So like tangent theorem to solve this ?

You can use the Pythagorean Theorem since it's indicated in the problem that the triangle is right (has a 90 degree angle in it).

So what you can do is use: a^2 + b^2 = c^2

8^2 + x^2 = 16^2

64 + x^2 = 256

x^2 = 192

x = 8√3

I don't think you can use tangent in this problem since no other interior angles are given.

1st Method- Pythagoras:

16² = x² + 8²  → 256 = x² + 64 → x² 256 - 64 →x² = 192 and x =√192 = 13.85

2nd Method- Trigo

sin Ф = 8/16 = 1/2 and sin⁻¹ (1/2) = 30°

now tan 30° = 8/x
 0.57735 = 8/x and x = 8/(0.57735) = 13.85

Answer: question no. 1 -a. yes, the sides are in the ratio 2 : 5.question no. 2 -b. 59°question no. 3 -c. angles x and y are remote interior angles.question no. 4 -d. 90°, 45° and 45°question no. 5 -b. mn and qrstep-by-step explanation: question no. 1 -rs : xu = 3 : 7.5 = 2 : 5rt : x? = 6.4 : 16 = 2 : 5st : u? = 6 : 15 = 2 : 5as the ratios of all sides are equivalent, the triangles must be congruent and the sides must be proportional.therefore, the answer is: a. yes, the sides are in the ratio 2 : 5.question no. 2 -in the second diagram, angle b is the only letter not shown. therefore, the number must be equivalent to angle b.if we substitute the value of angle b as being 22° in the first diagram, we can now use the angle sum of a triangle to obtain the value of angle a.angle sum of a triangle = 180°therefore: let angle a = aa + 99 + 22 = 180a = 180 - 99 - 22a = 59°therefore, the answer is: b. 59°question no. 3 -angles x and y don't share the vertex of the triangle with the exterior angle w. remote interior angles don't share a vertex of a triangle with an exterior angle. this means that angles x and y are remote interior angles.therefore, the answer is: c. angles x and y are remote interior angles.question no. 4 -the sum of all interior angles of a triangle must be equivalent to 180°.90°, 45° and 45° are the only set of angle measures that are equivalent to 180°.therefore, the answer is: d. 90°, 45° and 45°question no. 5 -the only pair of sides stated in the options which are corresponding are mn and qr.therefore, the answer is: b. mn and qr.