Congruent Triangles

In this tutorial, we’ll talk about congruent triangles – what congruency means, what are the conditions for congruency, properties of congruent triangles, and more.

Congruent Triangles - Meaning

Two triangles (or any geometric figures) are said to be congruent if they are of exactly the same shape and size.

Congruent Triangles
Congruent Triangles

So if two congruent triangles are aligned and kept over one another, they will have a perfect overlap.

For example, if you rotate the green triangle counter-clockwise such that the 18-side becomes horizontal, the two triangles can be aligned perfectly, one over the other).

How to Identify Congruent Triangles

To determine whether two triangles are congruent, we have five sets of conditions to check against. If we can prove that the two triangles fulfill any one set of conditions, it would prove they are congruent.

So let’s look at the different criteria.

If you need help with hash marks and bands and terms like corresponding parts or the included angle, I have explained them in the footnotes.

Side-Side-Side (SSS)

By the SSS condition, two triangles are congruent if the three sides in one triangle are the same as the sides in the other triangle.

SSS Congruency
SSS Congruency
(All three sides)

Side-Angle-Side (SAS)

By the SAS condition, two triangles are congruent if two sides and the included angle in one triangle are the same as two sides and the included angle in the other triangle.

SAS Congruency
SAS Congruency
(Two sides & the included angle)

Angle-Side-Angle (ASA)

By the ASA condition, two triangles are congruent if two angles and the included side in one triangle are the same as two angles and the included side in the other triangle.

ASA Congruency
ASA Congruency
(Two angles & the included side)

Angle-Angle-Side (AAS)

By the AAS condition, two triangles are congruent if two angles and one side in one triangle are the same as two angles and the corresponding side in the other triangle.

AAS Congruency
AAS Congruency
(Two angles & a (corresponding) side)

Hypotenuse-Leg (HL)

By the HL condition, two right triangles are congruent if the hypotenuse and one side in one triangle are the same as the hypotenuse and one side in the other triangle.

HL Congruency
HL Congruency
(Right angle, hypotenuse & a side)

The HL condition is also known as RHS (Right Angle-Hypotenuse-Side).

Congruent Triangles - Notation

The symbol for congruency is ≅. However, there’s something else to be careful about.

When writing the names of the two triangles, corresponding vertices must be in the same order/position.

For example, here we have two congruent triangles, and vertices A, B, and C correspond to Y, Z, and X, respectively.

Remember, corresponding vertices have equal angles and sides opposite to corresponding vertices are equal.

So the correct notation for their similarity would be △ ABC ≅ △ YZX.

Properties of Congruent Triangles

Now let’s look at the three most commonly used properties of similar triangles. We’ll use the triangles below as examples.

Property 1. Corresponding angles are equal.

m(∠A)=m(∠P)m(∠B)=m(∠R)m(∠C)=m(∠Q)\begin{align*} m(\angle A) = m(\angle P) \\[1em] m(\angle B) = m(\angle R) \\[1em] m(\angle C) = m(\angle Q) \end{align*}

Property 2. Corresponding sides are equal.

AB=PRBC=RQAC=PQ\begin{align*} AB = PR \\[1em] BC = RQ \\[1em] AC = PQ \end{align*}

Property 3. The areas of the two triangles are equal.

Area (△ABC)=Area (△PRQ)\text{Area } (\triangle ABC) = \text{Area } (\triangle PRQ)

Examples

Great! Now that we know the criteria for congruency and properties of congruent triangles, let’s use our understanding to solve a couple of examples.

Example

Prove that the two triangles in the figure below are congruent. And hence, find the lengths of XY and XZ.

Solution

Looking at △ XYZ, we have some information about two angles (Y and Z) and the included side. So that gives away a lot as far as proving the congruency is concerned.

In △ ABC and △ XYZ,

AC=YZm(∠A)=m(∠Z)m(∠C)=m(∠Y)\begin{align*} AC &= YZ \\[1em] m(\angle A) &= m(\angle Z) \\[1em] m(\angle C) &= m(\angle Y) \end{align*}

Therefore, △ ABC ≅ △ ZXY by ASA congruency.

For the second part of the question, we need to identify sides in △ ABC corresponding to XY and XZ.

Now the side corresponding to XY is BC (they are both opposite to equal angles marked by double bands). Similarly, the side corresponding to XZ is AB. So,

XY=BC=15XZ=AB=18\begin{align*} XY &= BC = 15 \\[1em] XZ &= AB = 18 \end{align*}
Example

Prove that a diagonal of a parallelogram divides it into two areas with equal areas.

Solution

We’ll start by drawing the following figure.

Here we have a parallelogram ABCD and one of its diagonals, AC. Also, we can see that AC divides the parallelogram into two triangles. We need to prove – they have the same area.

Alright, remember that opposite sides in a parallelogram are equal.

So, in △ ABC and △ ADC,

AB=CDBC=AD\begin{align*} AB &= CD \\[1em] BC &= AD \end{align*}

Also, we have a side common to both the triangles.

AC=ACAC = AC

Therefore, △ ABC ≅ △ ADC by SSS congruency. And because the two triangles are congruent, they must have the same area.

So we have proved that the diagonal AC divides the ▱ ABCD into two triangles with equal areas. Similarly, we can prove that the other diagonal BD does the same.

Congruency and Similarity

Congruency and similarity are two closely related concepts. So let’s see what’s common and what’s different between them.

The idea is simple. Congruency requires two triangles (or any geometric figures) to have exactly the same shape and size. Similarity, on the other hand, requires them to have exactly the same shape. They may or may not have the same size.

So if two triangles are congruent, they must be similar too. But the converse is not true.

And with that, we come to the end of this tutorial on congruent triangles. Until next time.

Footnotes

Corresponding Parts

Corresponding parts (sides or angles) refer to parts in two triangles that have the same relative position. You can think of them as matching parts.

It’s easier to explain with an example. So, here’s one.

In the figure below, we have two copies of the same triangle – although with different names and orientations. Also, BC in the first triangle is the same as PQ in the second, and vertex (or angle) C is the same as Q.

So which side in the second triangle corresponds to AB in the first? And which angle corresponds to B?

Well, AB is opposite C. And C and Q are corresponding angles. So PR – the side opposite Q – must correspond to AB.

Now, side BC corresponds to side PQ. And C (one angle on BC) corresponds to Q (one angle on PQ). So B (the other angle on BC) and P (the other angle on PQ) must be corresponding parts.

In the figure below, the corresponding sides/angles are in the same color.

Included Side/Angle

For any two sides in a triangle (or a polygon), the included angle is the one formed between those sides – at their intersection. For example, here B is the included angle between AB and BC.

Similarly, the included side is the side that’s common to two angles. In the figure below, AC is the included side for angles A and C.

Hash Marks and Bands

When comparing two triangles or even two sides or angles within a triangle, hash marks and/or bands are used to show which sides or angles are equal. For example –

Hash marks and bands