If the perimeter of two triangles is equal which of the following criteria can prove it congruent

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Let’s begin by understanding the meaning of congruent triangles.

Two triangles are congruent if their corresponding sides are congruent and their corresponding angle measures are congruent.

Now, let’s consider the following three triangles.

It does not matter if the triangles have different orientations or are flipped; we can still compare their side lengths and their angle measures. The corresponding sides and angles in these three triangles are highlighted below.

When all three sides are congruent and all three angle measures are congruent, then the three triangles are congruent.

This leads us to some important detail about the notation we use when we write congruence relationships. Congruent shapes can be related by the congruence symbol, ≅. However, the order in which we write the vertices of the shapes is very important because the congruence relationship itself indicates the vertices (and sides) that are congruent.

For example, in the figure above, we could write that △𝐴𝐵𝐶≅△𝐷𝐸𝐹≅△𝑃𝑄𝑅. If we wrote, for example, that △𝐴𝐵𝐶≅△𝐸𝐷𝐹, this would be incorrect since vertex 𝐴 corresponds to vertex 𝐷, not 𝐸. We could, however, write a number of different correct congruence statements; for example, △𝐵𝐶𝐴≅△𝐸𝐹𝐷≅△𝑄𝑅𝑃,△𝐶𝐴𝐵≅△𝐹𝐷𝐸≅△𝑅𝑃𝑄,△𝐵𝐴𝐶≅△𝐸𝐷𝐹≅△𝑄𝑃𝑅.

When it comes to determining if two triangles are congruent, there are some shorter methods we can use rather than establishing that all corresponding sides and angles are congruent. We will look at these congruence criteria and see why each of these criteria proves that two triangles are congruent.

The first criterion is the side-angle-side criterion, often abbreviated to SAS. Note that, in all of these criteria, the use of “S” or side means a pair of congruent sides and the use of “A” or angle means a pair of congruent angles.

Two triangles are congruent if two sides and the included angle in one triangle are congruent to the corresponding parts in the other triangle.

Let’s explore how knowing the measures of two sides and an included angle means that two triangles are congruent. We can take a triangle with two sides given as 5 cm and 6 cm and the included angle (the angle between the sides) that has a measure of 40∘.

Let’s now try and draw another triangle that has these same lengths and included angle measure. We will not constrain the size of the third side or the other two angles in any way.

When we begin sketching a triangle, we see that there will be only one way to create a complete triangle.

Hence, knowing that two triangles have two pairs of congruent sides and the included angle measure in each triangle is congruent is sufficient to show that the triangles themselves are congruent. This is the SAS congruency criterion.

The second criterion we can use to prove that two triangles are congruent is the angle-side-angle, or ASA, criterion.

Two triangles are congruent if two angles and the side drawn between their vertices in one triangle are congruent to the corresponding parts in the other triangle.

Let’s demonstrate why this is true with an example of a triangle that has two angles with measures 70∘ and 45∘. The side between these angles has a length of 6 cm.

Now, let’s see if we can draw a noncongruent triangle that also has these 3 properties. We will not constrain the lengths of the 2 other sides or the size of the third angle.

When we do so, however, we see that there will only be one possible triangle that we can draw; one that is identical to the original triangle. These two triangles are congruent. This can be extended for any two triangles with two congruent angles and two congruent sides.

Hence, knowing that two pairs of angles are congruent and that the included pair of sides is congruent is sufficient to prove that two triangles are congruent.

We will now see how we can apply these criteria in the following examples.

Determine whether the triangles in the given figure are congruent, and, if they are, state which of the congruence criteria proves this.

Answer

We recall that two triangles are congruent if their corresponding sides are congruent and corresponding angle measures are congruent. We can use a number of different criteria to help us establish if two triangles are congruent.

In the given figure, we have two pairs of congruent sides: 𝐴𝐶=𝐴′𝐶′=2.53,𝐵𝐶=𝐵′𝐶′=3.68.

We also have the included angles in both triangles, the angles between these sides, with the same measure: 𝑚∠𝐴𝐶𝐵=𝑚∠𝐴′𝐶′𝐵′=60.34.∘

The SAS congruence criterion states that two triangles are congruent if they have two congruent sides and an included congruent angle.

Hence, we can give the answer that the two triangles are congruent by the SAS congruence criterion.

Let’s now see another example.

Can you use SAS to prove the triangles in the given figure are congruent? Please state your reason.

Answer

Two triangles are congruent if their corresponding sides are congruent and corresponding angle measures are congruent. In the figure, we can see that there are two pairs of corresponding sides of equal length: 𝐴𝐵=𝐴′𝐵′=2.36,𝐴𝐶=𝐴′𝐶′=5.52.

