4. sketch two triangles (you do not have to provide the sketch). label the lengths of the sides of triangle a as 3, 4, and 5. label the lengths of the sides of triangle b as 5, 12, and 13. a. what is the sum of the measures of the acute angles of any right triangle? explain your reasoning. b. write the tangent ratios for the acute angles of triangle a. c. write the tangent ratios for the acute angles of triangle b. d. write a rule describing the relationship between the tangents of the acute angles of any right triangle.

The answer 
let 's consider the triangle A,
the height is 4
the basis is 3
the hypotenuse 5

A is a right triangle
.a) the sum of the measures of the acute angles of any right triangle is 90°
proof:
the sum of angles in a right triangle is 180° 
so x° +90° (right angle)= 180°, x is  the sum of the measures of the acute angles, x= 180° -90° = 90°
b) Write the tangent ratios for the acute angles of Triangle A
    by using definition, tan = opposite side / adjacent side
let's condider the acute angle 30°
 tan 30° = 3/4, because opposite side =3 and adjacent side=4
 for the other one, applying the some method we found:
 tan 60°= 4/5
we can find also he tangent ratios for the acute angles of Triangle B, by using the same method as given above:
tan 30° = 5/12, because opposite side =5 and adjacent side=12
 for the other one, applying the some method we found:
 tan 60°= 12/13
the main rule describing the relationship between the tangents of the acute angles of any right triangle is 
tangente =  opposite side / adjacent side