In this explainer, we will learn how to construct an angle to be congruent to a given angle and construct a line to be parallel to a given line.
Congruent angles are an important part of geometry since they are used in constructing congruent shapes and proving geometric properties using angle congruency. The ability to construct an angle congruent to another angle using a compass and straightedge is useful in both of these processes. It is important to note that we want to do this without using a protractor since a protractor will always have some errors in measurement.
To do this, we will use the fact that any two triangles with the same side lengths must have their corresponding angles congruent. This is called the SSS congruency criterion.
Before we duplicate an angle, it is worth noting that if the angle is a zero angle, a straight angle, or a full rotation, then the sides of the angle form either a line or a ray, and these are duplicated by just using the straightedge.
Letβs start with a given that we want to duplicate.
We are going to do this by constructing a congruent triangle. So, for now, we will assume that m.
We start by tracing a circle centered at that intersects the edges of the angle at two points we will label and .
We can connect these points with a straight line to construct triangle .
We want to construct a triangle congruent to triangle so that we duplicate the angle at . To do this, letβs start with a ray .
If we set the radius of our compass equal to and then trace a circle centered at , we can find a point of intersection between the circle and ray that we call . We note that since they are both radii of congruent circles.
We note that we need to trace this arc centered at further than the measure of angle . We note that every point on this arc is a distance of from . In particular, , so any point on this circle will give us a side congruent to .
Letβs now change the radius of the compass to be . We can then trace a circle centered at with this radius to find a point of intersection between the circles; we call this point .
We now sketch line segments and , and we note that and .
Since triangles and have three congruent sides, we can conclude, using the SSS congruency criterion, that the corresponding angles are congruent. In particular, this tells us that .
It is worth noting that this construction remains unchanged for an obtuse or a right angle at . In both cases, we construct the congruent triangle in exactly the same way. For example, for an obtuse angle at , we get the following.
We can extend this construction to reflex angles by noting that we can duplicate the smaller angle at the vertex, and this also duplicates the reflex angle. For example, if we want to duplicate the following reflex angle, we instead follow the process to duplicate the smaller angle at .
We then see that the reflex angle at is congruent to the reflex angle at .
This means we can use this process to duplicate any angle; however, we generally leave out the triangle construction and just use the arcs to find the duplicate angle.
How To: Duplicating an Angle with a Compass and Straightedge
We can construct a congruent angle to any with a compass and straightedge. First, if is a zero angle, a straight angle, or a full turn, then we just need to sketch a line and mark an angle with the same measure. Otherwise, we use the following steps:
- Sketch a ray ; will be the vertex of the congruent angle.
- Trace a circle centered at that intersects the edges of the angle at two points we will name and .
- Trace a circle of radius centered at . Label the point of intersection between the ray and circle .
- Set the radius of the compass equal to and then trace a circle centered at . Label the point of intersection between the two circles .
- We then have that .
Letβs now see an example of correctly identifying the first step in the construction of a congruent angle with a compass and straightedge.
Example 1: Correctly Describing the First Step to Construct a Congruent Angle
Fill in the blanks: The first step to construct a congruent angle to a given angle after drawing a ray is to draw from the vertex of the first angle, then to draw from the vertex of the second angle.
- a straight line, a straight line with a different length
- an arc with a suitable radius, a straight line
- an arc with a suitable radius, an arc with the same radius
- an arc with a suitable radius, an arc with a different radius
Answer
To construct a congruent angle with a compass and straightedge, we want to construct a congruent triangle since the corresponding angles will then be equal. To do this, we first sketch a ray that we will duplicate the angle onto. Then, we trace a circle at the vertex of the angle so that the two sides of the triangle have equal length.
We can then trace a circle of equal radius centered at the endpoint of the ray to get the following.
Finally, we trace a circle of radius centered at to construct congruent triangles .
Hence, the answer is option C.
The first step to construct a congruent angle to a given angle after drawing a ray is to draw an arc with a suitable radius from the vertex of the first angle, then to draw an arc with the same radius from the vertex of the second angle.
In our next example, we will determine which of a list of different constructions correctly duplicates an angle.
Example 2: Identifying the Construction of Congruent Angles
Which of these two figures shows the steps for constructing a congruent angle?
Answer
Letβs say these constructions are supposed to duplicate .
We recall that we construct a congruent angle with a compass and straightedge by sketching a ray , where will be the vertex of the congruent angle. We then trace a circle centered at that intersects the edges of the angle at two points, and . We trace a congruent circle centered at and label the point of intersection .
Finally, we set the radius of the compass equal to and trace a circle of this radius centered at . The point of intersection of these circles is point , and we have that .
We can see that this is the construction in Figure I.
For due diligence, we should also check if the construction in Figure II duplicates an angle. We note that the two congruent circles are traced with centers at and . This means that we have the following congruent line segments.
This does not duplicate the angle. In fact, we can note that this bisects the angle by adding in the following diagonal and point.