We also have a pair of corresponding angle measures that are congruent: 𝑚∠𝐴𝐶𝐵=𝑚∠𝐴′𝐶′𝐵′=25.22.∘

The SAS congruence criterion states that two triangles are congruent if they have two congruent sides and an included congruent angle. However, in this figure, the given angle in each triangle is not the included angle between the sides. To have the included angle here, we would need to know and compare 𝑚∠𝐵𝐴𝐶 and 𝑚∠𝐵′𝐴′𝐶′.

A suitable statement for the answer would reference the fact that we cannot say the triangles are congruent because the angle is not the appropriate angle. Therefore, the answer is no becausetheanglemustbecontainedbetweenthetwosides.

Let’s see another example.

Which congruence criteria can be used to prove that the two triangles in the given figure are congruent?

Answer

We recall that two triangles are congruent if their corresponding sides are congruent and corresponding angle measures are congruent. However, there are a number of congruence criteria that we can use to prove that two triangles are congruent.

In the figure, we have two triangles, △𝐴𝐵𝐶 and △𝐴′𝐵′𝐶′.

We are given the information that there is a pair of congruent sides and two pairs of congruent angles: 𝐴𝐵=𝐴′𝐵′=2.57,𝑚∠𝐴𝐵𝐶=𝑚∠𝐴′𝐵′𝐶′=65.03,𝑚∠𝐴𝐶𝐵=𝑚∠𝐴′𝐶′𝐵′=58.55.∘∘and

There is a congruency criterion (ASA) that relates two angles and a side: two triangles are congruent if they have two congruent angles and the included congruent side. However, in this figure, the side is not the included side since it does not lie between the two angles. So, we cannot immediately apply the ASA criterion.

However, as we have two given angles in each triangle, we can establish the third angle in each triangle using the fact that the internal angle measures in a triangle sum to 180∘.

Thus, in △𝐴𝐵𝐶, given 𝑚∠𝐴𝐵𝐶=65.03∘ and 𝑚∠𝐴𝐶𝐵=58.55∘, we can calculate 𝑚∠𝐵𝐴𝐶 as 𝑚∠𝐴𝐵𝐶+𝑚∠𝐴𝐶𝐵+𝑚∠𝐵𝐴𝐶=18065.03+58.55+𝑚∠𝐵𝐴𝐶=180123.58+𝑚∠𝐵𝐴𝐶=180𝑚∠𝐵𝐴𝐶=180−123.58=56.42.∘∘∘∘∘∘∘∘∘

In the same way, for triangle 𝐴′𝐵′𝐶′, given 𝑚∠𝐴′𝐵′𝐶′=65.03∘ and 𝑚∠𝐴′𝐶′𝐵′=58.55∘, we can calculate 𝑚∠𝐵′𝐴′𝐶′ as 𝑚∠𝐴′𝐵′𝐶′+𝑚∠𝐴′𝐶′𝐵′+𝑚∠𝐵′𝐴′𝐶′=18065.03+58.55+𝑚∠𝐵′𝐴′𝐶′=180123.58+𝑚∠𝐵′𝐴′𝐶′=180𝑚∠𝐵′𝐴′𝐶′=180−123.58=56.42.∘∘∘∘∘∘∘∘∘

We observe that we now have two angles and an included side in each triangle that are congruent since 𝑚∠𝐵𝐴𝐶=𝑚∠𝐵′𝐴′𝐶′=56.42𝐴𝐵=𝐴′𝐵′=2.57,𝑚∠𝐴𝐵𝐶=𝑚∠𝐴′𝐵′𝐶′=65.03.∘∘

Hence, we can give the answer that the congruence criterion that can be used to prove these two triangles are congruent is ASA.

Note that it is important here to demonstrate that the measures of the third angle in each triangle are equal.

As we saw demonstrated in the previous example, when we have two triangles with two pairs of corresponding angle measures that are congruent, then the third pair of corresponding angles will also be congruent. This is because, in each triangle, we subtract the same two angle measures from 180∘ (the sum of the internal angle measures in a triangle) and thus obtain a third pair of equal angles. Hence, the ASA congruence criterion can be extended such that Angle-Angle-Side (AAS) is also a congruence criterion. In this criterion, knowing that two pairs of corresponding angles are congruent and the nonincluded sides in each triangle are equal in length would prove that the pair of triangles are congruent.

We will now see the third criterion we can use for triangle congruence: the side-side-side (SSS) criterion.

Two triangles are congruent if each side in one triangle is congruent to the corresponding side in the other triangle.

The best way to consider why this criterion is true is to experiment with drawing a triangle congruent to another with 3 known side lengths. Consider a triangle with side lengths 5, 6, and 10 length units.

If we tried to construct a noncongruent triangle with the same side measures, we could start with a side length of 5 on the base and then construct two arcs of length 10 and 6 length units from either endpoint of the base.

However, there will be just one point of intersection. This will create a congruent triangle to the original triangle.

Even if we drew the base of 5 length units and constructed the sides of 10 and 6 length units on opposite sides, we would get a flipped but congruent triangle.

