In this explainer, we will learn how to identify triangles that have the same area when their bases are equal in length and the vertices opposite to these bases are on a parallel line to them.

To see why this result holds true, let’s consider the following scenario.

We have two parallel lines, and , and two triangles that share a base, and . We can prove that the areas of these triangles are equal by first recalling that the area of a triangle is given by half the length of its base multiplied by its perpendicular height. So, let’s add perpendicular lines from points and to find the perpendicular heights of each triangle.

We will call the points of intersection between these perpendicular lines and the parallel lines and and as shown. We can note that all the angles in are right angles, since and are right angles and . Thus, is a rectangle. Therefore, the lengths of and are equal. We can now show that the areas of the triangles are equal by finding expressions for their areas:

Since , we have

We have proven the following result.

### Theorem: Equality of Areas of Triangles on Parallel Lines

If two triangles share a base and the vertices opposite this base lie on a straight line parallel to the base, then they have equal areas.

Let’s now see an example of applying this theorem to find triangles with equal areas.

### Example 1: Finding the Areas of Triangles between Parallel Lines

Which of the following has the same area as ?

### Answer

We can answer this question by recalling that two triangles that share a base and have the vertex opposite the base lie on a straight line parallel to the base will have equal areas. We can then note that , so any triangle with a base of and a final vertex on will have an equal area to .

In particular, shares the base with , and its vertex lies on , so it has the same area as , which is option A.

In our next example, we will need to apply this theorem to determine the area of a triangle.

### Example 2: Finding the Areas of Triangles between Parallel Lines

Given that the area of , find the area of .

### Answer

Let’s start by marking the given triangle and the triangle whose area we wish to find on the given diagram.

We can split each of these triangles into two smaller triangles along the line to get the following.

Let’s consider the top two triangles first, as shown.

We note that these triangles share a base, , and their opposite vertices and both lie on a line parallel to the base. Hence, we know that these triangles have equal area.

Let’s now consider the bottom two triangles, as shown.

We can once again note that these two triangles share a base, , and their opposite vertices and both lie on a line parallel to the base. Hence, we know that these triangles have equal areas.

Since these triangles have the same areas and they combine to make the larger triangles, and , these must also have the same area.

Hence, the area of triangle is
568 cm^{2}.

In our next example, we will show that if two triangles lie on two parallel lines and they have bases of the same length, then they have the same area.

### Example 3: Identifying Triangles with Equal Areas between Parallel Lines

Given that , which of the following has the same area as ?

### Answer

We are given a pair of parallel lines, so we can use the fact that if two triangles share a base and the vertices opposite this base lie on a straight line parallel to the base, then they have equal areas. If we choose as the base of the triangle, then we can choose any point on as the final vertex of the triangle to find a triangle of equal area to . Hence, , , , , and all have the same area. However, none of these are options for this question.

Instead, let’s use the fact that the area of a triangle is half the length of its base multiplied by its perpendicular height. We choose as the base of the triangle and can add the perpendicular height to the diagram as shown.

Hence,

We can use the same method to determine the areas of and .

We add the perpendicular lines from the bases to the vertex and note that all of the green lines are parallel. We note that since each of these is a transversal of parallel lines, they also meet at right angles. Thus, they all form rectangles with sections of and , so each perpendicular line has the same length of . Therefore,

Finally, since , , and all have the same length, we can conclude that triangles , , and all have the same area.

Hence, has the same area as , which is option C.

In the previous example, we showed the following property.

### Property: Equality of Areas of Triangles on Parallel Lines

If two triangles lie on two parallel lines and they have bases of the same length, then they have the same area.

In our next example, we will consider how the median of a triangle splits the area of the original triangle.

### Example 4: Finding the Areas of Triangles with Congruent Bases

If the area of , find the area of .

### Answer

We want to determine the area of and to do this we are given the area of . This means that we will want to compare the areas of some triangles to that of . To do this, we recall that the area of a triangle is half the length of its base multiplied by its perpendicular height. If we choose to be the base of this triangle, we get the following.

We call the point on the perpendicular ; we can then see that

From the diagram, we can note that . In fact, this tells us that is a median of triangle . Since triangles and have the same base length, we can check to see if they have the same perpendicular height.

Choosing as the base, the perpendicular from to will also intersect at , so

Since these triangles combine to make , we have

We can apply the exact same reasoning to show that and have the same area. We see that both triangles have the same base length, since , and these bases lie on the same straight line. Finally, they share the vertex point , so the perpendicular distance from the base to will be the same for both triangles.

