Have you ever observed that two ATM cards issued by the same bank are identical?

Remember that whenever identical objects are to be produced, the concept of congruence is taken into consideration.

Congruence of triangles means that:

- All corresponding angle pairs are equal.
- All corresponding sides are equal.

However, in order to be sure that the two triangles are similar, we do not necessarily need to have information about all sides and all angles.

In this mini-lesson, we will learn about the SAS similarity theorem in the concept of the SAS rule of congruence, using similar illustrative examples.

Check out the interactive simulation to explore more congruent shapes and do not forget to try your hand at solving a few interesting practice questions at the end of the page.

**Lesson Plan**

**What Do You Mean by Side Angle Side?**

The word **"congruent"** means equal in every aspect or figure in terms of shape and size.

Congruence** **is the term used to describe the relation of two figures that are congruent.

Now let's discuss the congruence of triangles.

Look at \(\Delta ABC\) and \(\Delta PQR\) below.

These two triangles are of the same size and shape.

Thus, we can say that they are congruent.

They can be considered as congruent triangle examples.

We can represent this in a mathematical form using the congruent triangles symbol (≅).

\(\Delta ABC \cong \Delta PQR\) |

This means \(A\) falls on \(P\), \(B\) falls on \(Q\), and \(C\) falls on \(R\).

Also, \(AB\) falls on \(PQ\), \(BC\) falls on \(QR\) and \(AC\) falls on \(PR\).

This indicates that the corresponding parts of congruent triangles are equal.

**SAS Criterion**

Under this criterion, if the two sides of a triangle are equal to the two sides of another triangle, and the angle formed by these sides in the two triangles are equal, then these two triangles are congruent.

**The SAS Criterion stands for the 'Side-Angle-Side' triangle congruence theorem.**

**The SAS Congruence Rule **

**The SAS Congruence Rule**

**The Side-Angle-Side theorem of congruency states that, if two sides and the angle formed by these two sides are equal to two sides and the included angle of another triangle, then these triangles are said to be congruent.**

**Proof**

**Let's perform an activity to show the proof of SAS.**

**Given: \(AB=PQ\), \(BC=QR\), and \(\angle B=\angle Q\)**

**To prove: \(\Delta ABC \cong \Delta PQR\)**

**Place the triangle \(\Delta ABC\) over the triangle \(\Delta PQR\) such that B falls on Q and side AB falls along the side PQ.**

**Since \(AB=PQ\), so point A falls on point P.**

**Since \(\angle B=\angle Q\), so the side BC will fall along the side QR.**

**\(BC=QR\), so point C falls on point R.**

**Therefore, BC coincides with QR and AC coincides with PR.**

**So, \(\Delta ABC\) will coincide with \(\Delta PQR\).**

**Therefore, \(\Delta ABC \cong \Delta PQR\)**

**Hence, this completes the SAS proof.**

**The SAS Similarity Rule **

**The SAS Similarity Rule**

**The SAS similarity criterion states that If two sides of one triangle are respectively proportional to two corresponding sides of another, and if the included angles are equal, then the two triangles are similar.**

**Proof**

**Consider the following figure:**

**Given: **

**\(\frac{{DE}}{{AB}} = \frac{{DF}}{{AC}}\)****\(\angle D = \angle A\)**

**The SAS criterion tells us that \(\Delta ABC\) ~ \(\Delta DEF\). Let us see the justification of this.**

**To prove: ** \(\Delta DEF\) is similar to \(\Delta ABC\)

**Construction:**

**Take a point \(X\) on \(AB\) such that \(AX = DE\)****Through \(X\), draw segment \(XY\parallel BC\), intersecting \(AC\) at \(Y\)**

**Proof:** Since \(XY\parallel BC\), we can note that \(\Delta AXY\) ~ \(\Delta ABC\), and thus:

**\[\frac{{AX}}{{AB}} = \frac{{AY}}{{AC}}....(1)\]**

**Now, we will show that \(\Delta AXY\) and \(\Delta DEF\) are congruent. It is given that:**

**\[\frac{{DE}}{{AB}} = \frac{{DF}}{{AC}}....(2)\]**

**Since \(AX = DE\) (By construction) and from (1) and (2), we have:**

**\[\frac{{DE}}{{AB}} = \frac{{AX}}{{AB}} = \frac{{AY}}{{AC}} = \frac{{DF}}{{AC}}\]**

**Thus,**

**\[AY = DF\]**

**Now, by the SAS congruency criterion,**

**\[\begin{gathered}
&\Delta AXY \cong \Delta DEF \hfill \\
\Rightarrow &\Delta AXY \sim \Delta DEF \hfill \\
\end{gathered} \]**

**While we already have, \(\Delta AXY\) ~ \(\Delta ABC\),**

**This means that,**

**\(\Delta DEF\) and \(\Delta ABC\) are similar.**

**Hence Proved**.

**How Do You Solve a SAS Triangle?**

**How Do You Solve a SAS Triangle?**

**Observe the triangle ABC where**

**AB= 2 units, BC= 4 units and \(\angle ABC = 50°\).**

** **

**Now, to find the value of side AC, we will use the law of cosine.**

**The low of cosine gives the formula \(b^2=a^2+c^2-2ac\cos{B}\), where \(AB=c\); \(BC=a\); and \(AC=b\)**

**\[\begin{aligned}b^2&=a^2+c^2-2ac\cos{B}\\x^2&=4^2+2^2-2(2)(4)\cos{50^{\circ}}\\&=16+4-16(0.643)\\&=20-10.288\\x&=\sqrt{9.712}\\&=3.116\end{aligned}\]**

