Question

1. in the figure to the right, the two angles form a linear pair. find ( m angle a b d ).

Explanation:

Step1: Use the linear - pair property

Since \(\angle ABD\) and \(\angle DBC\) form a linear pair, we know that \(\angle ABD+\angle DBC = 180^{\circ}\) (by the definition of a linear pair of angles: two adjacent angles whose non - common sides are opposite rays and their sum is \(180^{\circ}\)).
Given \(\angle DBC=135^{\circ}\) and \(\angle ABD = 9x\), we substitute into the equation: \(9x + 135=180\).

Step2: Solve the equation for \(x\)

Subtract \(135\) from both sides of the equation \(9x+135 = 180\).
We get \(9x=180 - 135\), so \(9x=45\).
Divide both sides by \(9\): \(x=\frac{45}{9}=5\).

Step3: Find the measure of \(\angle ABD\)

Substitute \(x = 5\) into \(\angle ABD=9x\). Then \(\angle ABD=9\times5 = 45^{\circ}\).

Answer:

\(m\angle ABD = 45^{\circ}\)