27. “If two angles and a side of one triangle are equal to If two angles and a
side of another triangle then the two triangles must be congruent."
Is the statement true? Why?
Pls tell the answer in detail
Answers
Answer:
If there exits a correspondence between the vertices of two triangles such that two angles and the included side of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.
Step-by-step explanation:
For this purpose, consider
Δ
A
B
C
and
Δ
D
E
F
, where
B
C
=
E
F
,
∠
B
=
∠
E
and
∠
C
=
∠
F
. Note that this will also mean that
∠
A
=
∠
D
(can you see why?). Now, we have to show that these two triangles are congruent. Here is a step-by-step proof.
Step 1: If
A
B
is equal to
D
E
, then the two triangles would be congruent by the SAS congruence criterion. So, let us suppose that
A
B
is not equal to
D
E
. Then, one of them would be greater than the other – say that
A
B
>
D
E
.
✍Note: Refer SAS congruence criterion to understand Step 1.
Step 2: Mark a point on
A
B
(call it
G
), such that
G
B
=
D
E
, as shown below:
Incongruent triangles
Step 3: We note that
Δ
G
B
C
is congruent to
Δ
D
E
F
by the SAS criterion. This means that
∠
B
C
G
=
∠
F
.
Incongruent triangles checking
Step 4: Finally, we note that
∠
F
=
∠
C
(given), and so
∠
B
C
G
=
∠
C
.
enlightenedThink: Is that possible if
C
G
is in a different direction than
C
A
?
No, it’s not!
This means that
C
G
must be in the same direction as
C
A
, or in other words,
G
and
A
coincide, or:
G
B
=
A
B
=
D
E
. Thus, the two triangles are congruent by the SAS congruence criterion.
✍Note: An important aspect of the proof of the ASA congruence criterion we have encountered is that it builds on the SAS congruence criterion – it assumes the truth of the SAS congruence criterion.