Congruence of Triangles By ASA Criteria
Congruence of Triangles by ASA Criteria
Theorem: Two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other triangle.
Given: , in which
To Prove:
Proof: On comparing the sides AC and DE of , there are three possibilities:
(i) AC = DE (II) AC < DE (iii) AC > DE
|
Case (i): when AC = DE, then in AC = DE [ Assumed]
CB = EF [Given] Hence, |
 |
|
Case (ii) : When AC < DE, then we take point G on DE such that AC = GE , Join GF Now in AC = GE [ Assumed] CB = EF [ Given ]
Hence, Hence But Therefore From (1) and (2) we get Therefore AC = DE , Hence |
 |
Case (iii): When AC > DE, then we take point G on AC such that GC = DE
Then as in Case (ii) we can prove that A coincide with G i.e. AC = DE, Hence [ By SAS Criterion]
Hence in all the three cases
Illustration: AB is a line segment, AX and BY are two equal line segments drawn on opposite sides of line AB such that . If AB and XY intersect each other at P , prove that
(i) (ii) AB and XY bisect each other.
|
Solution: Since Similarly, we have BY at X and Y respectively] Thus, in triangles PAX and PBY, we have and, AX = BY So, by ASA congruence criterion, we have
Hence, |
 |
Illustration: AB is line segment and P is its mid - point. D and E are points on the same side of AB such that
BAD =
ABE and
EPA =
DPB.Show that
(i) DAP
EBP
(ii) AD = BE
Solution:
Given: AB is a line segment and P is its mid - points. D and E are points on the same side of AB such that BAD =
EPA =
DPB.
Prove that : (i) DAP
EBP
(ii) AD = BE
Proof: (i) Since, P is the mid - point of the line segment AB
In
DAP and
EBP,
AP = BP
Also, given DAP =
EBP
and EPA =
DPB
Adding EPD to both sides
EPA +
EPD =
EPD +
DPB
APD =
BPE
Thus, by ASA rule
DAP
EBP
(ii) Since, DAP
EBP (From above)
AD = BE (CPCT)
In the figure given below,
| |||
| Right Option : B | |||
| View Explanation | |||
l and m are two parallel lines intersected by another pair of parallel lines p and q as shown in the figure given below.
| |||
| Right Option : C | |||
| View Explanation | |||
In the below figure, if EF = QR then the congruence rule used for the congruency of the given triangles is ________________ | |||
| Right Option : B | |||
| View Explanation | |||
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