Triangle FGH is a right triangle. Angle G is a right angle, m∠F = 9x – 1, and m∠H = 3x + 7. What is m∠F?
The Correct Answer and Explanation is:
In a triangle, the sum of the interior angles is always 180 degrees. Since triangle FGH is a right triangle and angle G is the right angle, angle G measures 90 degrees.
Given:
- ∠G = 90°
- m∠F = 9x – 1
- m∠H = 3x + 7
To find the value of x, use the fact that:
m∠F + m∠H = 90° (since the other 90° is taken by the right angle)
So:
(9x – 1) + (3x + 7) = 90
Simplify the left-hand side:
9x – 1 + 3x + 7 = 90
12x + 6 = 90
Subtract 6 from both sides:
12x = 84
Divide both sides by 12:
x = 7
Now substitute x = 7 into the expression for m∠F:
m∠F = 9x – 1
m∠F = 9(7) – 1
m∠F = 63 – 1
m∠F = 62°
Explanation (300 words):
Triangles are fundamental shapes in geometry, each having three angles whose measures always add up to 180 degrees. In a right triangle, one of these angles measures exactly 90 degrees. The two remaining angles must then sum up to 90 degrees to maintain the total sum of 180 degrees.
Triangle FGH is labeled as a right triangle, with angle G specified as the right angle. That leaves angles F and H as the two acute angles. The measures of these angles are given in terms of a variable x: angle F is expressed as 9x – 1 and angle H as 3x + 7. Since the right angle at G takes up 90 degrees, angles F and H together must also add up to 90 degrees.
By combining the expressions for angle F and angle H and setting their sum equal to 90 degrees, it becomes possible to form a simple algebraic equation: (9x – 1) + (3x + 7) = 90. Solving this equation reveals the value of x, which turns out to be 7. This value is then substituted back into the expression for angle F, yielding m∠F = 62°.
This process demonstrates how algebra and geometric properties work together to solve unknown angle measures in a triangle. Recognizing the relationships among angles in right triangles is essential in both pure mathematics and applied settings like construction, navigation, and design.