### All Common Core: High School - Geometry Resources

## Example Questions

### Example Question #1 : Construct Geometric Figures: Ccss.Math.Content.Hsg Co.D.12

What must occur to construct two angles from a angle?

**Possible Answers:**

Bisect the in half.

Move the terminal ray of the angle up .

Move the terminal ray of the angle down .

Move the nonterminal ray of the angle down .

Move the nonterminal ray of the angle up .

**Correct answer:**

Bisect the in half.

To construct two angles from a angle a bisection must occur. Recall that bisection means to cut an angle into two equal parts.

Since

that makes a bisection of the angle equal two angles and thus is the solution.

Moving the terminal ray will not create two angles and thus it cannot be the solution.

Nonterminal is not the correct terminology and therefore, it cannot be the correct solution.

### Example Question #2 : Construct Geometric Figures: Ccss.Math.Content.Hsg Co.D.12

How are congruent lines identified in a parallelogram?

**Possible Answers:**

None of the other answers

Either single or double hash marks

Double hash marks

One single hash mark on one line and a double hash mark on the parallel line

Single hash marks

**Correct answer:**

Either single or double hash marks

Parallel lines by definition are lines that will never intersect one another. This means the slope of the lines are the same. When these lines make up a parallelogram figures that are created include squares, rectangles, diamonds, and rhombi.

Some images that depict parallelograms are as follows. Notice that either single or double hash marks can be used to identify that opposite lines are congruent.

### Example Question #3 : Construct Geometric Figures: Ccss.Math.Content.Hsg Co.D.12

Given the point , how would perpendicular lines be constructed?

**Possible Answers:**

Perpendicular lines are done by drawing a vertical line at and a horizontal line at .

Perpendicular lines are done by drawing a vertical line at and a horizontal line at .

Perpendicular lines are done by drawing a vertical line at and a horizontal line at .

Perpendicular lines are done by drawing a vertical line at and a horizontal line at .

Perpendicular lines are done by drawing a vertical line at and a horizontal line at .

**Correct answer:**

Perpendicular lines are done by drawing a vertical line at and a horizontal line at .

To construct perpendicular lines first recall what it means to be perpendicular.

Recall that perpendicular lines intersect at one point and have opposite, reciprocal slopes.

Plotting the given point is as follows:

Given this point, there are numerous combinations of perpendicular lines that can be made. Looking at the possible options the only pair that intersect at the point is when a vertical line is drawn at and a horizontal line at .

### Example Question #4 : Construct Geometric Figures: Ccss.Math.Content.Hsg Co.D.12

Determine whether the statement is true or false:

Lines that have fractional slopes cannot be perpendicular to other lines.

**Possible Answers:**

True

False

**Correct answer:**

False

For two lines to be perpendicular they must intersect at one point and have the opposite reciprocals on each other. Having slopes that are opposite reciprocals creates a angle between the two lines which is also by definition makes the two lines perpendicular.

Therefore, a line that has a fractional slope can also have a perpendicular line. If the line in question has a slope of then the perpendicular line would have a slope of .

Thus, the statement, "Lines that have fractional slopes cannot be perpendicular to other lines." is false.

### Example Question #5 : Construct Geometric Figures: Ccss.Math.Content.Hsg Co.D.12

Given the following ray, construct an angle that is .

**Possible Answers:**

None of the other answers.

**Correct answer:**

In order to construct a angle using a ray, first recall what a angle and ray are.

An angle in the most basic terms is when two rays having the same vertex, are going in different directions.

Now, recall that a angle occurs when the lines or rays of the angle intersect perpendicularly.

Since the problem gives the starting ray,

which is a horizontal ray, then to create a angle, a vertical ray needs to be constructed from the same vertex. To identify a angle it is common practice to draw a square at the vertex to show the angle measurement. This results in the final solution.

### Example Question #6 : Construct Geometric Figures: Ccss.Math.Content.Hsg Co.D.12

Given a line, what is the first step taken to construct a parallel line?

**Possible Answers:**

Take the negative reciprocal of the given line's slope.

Find the -intercept of the given line.

Identify the two lines intersection point.

Calculate the slope of the given line.

Find the -intercept of the given line.

**Correct answer:**

Calculate the slope of the given line.

To identify the first step that must be taken when constructing a parallel line, there is some information about the given line that needs to be clearly established first.

First recall what it means to be parallel. For lines to be parallel, their slopes must be equal and they must have different and intercepts. If two lines have the same intercepts and the same slope they are not parallel lines, they are the same line. Parallel lines mean that two lines will never intersect.