We will now see an example of how we can apply this criterion.

Determine whether the triangles in the given figure are congruent by applying SSS, SAS, or ASA. If they are congruent, state which of the congruence criteria proves this.

Answer

We are asked to determine if the triangles are congruent. We recall that congruent triangles have congruent corresponding sides and congruent corresponding angle measures. The SSS, SAS, and ASA criteria are three different criteria that we can use to prove that two triangles are congruent.

The SSS criterion states that two triangles are congruent if they have all three sides equal.

The SAS criterion states that two triangles are congruent if two sides and the included angle are equal.

The ASA criterion states that two triangles are congruent if two angles and the included side are equal.

We observe that, in the figure, we have no angle measures given nor any way in which we can calculate these. Therefore, it is likely that, if the triangles are congruent, we will demonstrate this using the SSS rule. We need to determine that there are 3 equal sides. We have 𝐴′𝐵′=𝐴𝐵=3,𝐵′𝐶′=𝐵𝐶=5,𝐴′𝐶′=𝐴𝐶=3.16.

Therefore, we can answer that the triangles are congruent using the SSS rule.

The fourth congruency criterion applies specifically, and only, to right triangles. Recall that the hypotenuse of a right triangle is the longest side and is always opposite the right angle.

Two right triangles are congruent if the hypotenuse and a side of one triangle are congruent to the corresponding parts in the other triangle.

The RHS congruency criterion is in fact a special application of the SSS criterion.

If we have a triangle with a 90∘ angle and can determine two sides in a triangle that are not the hypotenuse, then we can apply the SAS rule (with an angle of 90∘) to establish if it is congruent to another triangle with the same measurements. The RHS criterion is useful because it gives us an additional way to identify congruent (right) triangles.

It is important to note that the RHS rule only applies when the angle is 90∘. There is not an angle-side-side congruency criterion. Let’s consider why there is no such criterion.

As an example, we can take a triangle with sides 4 cm and 8 cm and a (nonincluded) angle of 30∘.

If we then constructed another triangle with the same properties, we would in fact find that there is more than one possible triangle we could create, as shown in the figure below.

Thus, angle-side-side is not sufficient to demonstrate congruency. If we have two sides and an angle, the angle must be the included angle between the two sides (the SAS criterion).

Let’s now look at one final example.

In the given figure, points 𝐿 and 𝑁 are on a circle with center 𝑂. Which congruence criterion can be used directly to prove that triangles 𝑂𝐿𝑀 and 𝑂𝑁𝑀 are congruent?

Answer

Two triangles are congruent if they have corresponding congruent sides and corresponding congruent angles. We can observe that the two triangles we are considering, △𝑂𝐿𝑀 and △𝑂𝑁𝑀, are both right triangles. There is a congruence criterion that is used in right triangles: the RHS criterion. This states that two triangles are congruent if they have a right angle and the hypotenuse and one other side are equal. Let’s see if we can apply this criterion here.

Note that we are not given any length measurements, but we can apply our knowledge of geometry to help. Firstly, as 𝑂 is the center of the circle, 𝑂𝐿 and 𝑂𝑁 are radii of the circle. This means that they will be congruent: 𝑂𝐿=𝑂𝑁.

Next, the side 𝑂𝑀 is a side that is common to both triangles; hence, 𝑂𝑀=𝑂𝑀.

Importantly, 𝑂𝑀 is the hypotenuse in each triangle.

Combined with the fact that we have a right angle in both triangles, we can apply the RHS criterion, stating the congruent angles and sides as 𝑚∠𝑂𝐿𝑀=𝑚∠𝑂𝑁𝑀,𝑂𝑀=𝑂𝑀,𝑂𝐿=𝑂𝑁.(rightangle)(hypotenuse)(side)

Hence, we can give the answer that the congruence criterion we can apply is RHS.

It is very common when we do have congruent triangles that there are multiple congruence criteria that can be applied. Thus, it is very important, when answering questions, that we clearly state the congruent sides and angles used as proof alongside the appropriate congruence criterion.

We now summarize the key points.

Two triangles are congruent if their corresponding sides are congruent and corresponding angle measures are congruent.
The congruence criteria allow us to more easily prove if triangles are congruent. The following are the congruence criteria:
- SAS: Two triangles are congruent if two sides and the included angle in one triangle are congruent to the corresponding parts in the other triangle.
- ASA: Two triangles are congruent if two angles and the side drawn between their vertices in one triangle are congruent to the corresponding parts in the other triangle.
- SSS: Two triangles are congruent if each side in one triangle is congruent to the corresponding side in the other triangle.
- RHS: Two right triangles are congruent if the hypotenuse and a side of one triangle are congruent to the corresponding parts in the other triangle.
There is no congruence criterion angle-side-side since noncongruent triangles can be created with equivalent measurements.