Question

question image (2x - 6)° (x + 10)° (x + 2) q. find the value of x

Explanation:

Assuming this is a triangle with angle bisector or exterior - interior angle relation, but from the given angles, if we assume it's a triangle and the angle \((2x - 9)^{\circ}\) is an exterior angle and \((x + 2)^{\circ}\) and \((x + 10)^{\circ}\) are interior angles, by the exterior angle theorem which states that the exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.

Step 1: Apply Exterior Angle Theorem

The exterior angle theorem formula is: Exterior angle \(=\) sum of two non - adjacent interior angles.
So, we have the equation:
\(2x-9=(x + 2)+(x + 10)\)

Step 2: Simplify the right - hand side

Simplify \((x + 2)+(x + 10)\):
\(x+2+x + 10=2x+12\)
So our equation becomes \(2x-9 = 2x+12\)

Wait, this gives \(- 9=12\) which is a contradiction. Maybe the angle \((2x - 9)^{\circ}\) is an interior angle and the other two are also interior angles and the sum of angles in a triangle is \(180^{\circ}\). Let's try that.

Step 1: Sum of angles in a triangle

The sum of interior angles of a triangle is \(180^{\circ}\). So:
\((2x-9)+(x + 2)+(x + 10)=180\)

Step 2: Combine like terms

Combine the \(x\) terms and the constant terms:
\(2x+x+x-9 + 2+10=180\)
\(4x+3 = 180\)

Step 3: Solve for \(x\)

Subtract 3 from both sides:
\(4x=180 - 3=177\)
\(x=\frac{177}{4}=44.25\)

Wait, maybe the first angle is \((2x - 9)^{\circ}\), the second is \((x + 2)^{\circ}\) and the third is \((x + 10)^{\circ}\). Let's re - check the addition:

\(2x-9+x + 2+x + 10=(2x+x+x)+(-9 + 2+10)=4x + 3\)

If we made a mistake in the angle labels, maybe the first angle is \((2x - 9)^{\circ}\), and the other two angles are such that \((2x-9)=(x + 10)+(x + 2)\) which we saw is wrong, or maybe the triangle is isoceles? Wait, perhaps the original problem has a typo, but assuming the sum of angles in a triangle:

If we assume the angles are \((2x-9)^{\circ}\), \((x + 2)^{\circ}\) and \((x + 10)^{\circ}\), then:

\(2x-9+x + 2+x + 10 = 180\)

\(4x+3=180\)

\(4x=177\)

\(x = 44.25\)

But maybe the first angle is \((2x + 9)^{\circ}\) (a possible typo). Let's try that:

\(2x + 9+x + 2+x + 10=180\)

\(4x+21 = 180\)

\(4x=159\)

\(x=\frac{159}{4}=39.75\)

Alternatively, if the exterior angle is \((2x - 9)^{\circ}\) and one interior angle is \((x + 10)^{\circ}\) and the other is \((x + 2)^{\circ}\), and the exterior angle is equal to the sum of the two remote interior angles, but we saw that gives a contradiction.

Wait, maybe the angle at the top is \((2x - 9)^{\circ}\) and the two base angles are \((x + 2)^{\circ}\) and \((x + 10)^{\circ}\). Let's re - calculate the sum:

\((2x-9)+(x + 2)+(x + 10)=2x-9+x + 2+x + 10=4x + 3\)

Set equal to 180:

\(4x=177\)

\(x = 44.25\)

Answer:

\(x = 44.25\) (or if there was a typo and the first angle was \((2x+9)^{\circ}\), \(x = 39.75\)). But based on the given problem as is, \(x=\frac{177}{4}=44.25\)