Question

1. find \\( \angle a b d \\)

Explanation:

Step1: Use the right - angle relationship

Since \(\angle ABD+(23x - 14)^{\circ}=90^{\circ}\) (complementary angles), and we know that the sum of \((15x + 18)+(23x-14)=90\) (because the two angles \((15x + 18)^{\circ}\) and \((23x - 14)^{\circ}\) form a right - angle).
First, combine like terms: \(15x+23x+18 - 14=90\).
\(38x+4 = 90\).

Step2: Solve for \(x\)

Subtract \(4\) from both sides: \(38x=90 - 4\).
\(38x=86\).
\(x=\frac{86}{38}=\frac{43}{19}\).

Step3: Calculate \(\angle ABD\)

Substitute \(x = \frac{43}{19}\) into \(\angle ABD=(15x + 18)^{\circ}\).
\(\angle ABD=15\times\frac{43}{19}+18\).
\(\angle ABD=\frac{645}{19}+18=\frac{645+18\times19}{19}=\frac{645 + 342}{19}=\frac{987}{19}=51^{\circ}\).

Answer:

\(51^{\circ}\)