Construct Geometric Figures: CCSS.Math.Content.HSG-CO.D.12 - Common Core: High School - Geometry
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Given the following ray, construct an angle that is 
.
Given the following ray, construct an angle that is
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.
In order to construct a
An angle in the most basic terms is when two rays having the same vertex, are going in different directions.
Now, recall that a
Since the problem gives the starting ray,
which is a horizontal ray, then to create a
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Given a line, what is the first step taken to construct a parallel line?
Given a line, what is the first step taken to construct a parallel 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.
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
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
"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.
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Given a line, what is the first step taken to construct a perpendicular line?
Given a line, what is the first step taken to construct a perpendicular line?
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.
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
"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.
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To construct a regular pentagon, what tools are required?
To construct a regular pentagon, what tools are required?
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"
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
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"
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Given two rays, how can a line be constructed?
Given two rays, how can a line be constructed?
Recall that a ray is a line segment that has a vertex at one end and extends at the other. The extension is depicted by an arrow. In order for two rays to become one line segment the vertices must be connected and the arrows going in the opposite direction. For the rays to be going in the opposite direction they must have 
angle between them.
Therefore, the correct answer is
"Connect the vertices of the rays and have their directions point 
from each other."
Recall that a ray is a line segment that has a vertex at one end and extends at the other. The extension is depicted by an arrow. In order for two rays to become one line segment the vertices must be connected and the arrows going in the opposite direction. For the rays to be going in the opposite direction they must have
Therefore, the correct answer is
"Connect the vertices of the rays and have their directions point
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What must occur to construct two 
angles from a 
angle?
What must occur to construct two
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.
To construct two
Since
that makes a bisection of the
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.
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How are congruent lines identified in a parallelogram?
How are congruent lines identified in a parallelogram?
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.
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.
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Given the point 
, how would perpendicular lines be constructed?
Given the point
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 
.
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
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Determine whether the statement is true or false:
Lines that have fractional slopes cannot be perpendicular to other lines.
Determine whether the statement is true or false:
Lines that have fractional slopes cannot be perpendicular to other lines.
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.
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
Therefore, a line that has a fractional slope can also have a perpendicular line. If the line in question has a slope of
Thus, the statement, "Lines that have fractional slopes cannot be perpendicular to other lines." is false.
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Determine whether the statement is true or false:
An equilateral triangle must have sides of equal length but the interior angles may differ.
Determine whether the statement is true or false:
An equilateral triangle must have sides of equal length but the interior angles may differ.
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.
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
Therefore, the statement "An equilateral triangle must have sides of equal length but the interior angles may differ." is false.
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Determine whether the statement is true or false:
In a parallelogram, opposite sides are always congruent.
Determine whether the statement is true or false:
In a parallelogram, opposite sides are always congruent.
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.
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.
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How is a square altered to result in a rhombus?
How is a square altered to result in a rhombus?
Both squares and rhombi are quadrilaterals and parallelograms. Quadrilaterals are four sided figures and parallelograms are figures that have opposite sides that are parallel.
Squares by definition contain four 
angles; rhombi on the other hand have two sets of opposite congruent angles. Therefore, for a square to to altered into a rhombus, the interior angles must be altered.
Both squares and rhombi are quadrilaterals and parallelograms. Quadrilaterals are four sided figures and parallelograms are figures that have opposite sides that are parallel.
Squares by definition contain four
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