Understanding the Triangle Angle Problem

The question gives us the expressions for the three angles of a triangle in terms of a variable \(x\). The angles are given as \((x - 6)^\circ\), \((x + 26)^\circ\), and \(8x^\circ\). We need to find the value of \(2x\).

To solve this, we need to use a fundamental property of triangles which relates their interior angles.

Using the Triangle Angle Sum Property

A key property of any triangle is that the sum of its interior angles is always equal to \(180^\circ\). This property holds true for all types of triangles.

We can set up an equation by adding the given expressions for the three angles and equating the sum to \(180^\circ\).

Setting up and Solving the Equation for x

The three angles are \((x - 6)\), \((x + 26)\), and \(8x\). According to the triangle angle sum property:

Sum of angles = \(180^\circ\)

$$(x - 6) + (x + 26) + 8x = 180$$

Now, let's simplify and solve this linear equation for \(x\):

1. Combine the terms involving \(x\): $$(x + x + 8x) + (-6 + 26) = 180$$ $$10x + 20 = 180$$
2. Subtract 20 from both sides of the equation: $$10x + 20 - 20 = 180 - 20$$ $$10x = 160$$
3. Divide both sides by 10 to find the value of \(x\): $$\frac{10x}{10} = \frac{160}{10}$$ $$x = 16$$

So, the value of \(x\) is 16.

Calculating the Value of 2x

The question asks for the value of \(2x\), not just \(x\). Now that we have found \(x = 16\), we can easily calculate \(2x\).

$$2x = 2 \times 16$$

$$2x = 32$$

Verifying the Triangle Angles

It's good practice to substitute the value of \(x = 16\) back into the original angle expressions to find the actual angle measures and ensure they sum up to \(180^\circ\).

• First angle: \(x - 6 = 16 - 6 = 10^\circ\)
• Second angle: \(x + 26 = 16 + 26 = 42^\circ\)
• Third angle: \(8x = 8 \times 16 = 128^\circ\)

Let's check the sum:

$$10^\circ + 42^\circ + 128^\circ = 52^\circ + 128^\circ = 180^\circ$$

The sum of the angles is indeed \(180^\circ\), which confirms our value of \(x = 16\) is correct and consequently, the value of \(2x = 32\) is also correct.

Revision Table: Triangle Angles Summary

Additional Information on Triangle Angles and Types

Triangles are classified based on their angles and sides. Understanding these classifications can be helpful in various geometry problems.

• Based on Angles:
  • Acute Triangle: All three angles are less than \(90^\circ\). (In our solved example, angles are \(10^\circ, 42^\circ, 128^\circ\). This is NOT an acute triangle.)
  • Right Triangle: One angle is exactly \(90^\circ\).
  • Obtuse Triangle: One angle is greater than \(90^\circ\) but less than \(180^\circ\). (Our example with \(128^\circ\) is an obtuse triangle.)
• Based on Sides:
  • Equilateral Triangle: All three sides are equal, and all three angles are \(60^\circ\).
  • Isosceles Triangle: Two sides are equal, and the angles opposite those sides are equal.
  • Scalene Triangle: All three sides are different lengths, and all three angles are different measures. (Our example with angles \(10^\circ, 42^\circ, 128^\circ\) is likely a scalene triangle).

Remember, the sum of interior angles is always \(180^\circ\) for any of these triangle types.