Question

this triangle is equilateral. find the value of x. (6x + 6)° (9x - 21)° x = ?

Explanation:

Step1: Recall equilateral triangle angle property

In an equilateral triangle, all interior angles are equal and each measures \(60^\circ\). So, the two given angles \((6x + 6)^\circ\) and \((9x - 21)^\circ\) must be equal to \(60^\circ\) (and also equal to each other since all angles in an equilateral triangle are equal). So we can set the two angle expressions equal to each other: \(6x + 6=9x - 21\).

Step2: Solve the equation for x

First, subtract \(6x\) from both sides of the equation:
\(6x + 6-6x=9x - 21-6x\)
Simplifying gives: \(6 = 3x-21\).

Then, add 21 to both sides:
\(6 + 21=3x-21 + 21\)
Simplifying gives: \(27 = 3x\).

Finally, divide both sides by 3:
\(\frac{27}{3}=\frac{3x}{3}\)
Which gives \(x = 9\). We can also verify by plugging \(x = 9\) back into the angle expressions: \(6(9)+6 = 54 + 6=60^\circ\) and \(9(9)-21=81 - 21 = 60^\circ\), which confirms the angles are equal (and \(60^\circ\) as required for an equilateral triangle).

Answer:

9