The horizontal arrows denote that segments RS and VT are parallel. This means that angleR and angleV are congruent. Similarly, angleS and angleT are congruent. They are denoted in blue and red color, respectively:
Therefore,
[tex]\Delta\text{TUV}\approx\Delta SUR[/tex]
by the AA-theorem (angle-angle theorem). That is because:
[tex]\begin{gathered} \angle R=\angle V \\ \angle S=\angle T \\ \angle U=\angle U \end{gathered}[/tex]
Now, lets find the missing measure x. Since the above triangles are similar, we have
[tex]\frac{x}{3}=\frac{x+1}{3+9}[/tex]
which gives
[tex]\frac{x}{3}=\frac{x+1}{12}[/tex]
By multiplying both side by 3, we get
[tex]x=\frac{x+1}{4}[/tex]
and by multiplying both side by 4, we have
[tex]4x=x+1[/tex]
then, x is given as
[tex]\begin{gathered} 4x-x=1 \\ 3x=1 \end{gathered}[/tex]
therefore, we obtain
[tex]x=\frac{1}{3}[/tex]
then, the missing side x measures 1/3.
Now, lets find y. In this case, we have
[tex]\frac{y}{3}=\frac{2}{9+3}[/tex]
then, it yields
[tex]\frac{y}{3}=\frac{2}{12}[/tex]
by multipluying both sides by 3, we have
[tex]\begin{gathered} y=3\cdot\frac{2}{12} \\ y=\frac{2}{4} \\ y=\frac{1}{2} \end{gathered}[/tex]
Therefore, the missing side y measures 1/2.
