A quadrilateral is a shape with four sides. The perimeter of a quadrilateral is the sum of the lengths of its four sides.

For parallel lines with a transversal, the following relationships apply:

• angles in the same position on different parallel lines equal each other (a° = w°, b° = x°, c° = z°, d° = y°)
• alternate interior angles are equal (a° = z°, b° = y°, c° = w°, d° = x°)
• all acute angles (a° = c° = w° = z°) and all obtuse angles (b° = d° = x° = y°) equal each other
• same-side interior angles are supplementary and add up to 180° (e.g. a° + d° = 180°, d° + c° = 180°)

Applying these relationships starting with w° = 15, the value of c° is 15.

Angles around a line add up to 180°. Angles around a point add up to 360°. When two lines intersect, adjacent angles are supplementary (they add up to 180°) and angles across from either other are vertical (they're equal).

A circle is a figure in which each point around its perimeter is an equal distance from the center. The radius of a circle is the distance between the center and any point along its perimeter. A chord is a line segment that connects any two points along its perimeter. The diameter of a circle is the length of a chord that passes through the center of the circle and equals twice the circle's radius (2r).

The first step to solve a quadratic expression that's not set to zero is to solve the equation so that it is set to zero:

a² - 12a + 71 = 5a - 1
a² - 12a + 71 + 1 = 5a
a² - 12a - 5a + 72 = 0
a² - 17a + 72 = 0

Next, factor the quadratic equation:

a² - 17a + 72 = 0
(a - 8)(a - 9) = 0

For this expression to be true, the left side of the expression must equal zero. Therefore, either (a - 8) or (a - 9) must equal zero:

If (a - 8) = 0, a must equal 8
If (a - 9) = 0, a must equal 9

So the solution is that a = 8 or 9