Lesson Video: Geometric Construction: Congruent Angles and Parallel Lines Mathematics • First Year of Preparatory School

Video Transcript

Geometric Construction: Congruent Angles and Parallel Lines

In this lesson, we will learn how to construct an angle to be congruent to a given angle using only a compass and a straight edge. We will then learn how to apply this process to construct a line parallel to another line through a given point.

Congruent angles are an important part of geometry since they’re used in constructing congruent shapes and in proving many geometric properties using congruency. The process of constructing an angle congruent to another angle is useful in both of these processes. It is important to note that we want to do this without using a protractor, since protractors will have errors in measurements, whereas a geometric construction will theoretically have no errors at all.

Let’s start with an angle 𝐴𝐵𝐶 that we want to duplicate. In this case, we can see that this angle is acute. To duplicate this angle, we want to apply the side-side-side congruency criterion for triangles. In other words, we are going to instead construct a triangle with angle 𝐴𝐵𝐶 as an internal angle and then duplicate this triangle, which will also duplicate the angle.

Let’s start by sketching a circle centered at 𝐵, which intersects the sides of the angle as shown. We only draw the arcs of this circle for simplicity. We can label the points of intersection 𝐴 prime and 𝐶 prime. We can now see that triangle 𝐴 prime 𝐵𝐶 prime has an internal angle with the same measure as angle 𝐴𝐵𝐶. We want to duplicate this triangle.

To duplicate this triangle, and hence the angle, we will start by sketching a ray in the plane. It is worth noting that we can sketch the ray in any orientation. It does not have to be in the same direction as 𝐵𝐶. However, for simplicity, we will sketch our ray in a similar direction as shown. We now trace a circle of radius 𝐶 prime 𝐵 centered at 𝐸. We can now see that line segments 𝐸𝐷 prime and 𝐵𝐶 prime are the same length. In fact, the distance between any point on this circle and 𝐸 is the same length as these lines.

Let’s now change the radius of our compass to be the length 𝐴 prime 𝐶 prime and trace a circle of this radius centered at 𝐷 prime. We can call the point of intersection between the circles 𝐹 as shown. We can now note that we have duplicated all of the sides of triangle 𝐴 prime 𝐵𝐶 prime as a new triangle 𝐹𝐸𝐷 prime. In particular, by the SSS criterion, this means that we have duplicated the angle at 𝐵.

We can write down the steps we followed in this process to describe how we can duplicate an angle 𝐴𝐵𝐶. We start by sketching a ray from 𝐸 through 𝐷 anywhere on the plane, where our construction will duplicate the angle such that 𝐸 is the vertex of the congruent angle. Next, we trace a circle centered at 𝐵 that intersects the sides of the angle. We will label these points of intersection 𝐴 prime and 𝐶 prime. Then, we trace a congruent circle of radius 𝐶 prime 𝐵 centered at 𝐸. We label the point of intersection between the ray from 𝐸 through 𝐷 and this circle 𝐷 prime. Finally, we trace a circle of radius 𝐴 prime 𝐶 prime centered at 𝐷 prime and label the point of intersection between the two circles 𝐹. Then, angle 𝐴𝐵𝐶 is congruent to angle 𝐹𝐸𝐷 prime.

There are a few things worth noting about this process. First, we started by assuming that our angle was acute. So we can ask if this process works for angles which are not acute. This process does indeed work for any angle. However, it is worth noting that if the angle we want to duplicate is a straight angle, a full turn, or the zero angle, then it is easier to just sketch a ray or line. We can then guarantee that the angle is duplicated.

We can show that this construction works to duplicate any reflex angle by noting that we can just use this construction to duplicate the acute angle. Thus, this will also duplicate the reflex angle.

There are two more cases we need to consider, that is, right angles and obtuse angles. The case for a right angle is the same as the case for an acute angle. So we will not show this in the video. Instead, let’s check the process for the obtuse angle shown.

First, we start by sketching any ray we will call 𝐸𝐷 anywhere in the plane. Second, we trace a circle centered at 𝐵 that intersects both sides of the angle. We call these points of intersection 𝐴 prime and 𝐶 prime. For simplicity, we only sketch the two arcs as shown. Third, we trace a circle of radius 𝐶 prime 𝐵 centered at 𝐸 and label the point of intersection between this circle and the ray 𝐷 prime. We need to sketch this arc to have larger measure than the angle we want to duplicate. Finally, we trace a circle of radius 𝐴 prime 𝐶 prime centered at 𝐷 prime and label the point of intersection between the two circles 𝐹. We can see that the triangles 𝐴 prime 𝐵𝐶 prime and 𝐹𝐸𝐷 prime are congruent by the side-side-side criterion. Hence, the angles at 𝐵 and 𝐸 have equal measure.