From here, look at the possible answers to decide which one is best.

"Identify the two lines intersection point." by definition of parallel lines this is false since parallel lines never intersect.

"Find the -intercept of the given line." and "Find the -intercept of the given line." could possible be considered among the steps to constructing parallel lines but not necessarily.

"Take the negative reciprocal of the given line's slope." Taking this would give the slope of a perpendicular line.

"Calculate the slope of the given line." in order to construct a parallel line, the slope must first be calculated on the given line. Therefore, this is the best answer.

### Example Question #7 : Construct Geometric Figures: Ccss.Math.Content.Hsg Co.D.12

Given a line, what is the first step taken to construct a perpendicular line?

**Possible Answers:**

Identify the two lines intersection point.

Calculate the slope of the given line.

Find the -intercept of the given line.

Either calculate the slope of the given line or identify the intersection point of the two lines.

Take the negative reciprocal of the given line's slope.

**Correct answer:**

Either calculate the slope of the given line or identify the intersection point of the two lines.

To identify the first step that must be taken when constructing a perpendicular line, there is some information about the given line that needs to be clearly established first.

First recall what it means to be perpendicular. For lines to be perpendicular, their slopes must be opposite reciprocals of each other and they must have one point of intersection.

From here, look at the possible answers to decide which one is best.

"Find the -intercept of the given line." doesn't necessarily help when constructing a perpendicular line.

"Take the negative reciprocal of the given line's slope." Taking this would give the slope of a perpendicular line but in order to do so the slope must be known of the given line.

"Calculate the slope of the given line." in order to construct a perpendicular line, the slope must be calculated on the given line.

"Either calculate the slope of the given line or identify the intersection point of the two lines." In order to construct a perpendicular line, the intersection point must be found and the slope of the given line must be found before the slope of the perpendicular line is found. Therefore, this is the best answer.

### Example Question #1 : Construct Geometric Figures: Ccss.Math.Content.Hsg Co.D.12

Determine whether the statement is true or false:

An equilateral triangle must have sides of equal length but the interior angles may differ.

**Possible Answers:**

False

True

**Correct answer:**

False

By definition an equilateral triangle has all sides that are of equal length and angles are also equal. All interior angles on an equilateral triangle equal regardless of the side lengths.

Therefore, the statement "An equilateral triangle must have sides of equal length but the interior angles may differ." is false.

### Example Question #122 : Congruence

Determine whether the statement is true or false:

In a parallelogram, opposite sides are always congruent.

**Possible Answers:**

False

True

**Correct answer:**

True

Using corresponding parts of congruent triangles proves this statement true.

A parallelogram is a four sided figure that has opposite parallel sides. Further more, when a parallelogram is divided into two triangles by drawing a line diagonally, they are equal triangles and thus the corresponding sides of each triangle are equal which in turn, makes the sides of the parallelogram congruent.

### Example Question #9 : Construct Geometric Figures: Ccss.Math.Content.Hsg Co.D.12

To construct a regular pentagon, what tools are required?

**Possible Answers:**

Straight Edge, Compass, Pencil, Paper, Measuring Device

Straight Edge, Compass, Pencil, Paper

Straight Edge, Pencil, Paper, Measuring Device

Compass, Pencil, Paper, Measuring Device

Straight Edge, Compass, Pencil, Measuring Device

**Correct answer:**

Straight Edge, Compass, Pencil, Paper, Measuring Device

In order to know which tools are required to construct a regular pentagon first recall the attributes of a pentagon and the possible geometric tools used to construct figures.

A regular pentagon by definition is a five sided figure. All sides of the figure are equal and every interior angle measures .

Geometric tools used to construct figures by hand include the follow:

Paper, Pencil, Compass, Ruler, Straight Edge, Protractor, Etc.

Geometric tools used to construct figures using technology include the following:

Graphing Calculators, Computer Programs, Etc.

Since the figure in question is a regular pentagon, to construct it by hand will require the use of a compass to ensure the angles are all equaled to . Also, a straight edge is required to ensure the lines are straight and a measuring device such as a rule to ensure the side lengths are equal.

Of the possible answer choices, three list the Straight Edge, Compass, and Measuring Device. If these tools are being used then the figure is being drawn by hand therefore, pencil and paper are also required. Therefore, the correct solution is:

"Straight Edge, Compass, Pencil, Paper, Measuring Device"