If \(\displaystyle l \parallel m\) and \(\displaystyle l \parallel n\) , then \(\displaystyle m \parallel n\), since two lines parallel to the same line are parallel to each other.

If \(\displaystyle m \angle1 + m \angle 2 = 180\), then \(\displaystyle m \parallel n\), since two same-side interior angles formed by transversal \(\displaystyle t\) are supplementary.

If \(\displaystyle \angle3 \cong \angle 4\), then \(\displaystyle m \parallel n\), since two alternate interior angles formed by transversal \(\displaystyle u\) are congruent.

However, \(\displaystyle \angle1 \cong \angle 5\) regardless of whether \(\displaystyle m\) and \(\displaystyle n\) are parallel; they are vertical angles, and by the Vertical Angles Theorem, they must be congruent.

By the Isosceles Triangle Theorem, two interior angles must be congruent. However, since a triangle cannot have two obtuse interior angles, the two missing angles must be the ones that are congruent. Since the total angle measure of a triangle is \(\displaystyle 180^{\circ }\) , each of the missing angles measures \(\displaystyle \frac{180-120}{2} =30^{\circ }\).

Obtuse angles are greater than \(\displaystyle 90^{\circ}\).

Scalene is a designation for triangles that have one angle greater than \(\displaystyle 90^{\circ}\), but this figure is not a triangle.

Acute angles are less than \(\displaystyle 90^{\circ}\), right angles are \(\displaystyle 90^{\circ}\), and straight angles are \(\displaystyle 180^{\circ}\).

Therefore this angle is obtuse.

When two parallel lines are crossed by a third line (called the transversal), the measure of the angles follows a specific pattern. The pairs of angles inside the two lines and on opposite sides are called alternate interior angles. Alternate interior angles, such as \(\displaystyle \measuredangle B\) and \(\displaystyle \measuredangle C\), have the same degree measure. Therefore, the measure of \(\displaystyle \measuredangle C\) is \(\displaystyle 35^{\circ}\).

Upon reading the question, we're left with this spatial image of Mark in our heads. After adding in the given information, the image becomes more like

The hill Mark is running on can be seen in terms of a right triangle. This problem quickly becomes one that is asking for a mystery angle given that the two legs of the triangle are given. In order to solve for the angle of depression, we have to call upon the principles of the tangent function. Tan, Sin, or Cos are normally used when there is an angle present and the goal is to calculate one of the sides of the triangle. In this case, the circumstances are reversed. 

Remember back to "SOH CAH TOA." In this problem, no information is given about the hypotenuse and nor are we trying to calculate the hypotenuse. Therefore, we are left with "TOA." If we were to check, this would work out because the angle at Mark's feet has the information for the opposite side and adjacent side. 

Because there's no angle given, we must use the principles behind the tan function while using a fraction composed of the given sides. This problem will be solved using arctan (sometimes denoted as \(\displaystyle tan^{-1}\)). 

\(\displaystyle tan^{-1}\left(\frac{17}{25}\right)= angle_{depression}\)

\(\displaystyle tan^{-1}\left(\frac{17}{25}\right)={\color{Blue} 34.2^{\circ}}\)

Since the angles are supplementary, their sum is 180 degrees.  Because they are in a ratio of 1:4, the following expression could be written:

\(\displaystyle x+4x=180\)

\(\displaystyle 5x=180\)

\(\displaystyle x=36^{\circ}\)

The angles are equal. When two parallel lines are intersected by a transversal, the corresponding angles have the same measure.  

Since A and B are parallel, and the triangle is isosceles, we can use the supplementary rule for the two angles, \(\displaystyle 4x\) and \(\displaystyle 2x+30\) which will sum up to \(\displaystyle 180\). Setting up an algebraic equation for this, we get \(\displaystyle 4x+2x+30=180\). Solving for \(\displaystyle x\), we get \(\displaystyle x=25\). With this, we can get either \(\displaystyle 2(25)+30=80\) (for the smaller angle) or \(\displaystyle 4(25)=100\) (for the larger angle - must then use supplementary rule again for inner smaller angle). Either way, we find that the inner angles at the top are 80 degrees each. Since the sum of the angles within a triangle must equal 180, we can set up the equation as \(\displaystyle 80+80+y=180\)

\(\displaystyle y=20\) degrees.

Let x equal the measure of angle DPB. Because the measure of angle APC is eighty-one degrees larger than the measure of DPB, we can represent this angle's measure as x + 81. Also, because the measure of angle CPD is equal to the measure of angle DPB, we can represent the measure of CPD as x.

Since APB is a straight line, the sum of the measures of angles DPB, APC, and CPD must all equal 180; therefore, we can write the following equation to find x:

x + (x + 81) + x = 180

Simplify by collecting the x terms.

3x + 81 = 180

Subtract 81 from both sides.

3x = 99

Divide by 3.

x = 33.

This means that the measures of angles DPB and CPD are both equal to 33 degrees. The original question asks us to find the measure of angle CPB, which is equal to the sum of the measures of angles DPB and CPD.

measure of CPB = 33 + 33 = 66.

The answer is 66.

Let x equal the measure of angle ABC, let y equal the measure of the supplement of angle ABC, and let z equal the measure of the complement of angle ABC.

Because x and y are supplements, the sum of their measures must equal 180. In other words, x + y = 180. 

We are told that one-half of the measure of the supplement is equal to twice the measure of ABC. We could write this equation as follows:

(1/2)y = 2x.

Because x + y = 180, we can solve for y in terms of x by subtracting x from both sides. In other words, y = 180 – x. Next, we can substitute this value into the equation (1/2)y = 2x and then solve for x.

(1/2)(180-x) = 2x.

Multiply both sides by 2 to get rid of the fraction.

(180 – x) = 4x.

Add x to both sides.

180 = 5x.

Divide both sides by 5.

x = 36.

The measure of angle ABC is 36 degrees. However, the original question asks us to find the measure of the complement of ABC, which we denoted previously as z. Because the sum of the measure of an angle and the measure of its complement equals 90, we can write the following equation:

x + z = 90.

Now, we can substitute 36 as the value of x and then solve for z.

36 + z = 90.

Subtract 36 from both sides.

z = 54.

The answer is 54.