If the angles, in degrees, of a triangle are x, 3x + 20 and 6x, the triangle must be

Question

Accepted Answer

To determine why the triangle is obtuse, we need to analyze the given angles: $x$, $3x + 20$, and $6x$. The sum of the angles in a triangle is always 180 degrees. Therefore, we can set up the equation:

$$ x + (3x + 20) + 6x = 180 $$

Simplifying this equation:

$$ x + 3x + 20 + 6x = 180 $$
$$ 10x + 20 = 180 $$
$$ 10x = 160 $$
$$ x = 16 $$

Now, we can find the measures of the angles:

1. $ x = 16 $ degrees
2. $ 3x + 20 = 3(16) + 20 = 48 + 20 = 68 $ degrees
3. $ 6x = 6(16) = 96 $ degrees

The angles of the triangle are 16 degrees, 68 degrees, and 96 degrees. An obtuse triangle is defined as a triangle with one angle greater than 90 degrees. In this case, the angle measuring 96 degrees is greater than 90 degrees, confirming that the triangle is indeed obtuse. Therefore, Option 1 is correct.

Now, let's explain why the other options are incorrect:

2. **Acute**: An acute triangle has all angles less than 90 degrees. Since one of the angles in this triangle is 96 degrees, which is greater than 90 degrees, the triangle cannot be acute.

3. **Right**: A right triangle has one angle exactly equal to 90 degrees. None of the angles in this triangle are 90 degrees, so it cannot be a right triangle.

4. **Isosceles**: An isosceles triangle has at least two equal angles. In this triangle, the angles are 16 degrees, 68 degrees, and 96 degrees, all of which are different. Therefore, the triangle is not isosceles.

In conclusion, the triangle is obtuse because it has one angle greater than 90 degrees, specifically 96 degrees, which supports Option 1 as the correct answer.