Geometry - GMAT Quantitative

Two angles are complementary if the total of their degree measures is . Therefore, an angle complementary to a angle measures .

Lines are perpendicular if their slopes are negative reciprocals of each other. First we need to find the slope of the line in the question stem.

The negative reciprocal of 3 is , so our answer will have a slope of . Let's go through the answer choices and see.

: This line is of the form , where is the slope. The slope is 3, so this line is parallel, not perpendicular, to our line in question.

: The slope here is , also wrong.

: The slope of this line is . This is the reciprocal, but not the negative reciprocal, so this is also incorrect.

The line between the points :.

This is the correct answer! Let's check the last answer choice as well.

The line between points :, which is incorrect.

If two equations are perpendicular, then they will have inverse negative slopes of each other. So if we compare the slopes of the two equations, then we can find the answer. For the first equation we have

so the slope is .

So for the equations to be perpendicular, the other equation needs to have a slope of 3. For the second equation, we have

so the slope is .

Since the slope of the second equation is not equal to 3, then the lines are not perpendicular.

Statement 1 alone establishes by definition that , but does not establish any relationship between and .

By Statement 2 alone, since same-side interior angles are supplementary, , but no conclusion can be drawn about the relationship of , since the actual measures of the angles are not given.

Assume both statements are true. If two lines are parallel, then any line in their plane perpendicular to one must be perpendicular to the other. and , so it can be established that .

If transversal crosses two parallel lines and , then same-side interior angles are supplementary, so and are supplementary angles. Also, corresponding angles are congruent, so .

By Statement 1 alone, angles and are congruent as well as supplementary; by Statement 2 alone, and are also supplementary as well as congruent. Two angles that are both supplementary and congruent are both right angles, so from either statement alone, and intersect at right angles, so, consequently, .

To solve this equation, we want to begin by recalling how to find the slope of a perpendicular line. In this case, our original line is modeled by the following:

To find the slope of any line perpendicular to the above equation, we simply need to take the reciprocal of the first slope, and then change its sign. Our original slope is , so

becomes

.

If we flip , we get , and the opposite sign of a negative is a positive; hence, our slope is positive .

So, we know our perpendicular line should look something like this:

However, we need to find out what (our -intercept) is in order to complete our equation. To do so, we need to plug in the ordered pair we received in the question, , and solve for :

So, by putting everything together, we get our final equation:

This equation satisfies the conditions of being perpendicular to our initial equation and passing through .

In order for a line to be perpendicular to another line defined by the equation , the slope of line must be a negative reciprocal of the slope of line . Since line 's slope is in the slope-intercept equation above, line 's slope would therefore be .

In this instance, , so . Therefore, the correct solution is .

Supplementary angles have degree measures that total , so an angle supplementary to would measure .

Complementary angles have degree measures that total , so an angle complementary to would measure .

Congruent angles have degree measures that are equal, so an angle congruent to is .

This question requires careful application of distance formula, which is really a modified form of Pythagorean theorem.

Plug in everthing and solve:

So our answer is 156.6

Using the distance formula for the length of a line between two points, we can plug in the given values and determine the length of the line segment by calculating the distance between the two points:

Supplementary angles have degree measures that total . Since we have an angle, the supplementary angle would measure

Complementary angles have degree measures that total . Since we have an angle, the supplementary angle would measure

Congruent angles have the same degree measure, so an angle congruent to a angle would also measure .

Let be the common measure of , , , and

Then

The sum of the measures of the angles of a pentagon is degrees; this translates to the equation

or

The sum of the measures of the exterior angles of any polygon, one at each vertex, is . Since a regular polygon with twenty-four sides has twenty-four congruent angles, and therefore, congruent exterior angles, just divide:

This can more easily be explained if the shared side is extended in one direction, and the new angles labeled.

and are exterior angles of the regular polygons. Also, the measures of the exterior angles of any polygon, one at each vertex, total . Therefore,

Add the measures of the angles to get :

Each angle of a square measures ; each angle of a regular pentagon measures . To get , subtract:

.