Solving for the Value of x in a Triangle

The question provides the expressions for the three angles of a triangle and asks for the value of x.

We know a fundamental property of triangles: the sum of the interior angles of any triangle is always 180 degrees.

The given angles are:

• Angle 1: $(8x - 15)^\circ$
• Angle 2: $(6x - 11)^\circ$
• Angle 3: $(4x - 10)^\circ$

Using the property of the sum of angles in a triangle, we can set up the following equation:

$(8x - 15) + (6x - 11) + (4x - 10) = 180$

Now, we will solve this linear equation for x:

First, combine the terms involving x:

$8x + 6x + 4x = (8 + 6 + 4)x = 18x$

Next, combine the constant terms:

$-15 - 11 - 10 = -(15 + 11 + 10) = -36$

Substitute these combined terms back into the equation:

$18x - 36 = 180$

To isolate the term with x, add 36 to both sides of the equation:

$18x - 36 + 36 = 180 + 36$

$18x = 216$

Finally, divide both sides by 18 to find the value of x:

$x = \frac{216}{18}$

$x = 12$

So, the value of x is 12.

Let's verify this by finding the measure of each angle using $x=12$:

• Angle 1: $8(12) - 15 = 96 - 15 = 81^\circ$
• Angle 2: $6(12) - 11 = 72 - 11 = 61^\circ$
• Angle 3: $4(12) - 10 = 48 - 10 = 38^\circ$

Sum of the angles: $81^\circ + 61^\circ + 38^\circ = 180^\circ$. The sum is indeed 180 degrees, and all angles are positive, which confirms that $x=12$ is the correct value.

Revision Table: Triangle Angles and Solving Equations

Additional Information: Types of Triangles

Triangles can be classified based on their angles or sides.

Based on angles:

• Acute Triangle: All three angles are less than $90^\circ$. (The triangle in this problem with angles $81^\circ, 61^\circ, 38^\circ$ is an acute triangle).
• Right Triangle: One angle is exactly $90^\circ$. The other two angles are acute and sum to $90^\circ$.
• Obtuse Triangle: One angle is greater than $90^\circ$. The other two angles are acute.

Based on sides:

• Scalene Triangle: All three sides have different lengths, and all three angles have different measures.
• Isosceles Triangle: Two sides are equal in length, and the angles opposite those sides are equal in measure.
• Equilateral Triangle: All three sides are equal in length, and all three angles are equal in measure ($60^\circ$ each).