Answer: The required measure of angle ABD is 148°.
Step-by-step explanation: Given that ray BD bisects angle CBE, ray BC is transparent to ray BA.
Also,
[tex]m\angle CBD=(3x+25)^\circ,~~m\angle DBE=(7x-19)^\circ,~~m\angle ABC=90^\circ.[/tex]
We are to find the measure of angle ABD.
Since ray BD bisects angle CBE, so we have
[tex]m\angle CBD=m\angle DBE\\\\\Rightarrow (3x+25)^\circ=(7x-19)^\circ\\\\\Rightarrow 3x+25=7x-19\\\\\Rightarrow 7x-3x=25+19\\\\\Rightarrow 4x=44\\\\\Rightarrow x=\dfrac{44}{4}\\\\\Rightarrow x=11.[/tex]
So, we get
[tex]m\angle CBD=(3\times11+25)^\circ=58^\circ.[/tex]
Therefore,
[tex]m\angle ABD=m\angle ABC+m\angle CBD=90^\circ+58^\circ=148^\circ.[/tex]
Thus, the required measure of angle ABD is 148°.