We see that triangles and are congruent by the SSS criterion, so .
Hence, only Figure I shows the construction of a congruent angle.
Before we move on to constructing parallel lines, we first need to recall a property about transversals of parallel lines. Letβs first see a specific example of this property.
Example 3: Using Congruence of Corresponding Angles to Identify Geometric Relationships
True or False: is parallel to .
Answer
We begin by recalling that if the corresponding angles of a transversal of two lines are congruent, then the lines are parallel. Since we are given that and these are corresponding angles, we can conclude that is parallel to .
The property in the previous example holds for any corresponding congruent angles.
In our next example, we will use the construction of congruent angles and the property of corresponding congruent angles in a transversal making the line parallel to determine a geometric relationship.
Example 4: Determining a Geometric Property by Constructing a Congruent Angle
Draw where , , and . Point lies on such that is not on line segment . Draw congruent to , where is on the upper side of . Which of the following is true?
Answer
We want to start by sketching and there are a few ways of doing this. One way is to note that 3, 4, 5 is a Pythagorean triple, so is a right triangle. However, it is not necessary to notice this to sketch the triangle.
We can start by drawing a 3 cm long line segment . Next, we trace a 4 cm circle centered at and a 5 cm circle centered at ; then, the point of intersection between these circles is .
We now need to extend to make it a ray and then find the point on this ray, not on , such that is congruent to . We will do this by using the construction for congruent angles.
First, letβs clear the previous construction to make the process clearer. We will start with only triangle where we will mark .
We now trace a circle at that intersects the sides of the triangle as shown.
It is worth noting that the radius of this circle does not matter; however, it is easiest to choose a radius smaller that 1.5 cm. The choice of the radius will change where we sketch our point .
We then trace a congruent circle at to get a possible point as shown.
We now set the radius of our compass to be equal to the length of the line segment between the two points of intersection on the edges of the angle we want to duplicate.
We trace a circle of this radius centered at and label the point of intersection between the circles .
Adding labels for the two points of intersection on the edges of the angle we wanted duplicated, we have that triangles and are congruent. Hence, is congruent to (i.e., ).
We now recall that if a transversal of two lines makes the same angle with both lines, then the lines are parallel. Since and these are the angles that the transversal makes with and , then is parallel to .
Now, since is a transversal to a pair of parallel lines, we recall that the alternate interior angles and must be congruent.
Hence, , which is option A.
In the previous example, we used a property of parallel lines and our construction of congruent angles to show that two lines are parallel. We can generalize this process to construct a line parallel to any given line through a point.
For example, letβs say we want to construct a line parallel to through .
We sketch the line and the .
We can duplicate at . First, we trace a circle centered at and then trace a congruent circle centered at , labeling the points as shown.
We then set the radius of the compass to and trace a circle of this radius centered at . We can label the point of intersection between the circles , and we have that is congruent to .
Sketching the line , we can then note that since the corresponding angles are equal, we must have that is parallel to .
Hence, we can construct a line parallel to through by constructing an angle congruent to at .
In our final example, we will construct a line parallel to another line to show a useful geometric property.
Example 5: Finding the Measure of an Angle by Constructing a Parallel Line
In the following figure, , while cuts and at and respectively. Draw straight line , where and cuts and at and , respectively, on the right of . Find .
Answer
To sketch , we first need to choose where we want this line to intersect one of the given lines. We will choose a point on and then construct an angle congruent to at .
We start by tracing a circle centered at that intersects the edges of the angle and then we trace a congruent circle centered at , labeling the points of intersection as shown.
We then set the radius of the compass equal to and trace a circle of this radius centered at . We call the point of intersection between the arcs , and we have that is congruent to .
Hence, both of these angles have a measure of . We can also label the point of intersection between and as . This allows us to mark on the diagram.
We can see that is a transversal of the parallel lines and . This means that the corresponding angles must be congruent. Hence,
In the previous example, quadrilateral has opposite sides that are parallel, so it is a parallelogram. Letβs note two properties of parallelograms.
First, as we have seen in the previous example, the diagonally opposite angles in a parallelogram are congruent.
Second, we recall that a straight angle has a measure of . So, we can see that , and we know this is congruent to .
We can use this process to show that these properties hold true for any parallelogram.
Letβs finish by recapping some of the important points from this explainer.
Key Points
- We can construct a congruent angle to any
with a compass and straightedge. First, if
is a zero angle, a straight angle, or a full turn, then we just need to sketch
a line and mark an angle with the same measure. Otherwise, we use the following
steps:
- Sketch a ray ; will be the vertex of the congruent angle.
- Trace a circle centered at that intersects the edges of the angle at two points that we will name and .
- Trace a circle of radius centered at . Label the point of intersection between the ray and circle .
- Set the radius of the compass equal to and then trace a circle centered at . Label the point of intersection between the two circles .
- We then have that m.
- We can construct a line parallel to through point by constructing an angle congruent to at .
