In ΔABC, ray BD bisects ∠ABC and ray CE bisects ∠ACB.If seg AB≅seg AC then prove that ED||BC
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Answers
Answer:
ED ∥ BC
Step-by-step-explanation:
NOTE: Refer to the attachment for the diagram.
In figure, in △ABC,
Ray BD bisects ∠ABC - - [ Given ]
∴ By property of angle bisector theorem of triangle,
AB / BC = AD / DC - - ( 1 )
Now,
In △ABC,
Ray CE bisects ∠ACB - - [ Given ]
∴ By property of angle bisector theorem of triangle,
AC / BC = AE / BE - - ( 2 )
Now,
Seg AB ≅ seg AC - - - ( 3 ) [ Given ]
From ( 1 ), ( 2 ) & ( 3 ),
AB / BC = AC / BC - - ( 4 )
Now,
In △ABC,
From ( 1 ), ( 2 ) & ( 4 ),
AE / BE = AD / CD
∴ ED ∥ BC - - [ Converse of BPT ]
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Additional Information:
1. Angle bisector theorem:
When a ray bisects an angle, every point on the ray is equidistant from the both arms of the angle.
2. Angle bisector theorem of triangle:
When a ray bisects an angle of a triangle, the arms of the angle and the remaining two sides are in the proportion.
3. This theorem is based on Basic Proportionality Theorem ( BPT ).
4. Basic Proportionality Theorem:
1. This theorem is related to the sides of a triangle and a line parallel to a side.
2. This theorem says that,
In a triangle, if a line is parallel to any of three sides, then that line divides the other two sides in the equal ratios.
3. It is also known as BPT in short form.
5. Converse of BPT:
1. This theorem is used to show if a line is parallel to any one of the sides of the triangle.
2. This theorem say that,
If a line divides any two sides of a triangle in equal ratios, then line is parallel to the third side.
