Answer :
The values are [tex]m\angle \mathrm{ABC}=62^{\circ}[/tex], [tex]m\angle \mathrm{DBE}=62^{\circ}[/tex], [tex]m\angle \mathrm{CBE}=118^{\circ}[/tex] and [tex]m\angle \mathrm{ABD}=118^{\circ}[/tex]
Explanation:
It is given that [tex]\angle \mathrm{ABC}=4x+2[/tex] and [tex]\angle \mathrm{DBE}=5x-13[/tex]
The image having these measurements is attached below:
The angles ABC and DBE are vertically opposite.
Since, vertically opposite angles are equal, [tex]\angle \mathrm{ABC}=\angle \mathrm{DBE}[/tex]
Equating the values, we have,
[tex]\begin{aligned}4 x+2 &=5 x-13 \\2+13 &=5 x-4 x \\15 &=x\end{aligned}[/tex]
Thus, the value of x is 15. Let us substitute x in the equation to find [tex]\angle \mathrm{ABC}[/tex] and [tex]\angle \mathrm{DBE}[/tex]
Thus,
[tex]\begin{aligned}\angle A B C &=4(15)+2 \\&=60+2 \\&=62\end{aligned}[/tex]
Thus, [tex]m\angle \mathrm{ABC}=62^{\circ}[/tex]
Also, substituting x = 15 in [tex]\angle \mathrm{DBE}[/tex]
We have,
[tex]\begin{aligned}\angle DBE &=5(15)-13 \\&=75-13 \\&=62\end{aligned}[/tex]
Thus, [tex]m\angle \mathrm{DBE}=62^{\circ}[/tex]
Hence, the measure of [tex]\angle \mathrm{ABC}=62^{\circ}[/tex] and [tex]\angle \mathrm{DBE}=62^{\circ}[/tex]
To find the measure of [tex]\angle \mathrm{CBE}[/tex] and [tex]\angle \mathrm{ABD}[/tex]:
Since, the angles in a straight line add up to 180°
To find [tex]\angle \mathrm{CBE}[/tex], let us add the angles and equals to 180°
[tex]\angle \mathrm{CBE}+\angle \mathrm{DBE}=180[/tex]
Substituting the value of DBE, we have,
[tex]\angle \mathrm{CBE}+62=180[/tex]
Subtracting both sides by 62,
[tex]\angle \mathrm{CBE}=118[/tex]
Thus, the measure of [tex]\angle \mathrm{CBE}[/tex] is 118°
Since, [tex]\angle \mathrm{CBE}[/tex] and [tex]\angle \mathrm{ABD}[/tex] are vertically opposite, they are equal.
Thus, [tex]\angle \mathrm{ABD}=118[/tex]
Thus, the measure of [tex]\angle \mathrm{ABD}[/tex] is 118°
Hence, the values of the angles are [tex]m\angle \mathrm{ABC}=62^{\circ}[/tex], [tex]m\angle \mathrm{DBE}=62^{\circ}[/tex], [tex]m\angle \mathrm{CBE}=118^{\circ}[/tex] and [tex]m\angle \mathrm{ABD}=118^{\circ}[/tex]
