Two triangles cannot be said to be congruent as per which of the following congruency rules

There are five ways of finding two similar triangles. But the main question is how to find out that two triangles are congruent to each other? Thus, in simple words, we can say that two triangles are said to be congruent if they have the same three sides and angles. But we need not have to check out all these three angles and sides for knowing its congruence, just three of all these six is fine. Thus the five theorems of congruent triangles are SSS, SAS, AAS, HL, and ASA.

SSS – side, side, and side

This ‘SSS’ means side, side, and side which clearly states that if the three sides of both triangles are equal then, both triangles are congruent to each other.

For instance, triangle ABC has the side of 2cm * 3cm * 5cm and triangle PQR has sides of 2cm * 3cm * 5cm, then according to the SSS theorem, both these triangles ABC and QPR are congruent to each other.

In simple words, if all the three sides of one triangle are equal to all the three sides of another triangle, then both the triangles are congruent to each other.

SAS – side, angle, and side

This ‘SAS’ means side, angle, and side which clearly states that any of the two sides and one angle of both triangles are the same, then both triangles are considered congruent to each other.

For instance, if the triangle ABC measures 2cm * 3cm * 6cm with the angle of 75 degrees, and the triangle PQR measures 4cm * 3cm * 6cm with the angle of 75 degrees, then according to the SAS theorem of the triangle, both these triangles will be considered congruent triangles.

In simple words, if the two sides and one angle of one triangle is similar to the other triangle, then they are known as congruent to each other.

ASA – angle, side, and angle

This ‘ASA’ means angle, side, and angle which clearly states that two angles and one side of both triangles are the same, then these two triangles are said to be congruent to each other.

For instance, triangle ABC measures angle of 35 and 45 degree with one side of 5 cm and triangle PQR measures angle of 35 and 45 degree with one side of 5 cm, then both these triangles have the same measures, which as per the theorem of ASA, these two triangles are congruent to each other.

In simple words, if two angles and one side of one triangle is equal to the other triangle, then both these triangles are congruent.

AAS – angle, angle, and side

This ‘AAS’ means angle, angle, and sides which clearly states that two angles and one side of both triangles are the same, then these two triangles are said to be congruent to each other.

For instance, triangle ABC measures angle of 35 and 45 degree with one side of 5 cm and triangle PQR measures angle of 35 and 45 degree with one side of 5 cm, then both these triangles have the same measures, which as per the theorem of AAS, these two triangles are congruent to each other.

In simple words, if two angles and one side of one triangle is equal to the other triangle, then both these triangles are congruent.

HL – hypotenuse and leg

This ‘HL’ means hypotenuse and leg which clearly states that if hypotenuse and leg of both the triangles are the same, then these two triangles are said to be congruent to each other.

For instance, triangle ABC measures hypotenuse of 5 cm and leg of 4 cm and triangle PQR measures hypotenuse of 5 cm and leg of 4 cm, then both these triangles have the same measures, which as per the theorem of HL, these two triangles are congruent to each other.

Geo 5.5-5.7 Notes

1. Under the RHS congruence criterion, we consider two sides and an angle in two right triangles. Under SAS criterion also, we consider two sides and an angle. So, think and tell why do we need to have RHS congruency rule as a separate criterion to prove congruency of triangles?

Let us do an activity to understand the proof of RHS congruence theorem.

Try to draw two triangles \(\triangle ABC\) and \(\triangle PQR\) with any one of the angles as \(90^o\).

Are these triangles congruent?

Now, let's try to keep hypotenuse side equal in both the triangles along with one \(90^o\) angle.

Let's try to make the hypotenuse side of \(\triangle PQR\) equals to 10 units.

If we change the hypotenuse of a triangle, other side-lengths will also be changed to maintain the Pythagoras relation between the sides, i.e. \(\text{hypotenuse}^2=\text{base}^2+\text{perpendicular}^2\)

Can you see congruent triangles now?

Now, let's keep one more side equal in both the triangles and observe the result.

Are we getting congruent triangles now?

Yes, in the above image, \(\triangle ABC \cong \triangle PQR\).

