In triangle XYZ, XY¯¯¯¯¯¯¯¯=15, YZ¯¯¯¯¯¯¯=8, ∡X=(3w−35)°, and ∡Y=52°. In triangle RST, RS¯¯¯¯¯¯¯=15, RT¯¯¯¯¯¯¯=8, ∡R=(w+17)°, and ∡S=52°. Find the angle measure for each of the four angles whose measures are not known.

A. ∡R=43°, ∡T=85°, ∡X=43°, ∡Z=85°

B. ∡R=64°, ∡T=64°, ∡X=64°, ∡Z=64°

C. ∡R=35°, ∡T=93°, ∡X=35°, ∡Z=93°

D. ∡R=52°, ∡T=76°, ∡X=52°, ∡Z=76°

A triangle is a polygon with three sides and three angles. The types of triangles are scalene triangle, equilateral triangle, right angled triangle and obtuse triangle.

Cosine rule states that given a triangle with sides a, b, c and their corresponding angles opposite to the sides as A, B, C. Then:

[tex]a^2=b^2+c^2-2bc*cos(A)[/tex]

In triangle XYZ, we can find XZ using cosine rule:

[tex]XZ^2=XY^2+YZ^2-2(XY)(YZ)cosY\\\\subtituting:\\\\XZ^2=15^2+8^2-2(15)(8)cos(52)\\\\XZ^2=141.24\\\\XZ=11.88\\\\Using\ sine\ rule:\\\\\frac{XZ}{sin(Y)}=\frac{XY}{sin(Z)}\\\\\frac{11.88}{sin(52)}=\frac{15}{sin(Z)}\\\\sin(Z)=15*sin(52) /11.88\\\\sin(Z)=0.9946\\\\Z=sin^{-1}(0.9946)\\\\Z=85^0[/tex]

∠X + ∠Y + ∠Z = 180° (sum of angles in a triangle)

∠X + 52 + 85 = 180

∠X + 137 = 180

∠X = 43°

∠X = 3w - 35

3w - 35 = 43

3w = 78

w = 26

In triangle RST, using sine rule:

∠R = w + 17

∠R = 26 + 17

∠R = 43°

∠R + ∠S + ∠T = 180° (sum of angles in a triangle)

43 + 52 + ∠T = 180

∠T + 95 = 180

∠T = 85°