Let’s now see an example of identifying the correct construction of duplicating an angle.

Which of these two figures shows the steps for constructing a congruent angle? Figure I or figure II.

In this question, we’re given two figures and we need to determine which of the figures demonstrates the steps of duplicating an angle. We note that constructing means with a compass and a straight edge.

Let’s start by saying that these constructions are supposed to duplicate angle 𝐷𝐸𝐹. We can then recall that to duplicate this angle, we need to start by drawing a ray in the plane, say the ray from 𝐴 through 𝐺, where 𝐴 will be the vertex of the angle we duplicate. We can see that both constructions have such a ray if we add a point 𝐺 to each figure as shown.

The next step in our construction is to trace a circle centered at 𝐸 that intersects the sides of the angle we want to duplicate at two points we will label 𝐷 prime and 𝐹 prime. After this, we need to trace a congruent circle centered at 𝐴. In the diagrams, we only sketch an arc of the circle to keep the construction clean. We call the point of intersection between the ray and the circle 𝐵. The final step in the construction is to trace a circle of radius 𝐷 prime 𝐹 prime centered at 𝐵. We can call the point of intersection of the two circles 𝐶 as shown.

We can conclude that the angles are congruent because triangle 𝐹 prime 𝐸𝐷 prime is congruent to triangle 𝐶𝐴𝐵 by the side-side-side criterion. We see that this is only the case in the first figure.

For due diligence, we can check what the second construction gives us. We see that we have the arcs of two congruent circles centered at 𝐶 and 𝐵. This gives us the following pairs of lines of the same length, since they are radii of the congruent circles. If we connect 𝐴 to the point of intersection of these circles, 𝐻, as shown, then we can note that we have two congruent triangles. This means that all of the corresponding angles of the two triangles must be congruent. We can then note that this means that this gives us the angle bisector. Hence, the answer is that only figure I shows the steps for constructing a congruent angle.

Before we move on to constructing parallel lines, there is an important property of parallel lines we need to recall by looking at an example.

True or False: The line between 𝐹 and 𝐵 is parallel to the ray from 𝐶 through 𝐸.

In this question, we want to determine if a given line and ray are parallel by using a given figure. To do this, we can start by highlighting the line and ray in the question on the diagram as shown. We can then note that the line between 𝐴 and 𝐷 is a transversal of these two lines and the corresponding angles of the transversal are both congruent since they have the same measure. They both measure 55 degrees.

We can then recall that if the corresponding angles of a transversal of a pair of lines are congruent, then the lines themselves are parallel. So the answer must be true. The line between 𝐹 and 𝐵 is parallel to the ray from 𝐶 through 𝐸.

In our next example, we will use this property and our construction of congruent angles to show a geometric property.

Draw triangle 𝐴𝐵𝐶 where 𝐴𝐵 equals three centimeters, 𝐵𝐶 equals four centimeters, and 𝐴𝐶 equals five centimeters. Point 𝐷 lies on the ray from 𝐵 through 𝐶 such that 𝐷 is not on line segment 𝐵𝐶. Draw angle 𝐷𝐶𝐸 congruent to angle 𝐶𝐵𝐴, where 𝐸 is on the upper side of line segment 𝐵𝐶. Which of the following is true? Option (A) the measure of angle 𝐴 equals the measure of angle 𝐸𝐶𝐴. Option (B) the measure of angle 𝐴 is equal to the measure of angle 𝐵𝐶𝐴. Option (C), the measure of angle 𝐴 equals the measure of angle 𝐸𝐶𝐷. Or is it option (D) the measure of angle 𝐵 is equal to the measure of angle 𝐸𝐶𝐴?

In this question, we’re given a construction and we need to determine which of five given options is correct in the construction. We need to start by sketching triangle 𝐴𝐵𝐶 by using the given side lengths of the triangle. We could do this by noting that the lengths of three, four, and five centimeters make a Pythagorean triple. This is enough to know that it is a right triangle with a hypotenuse of five centimeters by using the side-side-side congruency criterion.

