To the nearest tenth of a degree, find the sizes of the acute angles in the 5-12-13 triangle and in the 9-12-15 triangle. This enables you to calculate the sizes of the angles in the 13 - 14-15 triangle. Show how to do it, then invent another example of this sort.

Step-by-step explanation:

A triangle whose sides are 5-12-13 is a right angle triangle because the sides form a Pythagoras triple. This means that

Hypotenuse² = opposite side² + adjacent side²

If hypotenuse = 13,

Opposite side = 12, then we can determine one acute angle by applying the sine trigonometric ratio

Sin θ = opposite side/adjacent side

Sin θ = 12/13 = 0.923

θ = Sin^-1 (0.923) = 67.4°

The other acute angle is

90 - 67.4 = 22.6°

For 9-12-15 triangle

Sin θ = 12/15 = 0.8

θ = Sin^-1 (0.8) = 53.1°

The other acute angle is

90 - 53.1 = 36.9°

For 13 - 14-15 triangle,

Sin θ = 14/15 = 0.933

θ = Sin^-1 (0.933) = 68.9°

The other acute angle is

90 - 68.9 = 21.1°

Another example would be 3-4-5

Sin θ = 4/5 = 0.933

θ = Sin^-1 (0.8) = 53.1°

The other acute angle is

90 - 53.1 = 36.9°