In the triangle below, suppose that m∠H=(x+9)\degree, m∠I=(2x−5)\degree, and m∠J=(5x)\degree. Find the degree measure of each angle in the triangle.
The Correct Answer and Explanation is:
Here are the degree measures of each angle in the triangle:
m∠H = 31°
m∠I = 39°
m∠J = 110°
Explanation
This problem requires finding the measure of each interior angle of a triangle when the angles are given as algebraic expressions. The solution is based on the fundamental geometric principle known as the Triangle Angle-Sum Theorem. This theorem states that the sum of the measures of the three interior angles of any triangle is always 180 degrees.
Step 1: Set up the equation.
According to the Triangle Angle-Sum Theorem, the sum of the measures of angles H, I, and J must equal 180°. This can be written as an equation:
m∠H + m∠I + m∠J = 180°
By substituting the given expressions for each angle into this equation, the following algebraic relationship is established:
(x + 9) + (2x – 5) + (5x) = 180
Step 2: Solve the equation for x.
To find the value of x, the equation must be simplified by combining like terms. First, combine all terms containing the variable x:
x + 2x + 5x = 8x
Next, combine the constant terms:
9 – 5 = 4
The simplified equation is now:
8x + 4 = 180
To isolate the variable, subtract 4 from both sides of the equation:
8x = 180 – 4
8x = 176
Finally, divide both sides by 8 to solve for x:
x = 176 / 8
x = 22
Step 3: Calculate the measure of each angle.
Now that the value of x is known, it can be substituted back into the original expression for each angle to find its specific degree measure.
- For m∠H:
m∠H = (x + 9)°
m∠H = (22 + 9)°
m∠H = 31° - For m∠I:
m∠I = (2x – 5)°
m∠I = (2 * 22 – 5)°
m∠I = (44 – 5)°
m∠I = 39° - For m∠J:
m∠J = (5x)°
m∠J = (5 * 22)°
m∠J = 110°
To verify the solution, check if the sum of the calculated angle measures is 180°:
31° + 39° + 110° = 70° + 110° = 180°.
The sum is correct, confirming the accuracy of the results.thumb_upthumb_down
