Question

What are the angle measures in a triangle with side measures 5, 12, 13?

Answer 1

These angles add up to 180 degrees, as is the case in any triangle.

Explanation:

The triangle with side measures 5, 12, and 13 is a right triangle. This is a special type of right triangle known as a Pythagorean triple, where the sides are in the ratio of 5:12:13. In a right triangle, one of the angles is a 90-degree (right) angle. To find the measures of the other two angles, you can use trigonometric ratios.

1. The right angle is 90 degrees.

2. The angle opposite the side with a length of 5 units can be found using the sine function:

sin(θ) =opposite divided by hypotenus = 5 divided by 13

\theta = \sin^{-1}\left(\frac{5}{13})

Calculate θ, which is the measure of the smaller acute angle.

3. The remaining angle, which is opposite the side with a length of 12 units, is the complement of θ, because the sum of the two smaller angles in a right triangle is 90 degrees:

Complement of θ=90∘ − θ

Now, calculate θ and its complement: θ≈22.62 ∘

The larger angle (complement of θ) is approximately:

Complement of θ≈90 ∘ −22.62 ∘ ≈67.38 ∘

So, the measures of the angles in the triangle are approximately:

• 90 degrees (right angle)

• 22.62 degrees

• 67.38 degrees

These angles add up to 180 degrees, as is the case in any triangle.

Answer 2

Angles are:

• 90°
• 22.62°
• 67.38°

Explanation:

To calculate the angle measures, we can use the Pythagorean theorem to confirm that the triangle is a right triangle, and then use the sine and cosine functions to calculate the measures of the acute angles.

Pythagorean theorem:

`\sf a^2 + b^2 = c^2`

where a and b are the lengths of the legs of the triangle, and c is the length of the hypotenuse.

Substituting the side lengths of the triangle, we get:

`\sf 5^2 + 12^2 = 13^2`

`\sf 25 + 144 = 169`

`\sf 169 = 169`

Therefore, the triangle is a right triangle, with the 5 and 12 sides being the legs and the 13 side being the hypotenuse.Sine and cosine functions:

The sine and cosine functions can be used to calculate the measures of the acute angles in a right triangle, given the lengths of the sides.

The sine function is defined as the ratio of the opposite side to the hypotenuse:

`\sf sin(\theta) = \frac{\textsf{opposite}}{\textsf{hypotenuse}}`

The cosine function is defined as the ratio of the adjacent side to the hypotenuse:

`\sf cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}`

Substituting the side lengths of the triangle, we get:

`\sf sin(\theta_1) = (5)/(13)`

`\sf cos(\theta_1) = (12)/(13)`

`\sf sin(\theta_2) = (12)/(13)`

`\sf cos(\theta_2) = (5)/(13)`

Using a calculator, we can solve the measures of the acute angles:

Let's take sine angle only:

`\sf theta_1 = sin^(-1) \left((5)/(13)\right) \approx 22.62^\circ`

`\sf theta_2 = sin^(-1) \left((12)/(13)\right) \approx 67.38^\circ`

Therefore, the angle measures in a triangle with side measures 5, 12, and 13 are: .

• 90°
• 22.62°
• 67.38°