Question

Please help with explanation.............................................

Answer 1

Answer:

Easy!

GIVEN:

ABCD is a square.

∴∠A=∠B=∠C=∠D=90°.

APB is equilateral. Each angle of an equilateral triangle is 60°.

∴∠PAB=∠ABP=∠BPA=60°.

TO FIND:

$\angle x$

SOLUTION:

Now, see ∠A.

∠A=∠PAB+∠PAD.

Here, ∠A=90°, ∠PAB=60°.

90°=60°+∠PAD

∠PAD=30°

Now, join DP.

Here, we can notice that:

AP=BP (Equilateral triangle)

AD=BC (Sides of a square are equal)

∠PBC=∠PAD=30° (Proved above)

Thus, ΔAPD≅ΔPBC.

CPCT: DP=PC, ∠BPC=∠APD ----(1)

Now, In ΔAPD, AP=AB. But, AB=AD. Thus, AP=AD. So, ΔAPD is an isosceles triangle. And we know that base angles are equal. Thus:

∠ADP=∠APD.

Here, in ΔAPD,

∠APD+∠PAD+∠ADP=180° [∠PAD=30° and ∠ADP=∠APD]

2∠APD+30=180

∠APD=(180-30)/2

∠APD=75°.

Now, see (1). We proved that ∠BPC=∠APD. thus,

∠BPC=75°.

Here, 75% of our work is done.

We know that ∠DPC+∠BPC+∠BPA+∠APD=360°.

Substitute ∠BPA=60°, ∠BPC=∠APD=75°.

∠DPC+75+75+60=360

∠DPC=150°.

Now, look at ΔDPC. Again, look at (1).

DP=PC. ∴ ΔDPC is isosceles.

Thus, the base angles are equal. ∴∠PDC=∠PCD

∠DPC+∠PDC+x=180° [∠DPC=150°, ∠PDC=x]

x+x+150=180

2x=30

x=30÷2.

x=15°.

This was a pretty long sum, this took a lot of time and effort. I would appreciate if you would mark me brainliest. Thank you!

HOPE THIS HELPS :D