The angles in a triangle sum to 180 degrees. This makes the middle angle 60 degrees.

Since all the answer choices are in trigonometric form, we know we must not necessarily solve for the exact value (although we can do that and calculate each choice to see if it matches). The first step is to determine the length of the ground between the bottom of the ladder and the wall via the Pythagorean Theorem: "x² + 15² = 17²"; x = 8. Using trigonometric definitions, we know that "opposite/adjecent = tan(theta)"; since we have both values of the sides (opp = 15 and adj = 8), we can plug into the tangential form tan(theta) = 15/8. However, since we are solving for theta, we must take the inverse tangent of the left side, "tan^-1". Thus, our final answer is

\(\displaystyle \theta =\tan^{-1}\left( \frac{15}{8}\right)\)

By rule, this is a 3-4-5 right triangle. Sine = (the opposite leg)/(the hypotenuse). This gives us 3/5.

Given that it is a right triangle, either angle A or B has to be 90 degrees. The other angle then must be less than 30 degrees, given that C is greater than 60 because there are 180 degrees in a triangle.

Example:

If angle C is 61 degrees and angle A is 90 degrees, then angle B must be 29 degrees in order for the angle measures to sum to 180 degrees.

Solving this problem quickly requires that we recognize how to break apart our ratio.

The Triangle Angle Sum Theorem states that the sum of all interior angles in a triangle is \(\displaystyle 180^{\circ}\). Additionally, the Right Triangle Acute Angle Theorem states that the two non-right angles in a right triangle are acute; that is to say, the right angle is always the largest angle in a right triangle.

Since this is true, we can assume that \(\displaystyle 90^{\circ}\) is represented by the largest number in the ratio of angles. Now consider that the other two angles must also sum to \(\displaystyle 90^{\circ}\). We know therefore that the sum of their ratios must be divisible by \(\displaystyle 90^{\circ}\) as well.

Thus, \(\displaystyle 5:13 = 90\).

To find the value of one angle of the ratio, simply assign fractional value to the sum of the ratios and multiply by \(\displaystyle 90^{\circ}\).

\(\displaystyle 5+13 = 18\), so:

\(\displaystyle \frac{5}{18}\cdot 90^{\circ} = \frac{450}{18} = 25^{\circ}\)

Thus, the shortest angle (the one represented by \(\displaystyle 5\) in our ratio of angles) is \(\displaystyle 25^{\circ}\).

The Triangle Angle Sum Theorem states that the sum of all interior angles in a triangle must be \(\displaystyle 180^{\circ}\). We know that a right triangle has one angle equal to \(\displaystyle 90^{\circ}\), and we are told one of the acute angles is \(\displaystyle 42^{\circ}\).

The rest is simple subtraction:

\(\displaystyle 180^{\circ} - (90^{\circ} + 42^{\circ}) = 48^{\circ}\)

Thus, our missing angle is \(\displaystyle 48^{\circ}\).

The word similar means, comparable in measurement, but not equal. The best sides way to compare these two triangles is by looking at the diagonal side of the triangle since it cannot be mistaken for any of the other sides of either triangle. If the larger triangle has a measurement of 7 and the smaller triangle has a measurement of 3.5 for their diagonal sides, then that means the ratio of the larger triangle to the smaller one is \(\displaystyle 2\).

\(\displaystyle 7 \div 3.5 = 2\)

This means that segment EF must be similar to segment AC (look at the orientation). So, since segment DE is similar to segment CB, divide 5 by two to get your answer.

\(\displaystyle 5 \div 2 = 2.5\)

1. Since the two triangles are similar, find the ratio of the two triangles to each other:

In this case, both triangles are multiples of a \(\displaystyle 3:4:5\) special right triangle.

\(\displaystyle 6:8:10\)

and

\(\displaystyle 18:24:x\)

The ratio of the triangles is \(\displaystyle \frac{6}{18}=3\)

2. Use the ratio you found to solve for \(\displaystyle x\), or the length of the missing side:

\(\displaystyle \frac{10}{x}=3\)

\(\displaystyle (10)(3)=x=30\)

1. Find the length of the missing hypotenuse:

You can do this one of two ways:

1) Using the special 3-4-5 right triangle, you can infer that the missing hypotenuse is 5.

2) By using the Pythagorean Theorem, you can solve for the length of the hypotenuse:

\(\displaystyle a^{2}+b^{2}=c^{2}\)

In this case:

\(\displaystyle 3^{2}+4^{2}=c^{2}\)

\(\displaystyle 9+16=25=c^{2}\)

\(\displaystyle c=5\)

2. Using the meaning of congruent (the exact same three angles and sides), determine if these two triangles fit this meaning:

They do, because both are 3-4-5 right triangles, and thus must have corresponding and equal angles due to trigonometric properties.

We can model the side lengths of the triangle as 3x, 4x, and 5x. We know that perimeter is 3x+4x+5x=48, which implies that x=4. This tells us that the legs of the right triangle are 3x=12 and 4x=16, therefore the area is A=1/2 bh=(1/2)(12)(16)=96.