However, it is not necessary to notice this to construct this triangle. In general, we can construct a triangle from its lengths by using circles. We start by drawing one of its sides. Let’s say 𝐴𝐵, which has a length of three centimeters. We can then trace a circle of radius four centimeters centered at 𝐵 and a circle of radius five centimeters centered at point 𝐴. Either point of intersection between these circles can be our point 𝐶. We choose 𝐶 as shown. We note that triangle 𝐴𝐵𝐶 has the desired lengths.

We now want to extend the line segment 𝐵𝐶 to be the ray from 𝐵 through 𝐶 so that we can find the point 𝐷 on this ray to make the angle 𝐷𝐶𝐸 congruent to the angle 𝐶𝐵𝐴. To do this, let’s start by highlighting the angle 𝐶𝐵𝐴 that we want to be congruent to the angle 𝐷𝐶𝐸. We know that this is a right angle since this is a Pythagorean triple. However, it is not necessary to use this to answer the question.

Instead, we can duplicate this angle of vertex 𝐶 by using our construction for duplicating an angle at a point. To duplicate this angle, we start by tracing a circle at 𝐵 with radius less than three centimeters. We call the points of intersection with the sides of the angle 𝐴 prime and 𝐶 prime. It is worth noting that it is easier to choose a radius below 1.5 centimeters.

Next, we trace a circle of the same radius centered at 𝐶. We label the point of intersection between the ray from 𝐵 through 𝐶 and the circle that is above 𝐶 as point 𝐷. Now, we can duplicate the angle shown at 𝐶 by tracing a circle of radius 𝐴 prime 𝐶 prime centered at 𝐷 and labeling the point of intersection between the circles 𝐸 as shown. We then know that angle 𝐷𝐶𝐸 is congruent to angle 𝐶𝐵𝐴. We can also see that line 𝐵𝐶 is a transversal of the lines 𝐶𝐸 and 𝐴𝐵. And this transversal makes the same angle with both lines. We can recall that this means that the two lines must be parallel.

We can then note that the line between 𝐴 and 𝐶 is also a transversal between these parallel lines. So the alternating angles it makes with each line must be congruent. This gives us that angle 𝐸𝐶𝐴 must have the same measure as angle 𝐴, which we can see is option (A).

We can combine our construction of a congruent angle with the property that two lines are parallel if they have corresponding congruent angles in a transversal to construct a line parallel to any other line through any point. For instance, let’s say we want to construct a line parallel to 𝐴𝐵 through the point 𝐶 that is not on this line as shown.

To do this, we first need a transversal so that we can show that the transversal will have congruent corresponding angles with our pair of lines. We can sketch the line between 𝐴 and 𝐶 to be this transversal. We can see that the angle that line 𝐴𝐶 makes with line 𝐴𝐵 is angle 𝐵𝐴𝐶 as shown. We want to duplicate this angle at 𝐶. We duplicate this angle by tracing a circle at 𝐴 that intersects the sides of the angle at 𝐶 prime and 𝐵 prime as shown. We then trace a congruent circle centered at 𝐶 and call the point of intersection between the circle and the line 𝐷.

Now, we trace a circle of radius 𝐵 prime 𝐶 prime centered at 𝐷 and call the point of intersection between the circles 𝐸 as shown. We then know that angle 𝐵𝐴𝐶 is congruent to angle 𝐷𝐶𝐸. Since the corresponding angles in this pair of parallel lines cut by a transversal have equal measure, we can conclude that the lines must be parallel.

In our final example, we will use this idea of constructing a line parallel to another line to show a useful geometric property.

In the following figure, the line between 𝐴 and 𝐵 is parallel to the line between 𝐶 and 𝐷, while the line between 𝐸 and 𝐹 cuts the line between 𝐴 and 𝐵 and the line between 𝐶 and 𝐷 at 𝑋 and 𝑌, respectively. Draw straight line 𝑀𝑁, where the line between 𝑀 and 𝑁 is parallel to the line between 𝐸 and 𝐹 and cuts the line between 𝐴 and 𝐵 and the line between 𝐶 and 𝐷 at 𝑂 and 𝑃, respectively, on the right side of the line between 𝐸 and 𝐹. Find the measure of the angle 𝑂𝑃𝑌.

In this question, we’re given a lot of information about a given figure. We can start by adding the extra information onto the figure. First, we are told that the lines between 𝐴 and 𝐵 and 𝐶 and 𝐷 are parallel. This makes the line between 𝐸 and 𝐹 a transversal of this pair of parallel lines.

