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CBSE Class 9 Mathematics Question 7 of 21

Triangles — Question 7

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Question 7

AB is a 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) Δ DAP ≅ Δ EBP

(ii) AD = BE

AB is a 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. NCERT Class 9 Mathematics CBSE Solutions.
Answer

Given :

P is the mid-point of AB.

∴ AP = BP ......(1)

Given,

⇒ ∠BAD = ∠ABE .....(2)

From figure,

⇒ ∠BAD = ∠PAD and ∠ABE = ∠PBE

Substituting values in equation (2), we get :

⇒ ∠PAD = ∠PBE ........(3)

(i) Given,

⇒ ∠EPA = ∠DPB .........(4)

Adding ∠DPE to both sides of the above equation,

⇒ ∠EPA + ∠DPE = ∠DPB + ∠DPE

∴ ∠DPA = ∠EPB ......(5)

In Δ DAP and Δ EBP,

⇒ ∠PAD = ∠PBE [From (3)]

⇒ AP = BP [From (1)]

⇒ ∠DPA = ∠EPB [From (5)]

∴ Δ DAP ≅ Δ EBP (By A.S.A. congruence rule)

Hence, proved that Δ DAP ≅ Δ EBP.

(ii) As,

Δ DAP ≅ Δ EBP

We know that,

Corresponding parts of congruent triangles are equal.

∴ AD = BE (By C.P.C.T.)

Hence, proved that AD = BE.