Question

refer to the figure. if ( mangle 2=(a + 15)^{circ} ) and ( mangle 3=(a + 35)^{circ} ), find the value of ( a ) such that ( overrightarrow{hl}perpoverrightarrow{hj} ).

Explanation:

Step1: Recall the definition of perpendicular lines

If \(\overrightarrow{HL}\perp\overrightarrow{HJ}\), then \(m\angle JHL = 90^{\circ}\), so \(m\angle2 + m\angle3=90^{\circ}\).

Step2: Substitute the given angle measures into the equation

Given \(m\angle2=(a + 15)^{\circ}\) and \(m\angle3=(a + 35)^{\circ}\), we substitute into \(m\angle2 + m\angle3 = 90^{\circ}\).
We get \((a + 15)+(a + 35)=90\).

Step3: Simplify the left - hand side of the equation

Combine like terms: \(a+a+15 + 35=90\), which simplifies to \(2a+50 = 90\).

Step4: Solve for \(a\)

Subtract 50 from both sides: \(2a+50-50=90 - 50\), so \(2a=40\).
Divide both sides by 2: \(a=\frac{40}{2}\).

Answer:

\(20\)