The base is equal to 6.

The height of an quilateral triangle is equal to , where is the length of the base.

We know that the sum of all the angles must be 180 and we already know one angle is 90, leaving the sum of x and y to be 90.

Solve for x to find y.

One third of 21 is 7. Four less than  7 is 3. So if angle x is 3 then that leaves 87 for angle y.

The first thing to notice is that this is a 30^o:60^o:90^o triangle. If you draw a diagram, it is easier to see that the angle that is asked for corresponds to the side with a length of 4. This will be the smallest angle. The correct answer is 30.

In the figure above, angle ADB is a right angle. Because side AC is a straight line, angle CDB must also be a right angle.

Let’s examine triangle ADB. The sum of the measures of the three angles must be 180 degrees, and we know that angle ADB must be 90 degrees, since it is a right angle. We can now set up the following equation.

x + y + 90 = 180

Subtract 90 from both sides.

x + y = 90

Next, we will look at triangle CDB. We know that angle CDB is also 90 degrees, so we will write the following equation:

y – 10 + 2x – 20 + 90 = 180

y + 2x + 60 = 180

Subtract 60 from both sides.

 y + 2x = 120

We have a system of equations consisting of x + y = 90 and y + 2x = 120. We can solve this system by solving one equation in terms of x and then substituting this value into the second equation. Let’s solve for y in the equation x + y = 90.

x + y = 90

Subtract x from both sides.

y = 90 – x

Next, we can substitute 90 – x into the equation y + 2x = 120.

(90 – x) + 2x = 120

90 + x = 120

x = 120 – 90 = 30

x = 30

Since y = 90 – x, y = 90 – 30 = 60.

The question ultimately asks us to find the positive difference between the measures of ACB and CBD. The measure of ACB = 2x – 20 = 2(30) – 20 = 40 degrees. The measure of CBD = y – 10 = 60 – 10 = 50 degrees. The positive difference between 50 degrees and 40 degrees is 10. 

The answer is 10.

For this problem, remember that the sum of the degrees in a triangle is .

That means that .

Plug in our given values to solve:

Subtract  from both sides:

In any given triangle, the sum of any two sides is greater than the third.  The incorrect answers have the sum of two sides equal to the third.

There are 180 degrees in every triangle. Since this triangle is a right triangle, one of the angles measures 90 degrees.

Therefore, .

To solve we must first write out what is:

Now,we can simplify. However, notice that when subtracting these terms, we subtract all terms in the parentheses. Remember when we subtract a negative number, it is the same as adding the number. This is illustrated in the simplification below.

This simplifies to 

Now we can combine like terms. Let's put those together and then simplify

When simplifying the addition or subtraction of polynomials, we want to combine like terms. First, when we have a negative sign outside our parentheses, we know that we need to distribute that negative; think of it as an imaginary  and use the distributive property).

Then, we combine our like terms. Be careful when subtracting.

Rearrange the expression.

Combine terms and simplify.

Combine like terms.

Add the terms together.