To find the value of $x$, we need to use the
Exterior Angle Theorem.
Step 1: Understand the Rule
The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.
*
Exterior Angle: $(6x - 9)^\circ$
*
Remote Interior Angles: $(2x + 14)^\circ$ and $(3x + 2)^\circ$
So, the equation is:
$$ \text{Exterior Angle} = \text{Interior Angle 1} + \text{Interior Angle 2} $$
Step 2: Set Up the Equation
Substitute the expressions from the diagram into the formula:
$$ 6x - 9 = (2x + 14) + (3x + 2) $$
Step 3: Solve for $x$
First, combine like terms on the right side of the equation:
$$ 6x - 9 = 2x + 3x + 14 + 2 $$
$$ 6x - 9 = 5x + 16 $$
Next, get all the $x$ terms on one side. Subtract $5x$ from both sides:
$$ 6x - 5x - 9 = 16 $$
$$ x - 9 = 16 $$
Finally, isolate $x$ by adding 9 to both sides:
$$ x = 16 + 9 $$
$$ x = 25 $$
Step 4: Check the Answer
Let's plug $x = 25$ back into the original expressions to make sure it works.
* Exterior Angle: $6(25) - 9 = 150 - 9 = 141^\circ$
* Interior Angle 1: $2(25) + 14 = 50 + 14 = 64^\circ$
* Interior Angle 2: $3(25) + 2 = 75 + 2 = 77^\circ$
Check if the sum of the interior angles equals the exterior angle:
$$ 64 + 77 = 141 $$
Since $141 = 141$, the answer is correct.
Final Answer:
x = 25
Parent Tip: Review the logic above to help your child master the concept of exterior angle theorem worksheet.
