1. **Problem statement:** We have two congruent triangles, $\triangle ABC$ and $\triangle ABD$. Given: $AC=10$, $AB=6$, $\angle BAC=53^\circ$, $\angle ACB=37^\circ$, $BC=8$ (perpendicular). Triangle $ABD$ shares side $AB=6$, $\angle B$ is right angle in $\triangle ABD$, and $BD$ is perpendicular to $AB$. We need to find the length of $BD$ and the measure of $\angle DAB$. 2. **Step 1: Understand congruence and given data.** Since $\triangle ABC \cong \triangle ABD$, corresponding sides and angles are equal. In $\triangle ABC$, $BC$ is perpendicular to $AC$ and $BC=8$. In $\triangle ABD$, $BD$ is perpendicular to $AB$ and $AB=6$. 3. **Step 2: Find $BD$.** Since $\triangle ABD$ is right angled at $B$, with $AB=6$ and $BD$ perpendicular to $AB$, $BD$ is the height from $B$ to $AD$. Because $\triangle ABC \cong \triangle ABD$, side $BC$ corresponds to side $BD$. Given $BC=8$, so $BD=8$. 4. **Step 3: Find $\angle DAB$.** Since $\triangle ABC \cong \triangle ABD$, $\angle BAC$ corresponds to $\angle DAB$. Given $\angle BAC=53^\circ$, so $\angle DAB=53^\circ$. **Final answers:** $$BD=8$$ $$\angle DAB=53^\circ$$