Question

what is the value of s? s =

Explanation:

Step1: Identify angle - sum property

In right - triangle $ABC$ and right - triangle $DCB$, since $AB = DC$ and $BC$ is common, $\triangle ABC\cong\triangle DCB$ (by Hypotenuse - Leg congruence in right - triangles). Then $\angle DBC=\angle ACB$. In right - triangle $ABC$, $\angle ABC = 90^{\circ}$.
We know that $\angle ABD+\angle DBC = 90^{\circ}$. Also, in the given figure, $\angle ABD = 7s$ and $\angle DBC=s + 42^{\circ}$.

Step2: Set up equation

Since $\angle ABD+\angle DBC = 90^{\circ}$, we can write the equation $7s+(s + 42^{\circ})=90^{\circ}$.

Step3: Simplify the equation

Combine like terms: $7s+s+42^{\circ}=90^{\circ}$, which gives $8s+42^{\circ}=90^{\circ}$.
Subtract $42^{\circ}$ from both sides: $8s=90^{\circ}- 42^{\circ}$, so $8s = 48^{\circ}$.

Step4: Solve for $s$

Divide both sides by 8: $s=\frac{48^{\circ}}{8}=6^{\circ}$.

Answer:

$6$