Next, we’re told that we need to construct a line between two points, 𝑀 and 𝑁, on the figure, where this line is parallel to the line between 𝐸 and 𝐹. And it intersects the line between 𝐴 and 𝐵 at 𝑂 and the line between 𝐶 and 𝐷 at 𝑃 and is on the right of the line between 𝐸 and 𝐹.

To construct this line, we first need to choose a point of intersection between the line and one of the lines 𝐴𝐵 and 𝐶𝐷. Let’s choose the point 𝑂 as shown. We need to make sure that this point is on the line between 𝐴 and 𝐵 and that this point is to the right of the line between 𝐸 and 𝐹. We can then recall that we can construct a line parallel to another line through a point by duplicating the angle at this point. So we will duplicate angle 𝐴𝑋𝐸 at the point 𝑂.

To do this, we first need to trace a circle centered at 𝑋 that intersects the two sides of the angle as shown. We will call these points of intersection 𝐴 prime and 𝐸 prime. Next, we trace a congruent circle centered at 𝑂. And we will call the point of intersection between the circle and the line that is to the left of 𝑂 𝑋 prime as shown. We now trace a circle of radius 𝐴 prime 𝐸 prime centered at 𝑋 prime and call the point of intersection between the circle shown 𝑀.

We can now note that triangle 𝐴 prime 𝑋𝐸 prime is congruent to triangle 𝑋 prime 𝑂𝑀 by the side-side-side criterion. Hence, angle 𝐴 prime 𝑋𝐸 prime is congruent to angle 𝑋 prime 𝑂𝑀. We can add to the diagram that these angles both have measure 80 degrees. We can also call the point of intersection between line 𝑀𝑂 and line 𝐶𝐷 𝑃 as shown. And we can also add a point onto the line called 𝑁. We can also note that the line between 𝐴 and 𝐵 is a transversal of the lines between 𝑋 and 𝑌 and 𝑂 and 𝑃 with congruent corresponding angles. So they must be parallel.

We want to find the measure of angle 𝑂𝑃𝑌. And we can mark this angle on the diagram as shown. We can note that the line between 𝑂 and 𝑃 is a transversal of parallel lines. So the corresponding angles must be congruent. Hence, we can conclude that the measure of angle 𝑂𝑃𝑌 must be equal to 80 degrees.

In the previous example, we can actually note many more useful properties of this type of construction. First, we can note that quadrilateral 𝑂𝑃𝑌𝑋 has opposite sides parallel. So it is a parallelogram. We also know that diagonally opposite angles in a parallelogram are congruent. So we can note that the measure of angle 𝑌𝑋𝑂 must also be 80 degrees. Similarly, we can note that a straight angle has measure 180 degrees. We can use this to find that the measure of angle 𝑋𝑂𝑃 is 100 degrees and that the diagonally opposite angle 𝑋𝑌𝑃 must have the same measure. We can use this process to prove these properties will hold true for any parallelogram.

Let’s now go over the key points of this lesson. First, we saw that we can duplicate an angle with a compass and a straight edge. In particular, if the angle we want to duplicate is a zero angle, straight angle, or full turn, then we only need to use the straight edge to duplicate the angle.

However, if we want to duplicate a different angle, then we need to follow four steps. The first step is to sketch a ray anywhere in the plane. Let’s say we sketch a ray from 𝐸 through 𝐷 as shown. Second, we need to trace a circle centered at point 𝐵 such that it intersects both sides of the angle. We will label these points of intersection 𝐴 prime and 𝐶 prime as shown. Third, we trace a congruent circle centered at point 𝐸 and label the point of intersection between the circle and the ray 𝐷 prime. Finally, we trace a circle of radius 𝐴 prime 𝐶 prime centered at 𝐷 prime and call the point of intersection between the two circles 𝐸 prime.

We can note that triangle 𝐴 prime 𝐵𝐶 prime and triangle 𝐷 prime 𝐸𝐸 prime are congruent by the side-side-side congruency criterion. So the measure of angle 𝐴𝐵𝐶 is equal to the measure of angle 𝐷 prime 𝐸𝐸 prime.

Finally, we saw that we can apply this construction process to construct a line parallel to another line through a point. In particular, if we want to construct a line parallel to a given line between 𝐴 and 𝐵 through a given point 𝐶, we can do this by duplicating the angle 𝐵𝐴𝐶 at the point 𝐶.