Question

triangle pqr similar to triangle xyz, angle Y + angle Z is equal to 90° and xy:yz = 2/3 then determine the angles of Triangle PQR​

Answer 1

Angle Y+Angle Z = 90° ( given).

Angle X = 180°- 90°= 90°.

Triangle PQR and Triangle XYZ are similar (given),

Therefore all angles and all sides of both Triangle will be equal.

So here:---

In Triangle PQR

Angle P= Angle X =90°,

Angle ( Q+R) = ( Y +Z) = 90°.

Answer 2

∠P = X = 90° ,      ∠Q = Y = Sin⁻¹(√5/3) = 48.19° ,       ∠R= Z = Sin⁻¹(2/3) = 41.81°    ΔPQR ≈ Δ XYZ   ∠Y + ∠Z = 90°

Step-by-step explanation:

in Δ xyz

∠Y + ∠Z = 90°

=> ∠X = 90°

Using Sine formulae

xy/SinZ  = yz/SinX

=> xy/yz =SinZ/SinX

SinX = Sin90° = 1

xy/yz = 2/3

=>  2/3  = SinZ /1

=> SinZ = 2/3

=> Z = Sin⁻¹(2/3) = 41.81°

Sin²Z  + Cos²Z = 1

CosZ = Cos(90 - Y) = SinY

=> (2/3)² + Sin²Y = 1

=> Sin²Y =  1  - 4/9

=> Sin²Y =  5/9

=> SinY = √5 / 3

=> Y = Sin⁻¹(√5/3) = 48.19°

   X = 90°

ΔPQR ≈ Δ XYZ

=>  ∠P = X = 90°

    ∠Q = Y = Sin⁻¹(√5/3) = 48.19°

     ∠R= Z = Sin⁻¹(2/3) = 41.81°