Solve for x in a triangle given three angles, one of which is an expression in term...

1. Identify the problem type: This is an algebra problem involving the properties of a triangle.
2. Recall the triangle angle sum theorem: The sum of the interior angles of any triangle is always $180^\circ$.
3. Set up the equation: Based on the theorem, we can write an equation by adding the three given angles and setting the sum equal to $180^\circ$. The angles are $65^\circ$, $36^\circ$, and $(5x+14)^\circ$.
$$65 + 36 + (5x + 14) = 180$$
4. Combine constant terms: Add the known angle measures together.
$$101 + (5x + 14) = 180$$
$$115 + 5x = 180$$
5. Isolate the term with x: Subtract $115$ from both sides of the equation.
$$5x = 180 - 115$$
$$5x = 65$$
6. Solve for x: Divide both sides by $5$.
$$x = \frac{65}{5}$$
$$x = 16$$
7. Verify the answer (optional but recommended): Substitute $x=16$ back into the expression for the third angle to ensure the sum is $180^\circ$.
Angle 1: $65^\circ$
Angle 2: $36^\circ$
Angle 3: $5x + 14 = 5(16) + 14 = 80 + 14 = 94^\circ$
Sum: $65^\circ + 36^\circ + 94^\circ = 101^\circ + 94^\circ = 195^\circ$.
There seems to be a calculation error in the verification step. Let's recheck step 4.
Step 4 recheck: $65 + 36 = 101$. $101 + 14 = 115$. So $115 + 5x = 180$. This is correct.
Step 5 recheck: $180 - 115 = 65$. So $5x = 65$. This is correct.
Step 6 recheck: $65 / 5 = 13$. So $x=13$.
Let's re-verify with $x=13$.
Angle 1: $65^\circ$
Angle 2: $36^\circ$
Angle 3: $5x + 14 = 5(13) + 14 = 65 + 14 = 79^\circ$
Sum: $65^\circ + 36^\circ + 79^\circ = 101^\circ + 79^\circ = 180^\circ$. This is correct.
The previous calculation for $65/5$ was incorrect. $65/5 = 13$.
Therefore, $x=13$.