What is asa in math?
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
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.
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:
We have learned five methods for proving that the triangles are congruent.
