Question

find the value of x. then find the measure of each angle and write the answers in ascending order. 37° (5x + 62)° (3x + 47)°

Explanation:

Step1: Apply the exterior angle theorem

The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So, \(3x + 47=(5x + 62)+37\).

Step2: Simplify the equation

First, simplify the right - hand side: \((5x + 62)+37 = 5x+99\). Then the equation becomes \(3x + 47=5x + 99\).
Subtract \(3x\) from both sides: \(47 = 2x+99\).
Subtract \(99\) from both sides: \(47−99 = 2x\), so \(- 52=2x\).
Divide both sides by \(2\): \(x=-26\).

Step3: Calculate the angles

For the angle \((5x + 62)^{\circ}\), substitute \(x =-26\): \(5\times(-26)+62=-130 + 62=-68^{\circ}\). This is not possible for an angle measure. There is a mistake. Let's re - check the exterior angle theorem.
The correct exterior angle theorem formula is: \(3x + 47=37+(5x + 62)\) (wrong above, should be \(3x + 47\) (exterior angle) \(=37+(5x + 62)\) (sum of non - adjacent interior angles)).
Simplify: \(3x+47=5x + 99\) (same as before, wrong).
The correct formula is: The exterior angle \(3x + 47\) and the non - adjacent interior angles are \(37^{\circ}\) and \((180-(5x + 62))^{\circ}\) (using the linear pair for the angle adjacent to \(5x + 62\)).
Another way: Using the angle sum property of a triangle. Let the third interior angle be \(y\), \(y = 180-(5x + 62)=118 - 5x\). Then \(37+(118 - 5x)+(3x + 47)=180\) (angle sum of a triangle).
Simplify: \(37+118-5x+3x + 47=180\).
\(202-2x=180\).
Subtract \(202\) from both sides: \(-2x=180 - 202=-22\).
Divide by \(-2\): \(x = 11\).

Step4: Calculate the angles

For \(x = 11\):
The angle \((3x + 47)^{\circ}=(3\times11 + 47)^{\circ}=(33 + 47)^{\circ}=80^{\circ}\).
The angle \((5x + 62)^{\circ}=(5\times11+62)^{\circ}=(55 + 62)^{\circ}=117^{\circ}\).
The third angle (using angle sum of triangle \(180-(80 + 37)=63^{\circ}\)).

Answer:

\(x = 11\), the angles (in ascending order) are \(37^{\circ},63^{\circ},80^{\circ}\)