In triangle FGH, the measure of angle F is twice as large as the measure of angle G and the measure of angle G is 3 times the measure of angle H. Find the measure of angle G
The Correct Answer and Explanation is:
We are given a triangle FGH with the following relationships between the angles:
- The measure of angle F is twice the measure of angle G
- The measure of angle G is three times the measure of angle H
Let us use algebra to find the measure of angle G.
Step 1: Define the unknown
Let the measure of angle H be x degrees.
From the given:
- Angle G is 3 times angle H → G = 3x
- Angle F is twice angle G → F = 2 × 3x = 6x
Step 2: Use the triangle angle sum rule
In any triangle, the sum of the interior angles is always 180 degrees. So:
F + G + H = 180
Substitute the expressions:
6x + 3x + x = 180
Combine like terms:
10x = 180
Step 3: Solve for x
Divide both sides by 10:
x = 18
Step 4: Find the measure of angle G
We had defined G = 3x
So G = 3 × 18 = 54 degrees
Final Answer:
The measure of angle G is 54 degrees.
Explanation
To solve problems involving angles in a triangle, it is essential to remember that the sum of the interior angles in any triangle is always 180 degrees. This property provides a reliable equation that can be used in combination with given relationships between the angles.
In this triangle FGH, we were told that angle F is twice the size of angle G, and angle G is three times the size of angle H. By assigning a variable to the smallest angle (angle H), we set H = x. This made it easier to express the other angles in terms of x: G = 3x and F = 6x. This approach simplifies complex word problems and allows us to reduce everything to a single variable.
Once the angles were expressed in terms of x, the next step was to use the triangle angle sum rule. Adding the expressions for F, G, and H and setting the sum equal to 180 gave us a solvable equation: 6x + 3x + x = 180. Solving that gave us x = 18, which represents angle H.
From there, we used the value of x to find the measure of angle G, which is 3x. Multiplying 18 by 3 gave us 54 degrees.
This method demonstrates the power of algebra in solving geometric problems and shows the importance of understanding relationships between angles in a triangle.
