Question

1. in the diagram below ( p parallel q ), and ( mangle 7=(x^{2}+90)^{circ} ) and ( mangle 4=(35 - 6x)^{circ} ). determine ( mangle 3 ).

Explanation:

Step1: Use the property of parallel lines

Since \(p\parallel q\), \(\angle4\) and \(\angle7\) are supplementary (consecutive interior angles). So \(m\angle4 + m\angle7=180^{\circ}\).
Substitute \(m\angle7=(x^{2}+90)^{\circ}\) and \(m\angle4=(35 - 6x)^{\circ}\) into the equation:
\(x^{2}+90+35 - 6x = 180\)
\(x^{2}-6x + 125-180=0\)
\(x^{2}-6x - 55 = 0\)

Step2: Solve the quadratic equation

Factor the quadratic equation \(x^{2}-6x - 55=(x - 11)(x + 5)=0\)
Set each factor equal to zero:
\(x-11 = 0\) gives \(x = 11\); \(x + 5=0\) gives \(x=- 5\)
If \(x = 11\), \(m\angle4=35-6\times11=35 - 66=-31\) (rejected since angle measure cannot be negative)
If \(x=-5\), \(m\angle4=35-6\times(-5)=35 + 30 = 65^{\circ}\)

Step3: Find \(m\angle3\)

Since \(\angle3\) and \(\angle4\) are vertical angles, \(m\angle3=m\angle4\)

Answer:

\(65^{\circ}\)