅ △CBJ.
Use the ASA congruence theorem to prove that △ABK ≅ △CBJ.
  1. Use the AAS congruence theorem to prove that △XWV ≅ △ZWU.
  1. Use the AAS congruence theorem to prove that △NMK ≅ △LKM.
Use the AAS congruence theorem to prove that △NMK ≅ △LKM.
  1. Prove that △FDE ≅ △BCD ≅ △ABF from the given figure.
Prove that △FDE ≅ △BCD ≅ △ABF from the given figure
  1. If m∥n, find the value of x.
 If m∥n, find the value of x.
  1. Prove that △HJK ≅ △LKJ from the given figure.
Prove that △HJK ≅ △LKJ from the given figure.

Concept Summary

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

Concept Summary

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.

Comments:

Related topics

Composite Figures – Area and Volume

A composite figure is made up of simple geometric shapes. It is a 2-dimensional figure of basic two-dimensional shapes such as squares, triangles, rectangles, circles, etc. There are various shapes whose areas are different from one another. Everything has an area they occupy, from the laptop to your book. To understand the dynamics of composite […]

Read More >>
special right triangles_01

Special Right Triangles: Types, Formulas, with Solved Examples.

Learn all about special right triangles- their types, formulas, and examples explained in detail for a better understanding. What are the shortcut ratios for the side lengths of special right triangles 30 60 90 and 45 45 90? How are these ratios related to the Pythagorean theorem?  Right Angle Triangles A triangle with a ninety-degree […]

Read More >>
simplify algebraic expressions

Ways to Simplify Algebraic Expressions

Simplify algebraic expressions in Mathematics is a collection of various numeric expressions that multiple philosophers and historians have brought down. Talking of algebra, this branch of mathematics deals with the oldest concepts of mathematical sciences, geometry, and number theory. It is one of the earliest branches in the history of mathematics. The study of mathematical […]

Read More >>
solve right triangles

How to Solve Right Triangles?

In this article, we’ll learn about how to Solve Right Triangles. But first, learn about the Triangles.  Triangles are made up of three line segments. These three segments meet to form three angles. The lengths of the sides and sizes of the angles are related to one another. If you know the size (length) of […]

Read More >>

Other topics