In figure, AD ⊥ CD and CB ⊥ CD.
Question:

In figure, AD ⊥ CD and CB ⊥ CD. If AQ = BP and DP = CQ, prove that ∠DAQ = ∠CBP.

 

Solution:

Given in the figure, AD ⊥ CD and CB ⊥ CD.

And AQ = BP and DP = CQ,

To prove that ∠DAQ = ∠CBP

Given that DP = QC

Add PQ on both sides

DP + PQ = PQ + QC

DQ = PC … (i)

Now, consider triangle DAQ and CBP,

We have

∠ADQ = ∠BCP = 90°   [given] AQ = BP [given]

And DQ = PC [From (i)]

So, by RHS congruence criterion, we have

ΔDAQ ≅ ΔCBP

Now,

∠DAQ = ∠CBP   [Corresponding parts of congruent triangles are equal]

Hence proved

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