Answer:
34. x = 3 1/3
35. x = 5.2
Explanation:
Given figures involving triangles with angle bisectors and parallel segments, you want to solve for x.
In each case, you need to make use of two proportional relationships. An angle bisector divides the triangle proportionally. An segment parallel to one side of the triangle divides it proportionally.
34.
Using the angle bisector relation, you have ...
QP/PT = QS/ST
x/3 = 5/ST ⇒ x = 15/ST
Using the parallel segment relation, you have ...
PT/QR = ST/SR
3/2 = ST/3 ⇒ ST = 9/2
Using the value of ST in the equation for x gives ...
x = 15/(9/2) = 30/9 = 10/3
x = 3 1/3
35.
Using the angle bisector relation, you have ...
EF/ED = CF/CD
7.2/9 = CF/6 ⇒ CF = 6(7.2/9) = 4.8
Using the parallel segment relation, you have ...
CB/BA = CF/FE
x/7.8 = 4.8/7.2
x = 7.8(4.8/7.2)
x = 5.2
__
Additional comment
In each case, we assigned a numerical value to the intermediate variable (ST, CF). We didn't actually need to do that.