Question

If ray BD bisects angle CBE, ray BC is transparent to ray BA, angle CBD = (3x + 25) degrees, and angle DBE = (7x - 19) degrees, find angle ABD

Answer 1

Check the picture below.

Answer 2

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,

$m\angle CBD=(3x+25)^\circ,~~m\angle DBE=(7x-19)^\circ,~~m\angle ABC=90^\circ.$

We are to find the measure of angle ABD.

Since ray BD bisects angle CBE, so we have

$m\angle CBD=m\angle DBE\\\\\Rightarrow (3x+25)^\circ=(7x-19)^\circ\\\\\Rightarrow 3x+25=7x-19\\\\\Rightarrow 7x-3x=25+19\\\\\Rightarrow 4x=44\\\\\Rightarrow x=\dfrac{44}{4}\\\\\Rightarrow x=11.$

So, we get

$m\angle CBD=(3\times11+25)^\circ=58^\circ.$

Therefore,

$m\angle ABD=m\angle ABC+m\angle CBD=90^\circ+58^\circ=148^\circ.$

Thus, the required measure of angle ABD is 148°.