We can place both the triangles on each other without any gaps and overlaps.

Hence, \(\triangle ABC \cong \triangle PQR\) using RHS congruency rule.

State and proof whether the given triangles are congruent or not.

In the given triangles, \(\triangle ZXY\) and \(\triangle PQR\)

\[\text{side}\ XZ=\text{side}\ PQ\]

\[\text{side}\ YZ=\text{side}\ PR\]

\[\angle ZXY=\angle PQR=90^o\]

\(\therefore \triangle ZXY \cong \triangle PQR\), by RHS congruence criterion.

1. RHS congruency criterion is applicable only in right-angled triangles.
2. Under RHS rule, we consider only the hypotenuse and one corresponding side of the given two right triangles to prove the congruency of triangles.
3. Two congruent triangles are always equal in area.
4. When we place two congruent right-angled triangles on one another, there are no gaps and overlaps. They completely fit each other.

Now, look at some RHS criteria examples for a deeper understanding.

Solved Examples

In the given isosceles triangle \(\triangle PQR\), prove that the altitude \(PO\) bisects the base of the triangle \(QR\).

Solution

In the given triangle \(\triangle PQR\), there are two small right-angled triangles formed and those are \(\triangle POQ\) and \(\triangle POR\),

Altitude \(PO\) bisects \(QR\) when \(OQ=OR\).

So, let us prove that \(\triangle POQ \cong \triangle POR\).

         \(PQ=PR\) (given as equal)

\(PO=PO\) (common)

\(\angle POQ=\angle POR=90^o\)

So, by RHS congruence criterion,

\(\triangle POQ \cong \triangle POR\)

\(OQ=OR\) (by CPCT)

\(\therefore\) Altitude of triangle \(\triangle PQR\) bisects the base \(QR\) of the triangle.

In the given triangle, \(\triangle ABD\), if \(AC\) bisects side \(BD\) and \(CE=CF\), prove that the area of triangles \(\triangle BCE\) and \(\triangle DCF\) are equal.

Solution

We know that area of two congruent triangles is always equal.

So, to prove that the triangles \(\triangle BCE\) and \(\triangle DCF\) are equal, we just need to prove that they are congruent triangles.

\(\triangle BCE\) and \(\triangle DCF\) are right triangles, in which,

                              \(\begin{align} CB=CD\end{align}\) (as AC bisects BD)

\(\begin{align}CE=CF\end{align}\)

          \(\begin{align}\angle CEB=\angle CFD=90^o\end{align}\)

\(\therefore \triangle BCE \cong \triangle DCF\) (by RHS congruence criterion)

Hence, \(\triangle BCE\) and \(\triangle DCF\) are equal in area.

Interactive Questions on RHS

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

The mini-lesson targeted the fascinating concept of RHS. The math journey around RHS started with what a student already knew and went on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. 

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FAQs on RHS

1. What are the Rules of Congruency?

There are 5 main rules of congruency of triangles and those are:

• SSS (side side side) rule
• ASA (angle side angle) rule
• SAS (side angle side) rule
• AAS (angle angle side) rule
• RHS (right-angle hypotenuse side) rule

2. What is RHS criterion in triangles?

If the hypotenuse and side of one right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle, the two triangles are congruent.

3. How do you use RHS rule?

Under RHS rule, we show that in two right triangles, the length of the hypotenuse is equal along with the length of another corresponding side of the triangles. If we can prove this, that means the given triangles are congruent, otherwise not.

4. What is the difference between SAS and RHS?

In SAS rule, we consider the angle between two sides that we are equal, but in RHS, the placement of angle does not matter as we take hypotenuse and any one of the other two corresponding sides.

5. What is the full form of RHS congruence rule?

Full form of RHS is right-angle hypotenuse side.

6. Where do we use RHS congruence rule?

RHS rule is used to find the length of the missing side in right triangles, to find the area of triangles, to design buildings and towers, etc.

7. How to proof RHS congruence criterion?

RHS congruence criterion can be proved by keeping one side and hypotenuse of two right-angled triangles equal.