An acute angle measures less than 90°, a right angle measures 90°, and an obtuse angle measures more than 90°.

The sum of the interior angles of a triangle is 180°:
180° = a° + b° + c°
c° = 180° - a° - b°
c° = 180° - 26° - 31° = 123°

The perimeter of a triangle is the sum of the lengths of its sides:

p = x + y + z
p = 11cm + 9cm + 6cm = 26cm

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° = 17, the value of x° is 163.

The first step to solve a quadratic equation that's set to zero is to factor the quadratic equation:

z² + 12z + 32 = 0
(z + 4)(z + 8) = 0

For this expression to be true, the left side of the expression must equal zero. Therefore, either (z + 4) or (z + 8) must equal zero:

If (z + 4) = 0, z must equal -4
If (z + 8) = 0, z must equal -8

So the solution is that z = -4 or -8