prove that if in two triangles,corresponding angles are equal ,then their corresponding sides in the same ratio and hence the two triangles are similar
Answers
Answer:
Step-by-step explanation:
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
theorem on similarity of triangles
Construction: ABC is a triangle. DE || BC and DE intersects AB at D and AC at E.
Join B to E and C to D. Draw DN ⊥ AB and EM ⊥ AC.
To prove:
A
D
D
B
=
A
E
E
C
Proof:
ar AEM
=
1
2
×
(
A
D
)
×
(
E
M
)
Similarly;
ar BDE
=
1
2
×
(
D
B
)
×
(
E
M
)
ar ADE
=
1
2
×
(
A
E
)
×
(
D
N
)
ar DEC
=
1
2
×
(
E
C
)
×
(
D
N
)
Hence;
ar ADE
ar BDE
=
1
2
×
(
A
D
)
×
(
E
M
)
1
2
×
(
D
B
)
×
(
E
M
)
=
A
D
D
B
Similarly;
ar ADE
ar DEC
=
A
E
E
C
Triangles BDE and DEC are on the same base, i.e. DE and between same parallels, i.e. DE and BC.
Hence, ar(BDE) = ar(DEC)
From above equations, it is clear that;
A
D
D
B
=
A
E
E
C
proved
Answer:
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