Hence, their areas are the same and so

Since is the combination of these triangles, we have

In our previous example, we showed two useful results. First, we saw that the median of a triangle will split the triangle into two triangles with the same area. Second, we saw that two triangles with congruent bases on the same straight line that share the opposite vertex will have equal areas since their perpendicular heights are equal. We can write these results formally as follows.

### Property: Equality of Areas of Triangles with Congruent Bases

Any median of a triangle will split the triangle into two triangles with the same area.

Any two triangles with congruent bases that lie on the same straight line and share a common vertex opposite the base have the same area.

In our next example, we will apply this property to find triangles of equal area to a given triangle.

### Example 5: Identifying the Triangles with the Same Area between Parallel Lines

Which triangle has the same area as ?

### Answer

We note that we are given that ; we can then recall that any two triangles with congruent bases that lie on the same straight line and share a common vertex opposite the base have the same area. Hence, and have the same area.

Thus far, we have concentrated on finding triangles with equal areas to a given triangle or using these results to determine areas. However, we can also ask the same questions in reverse. For example, if two triangles of equal area share a base and their vertices opposite the base lie on the same side, what can we say about these vertices?

To help us understand the situation, let’s first sketch this information.

We know that and have the same area; we can find expressions for the area of each triangle by using half the length of the base multiplied by the perpendicular height. Adding the perpendiculars to the diagram gives us the following.

We now have

Since the areas of the triangles are equal, we must have that . Next, we note that these lines are perpendicular to and are of the same length; hence, we must have that is a rectangle and in particular this means that . We have proven the following result.

### Theorem: Vertices of Equal-Area Triangles Sharing a Base Are Aligned Parallel to Their Common Base

If two triangles share a base and have equal areas and the vertices opposite the base lie on the same side of the base, then these vertices lie on a straight line parallel to the base.

It is worth noting that this result also holds if the triangles have congruent bases on the same line. We can write this formally as follows.

### Theorem: Vertices of Equal-Area Triangles with Congruent Bases Are Aligned Parallel to Their Common Base

If two triangles have congruent bases on a straight line and have equal areas and the vertices opposite the base lie on the same side of the base, then these vertices lie on a straight line parallel to the base.

Let’s now see an example of applying this theorem to identify a geometric property from a diagram.

### Example 6: Triangles between Parallel Lines Sharing the Same Base

If the areas of and are the same, which of the following must be true?

### Answer

We start by noting that each of triangles and is composed of two smaller triangles; we can compare the area of these smaller triangles. Let’s start by comparing triangles and ; we can do this by adding the perpendicular distance from to to the diagram as shown.

We recall that any two triangles with congruent bases that lie on the same straight line and share a common vertex opposite the base have the same area. Hence,

Combining this result with the fact that triangles and have the same area means that triangles and must also have the same area.

We then recall that if two triangles have congruent bases on a straight line and have equal areas and the vertices opposite the base lie on the same side of the base, then these vertices lie on a straight line parallel to the base.

Hence, , which is option E.

In our final example, we will apply these theorems and properties to determine a geometric property from a given diagram.

### Example 7: Identifying a Geometric Property given that Two Triangles Have Equal Areas

Points , , and are collinear. If the areas of and are the same, which of the following must be true?

### Answer

We first note that is parallel to and is parallel to . Thus, is a parallelogram. Hence, its diagonal, , splits the parallelogram into two equal-area triangles, and . It is also worth noting that saying three points are collinear means that they all lie on the same straight line.

Therefore, since we can conclude that

We can also note that these triangles share the same base, .

We can then recall that if two triangles share a base and have equal areas and the vertices opposite the base lie on the same side of the base, then these vertices lie on a straight line parallel to the base.

Since triangles and have the same area, share a base , and have vertices on the same side of the base, we can conclude that the base is parallel to the line between the vertices opposite the base. That is , which is option B.

In the previous example, there are actually many different ways of determining the result. For example, we could use the fact that if the areas of and are the same, then their bases lying on the same line are congruent, which leads to , and as , so is a parallelogram.

Let’s finish by recapping some of the important points from this explainer.

### Key Points

- If two triangles share a base and the vertices opposite this base lie on a straight line parallel to the base, then they have equal areas.
- If two triangles lie on two parallel lines and they have the same base length, then they have the same area.
- The median of a triangle divides it into two parts equal in area.
- Triangles that have congruent bases on the same straight line and have a common vertex opposite the bases are equal in area.
- If two triangles share a base and have equal areas and the vertices opposite the base lie on the same side of the base, then these vertices lie on a straight line parallel to the base.