Find the missing values in the following figure

Step-by-step explanation:

Angles u° and 55° are the acute angles of a right triangle, so are complementary.

  u° = 90° -55° = 35°

Angles x° and y° are corresponding angles with 55° and u°, so are congruent to them, respectively.

  x° = 55°; y° = 35°

In summary:

  u = 35, x = 55, y = 35

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Corresponding sides of the triangles are proportional, so ...

  p/6 = 3/4

  p = 18/4 = 4.5 . . . . . multiply by 6

The Pythagorean theorem can be used to find z:

  z² = 4² +6² = 52

  z = √52 = 2√13 ≈ 7.2

The scale factor between the larger triangle and the smaller one is ...

  (3+4)/4 = 7/4

so ...

  w = 7/4·z = (7/2)√13 ≈ 12.6

In summary:

  p = 4.5; w ≈ 12.6; z ≈ 7.2

Comment on this problem figure

With the given side measurements, the angles would be more correctly described as 56.3° and 33.7°. The geometry shown cannot exist.

We presume you're to use corresponding side relationships to find the side lengths, and angle relationships to find the angles. Trig relations will relate sides to angles, but those are not needed (or useful) in this problem. Since the angles are not properly related to the sides, trig relationships can only introduce confusion into what is otherwise a straightforward problem.