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Standard VII

Mathematics

RHS Criteria for Congruency

25. Let ABC b...

Question

25. Let ABC be a triangle. Let D, E, F be points respectively on segments BC, CA, AB such that AD, BE, CF concurrent at the point K. Suppose BD/DC=BF/FA and angle ADB=angleAFC. Prove that angleABE=angleCAD.

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Similar questions

Q. Let

A B C

be a triangle. Let

D, E, F

be the points respectively on the segments

B C, C A, A B

such that

A D, B E, C F

concur at the point

K .

Suppose

\frac{B D}{D C} = \frac{B F}{F A}

and

∠ A D B = ∠ A F C

. Is

∠ A B E = ∠ C A D . ?

Q. Let

A B C

be a triangle. Let

B E

and

C F

be internal angle bisectors of

∠ B

and

∠ C

, respectively, with

E

A C

and

F

A B .

Suppose

X

is a point on the segment

C F

such that

A X ⊥ C F

, and

Y

is a point on the segment

B E

such that

A Y ⊥ B E

Prove that

X Y = \frac{(b + c - a)}{2}

, where

B C = a

C A = b

, and

A B = c

Q. Let

A B C

be an acute-angled triangle triangle, and let

D, E, F

be points on

B C, C A, A B

respectively such that

A D

is the median,

B E

is the internal angle bisector and

C F

is the altitude. Suppose

∠ F D E = ∠ C, ∠ D E F = ∠ A a n d ∠ E F D = ∠ B

. Prove that

A B C

is equilateral.

Q. Let

D, E, F

be points on the sides

B C, C A, A B,

respectively, of a triangle

A B C

such that

B D = C E = A F

and

∠ B D F = ∠ C E D = ∠ A F E

. Prove that

Δ A B C

is equilateral.

Let ABC be a triangle such that ∠B = 70° and ∠C = 40°. Suppose D is a point on BC such that AB = AD. Prove that AB > CD.

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