**Now apply the sine rule in the triangle ABC and calculate the value of \(\angle C\).**

**\[\begin{aligned}\frac{\sin{C}}{2}&=\frac{\sin{B}}{x}\\\frac{\sin{C}}{2}&=\frac{\sin{50^{\circ}}}{3.116}\\\sin{C}&=2\times\frac{0.766}{3.116}\\&=0.492\\\angle C&=\sin^{-1}{0.492}\\&=29.47^{\circ}\end{aligned}\]**

**Then to calculate the value of \(\angle A\) use the sum of interior angles of a triangle is \(180^{\circ}\):**

**\[\begin{aligned}\angle A+\angle B+\angle C&=180^{\circ}\\\angle A+50^{\circ}+29.47^{\circ}&=180^{\circ}\\\angle A&=180^{\circ}-79.47^{\circ}\\&=100.53^{\circ}\end{aligned}\]**

**Important Notes**

**Two triangles are congruent if their shape and size are exactly the same.****We represent the congruence of triangles using the symbol (≅).****If the two sides and the angle formed at their vertex of one triangle are equal to the two corresponding sides and the angle formed at their vertex of another triangle then the triangles are congruent by SAS Criterion for Congruence.**

**Solved Examples**

**Solved Examples**Example 1 |

** **

** **

**James wanted to know which congruency rule says that these triangles are congruent. Let's help him.**

**Solution**

**\[\begin{aligned}
&\text { Here, } E F = M O =3 \mathrm{\;in}\\
& F G = N O =4.5 \mathrm{\;in}\\
&\angle E F G=\angle M O N =110^{\circ}\\
&\triangle \mathrm{EFG}\cong\triangle \mathrm{MNO}(\text { By SAS rule })
\end{aligned}\]**

\(\therefore\) These triangles are congruent by the SAS rule |

Example 2 |

** **

** **

**Triangle ABC is an isosceles triangle and the line segment AD is the angle bisector of the angle A.**

**Can you prove that \(\Delta ADB\) is congruent to the \(\Delta ADC\) by using SAS rule?**

**Solution**

**The triangle, ABC is an isosceles triangle where it is given that \(AB=AC\).**

**Now the side AD is common in both the triangles \(\Delta ADB\) and \(\Delta ADC\).**

**As the line segment AD is the angle bisector of the angle A then it divides the \(\angle A\) into two equal parts.**

**Therefore, \(\angle BAD=\angle CAD\)**

**Now according to the SAS rule, the two triangles are congruent.**

**Hence,**

**\[\Delta ADB \cong \Delta ADC\]**

\(\therefore\) \(\Delta ADB \cong \Delta ADC\). |

Example 3 |

** **

** **

**Jolly was doing geometrical construction assignments in her notebook.**

**She drew an isosceles triangle \(PQR\) on a page.**

**She marked \(L\), \(M\) as the midpoints of the equal sides (PQ and QR) of the triangle and \(N\) as the midpoint of the third side.**

**She states that \( LN=MN \).**

**Is she right?**

**Solution**

**We will prove that**

**\[\Delta LPN \cong \Delta MRN\]**

**We know that \(\Delta PQR\) is an isosceles triangle and \(PQ=QR\).**

**Angles opposite to equal sides are equal.**

**Thus,**

**\[\angle QPR=\angle QRP\]**

**Since \(L\) and \(M\) are the mid points of \(PQ\) and \(QR\) respectively,**

**\[\begin{align}PL=LQ=QM=MR=\frac{QR}{2}\end{align}\]**

**\(N\) is the mid point of \(PR\), hence,**

**\[PN=NR\]**

**In \(\Delta LPN\) and \(\Delta MRN\),**

**\(LP=MR\)****\(\angle LPN=\angle MRN\)****\(PN=NR\)**

**Thus, by SAS Criterion of Congruence,**

**\[\Delta LPN \cong \Delta MRN\]**

**Since congruent parts of congruent triangles are equal, \(LN=MN\).**

Yes, she is right that \(LN=MN\) |

**Challenging Questions**

**Can you determine the difference between the SAS congruency rule and the SAS similarity rule?****Can you prove the SAS congruency rule using any other congruency rule?**

**Interactive Questions**

**Interactive Questions****Here are a few activities for you to practice. **

**Select/Type your answer and click the "Check Answer" button to see the result.**

**Let's Summarize**

**Let's Summarize****The mini-lesson targeted the fascinating concept of the SAS rule of congruence. The math journey around the SAS rule starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that is not only relatable and easy to grasp but will also stay with them forever. Here lies the magic with Cuemath.**

**About Cuemath**

**About Cuemath****At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!**

**Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.**

**Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.**

**FAQs**

**FAQs****1. How do you prove the SAS congruence rule?**

**1. How do you prove the SAS congruence rule?****If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are said to be congruent by the SAS congruence rule.**

**2. What is a SAS triangle?**

**2. What is a SAS triangle?****A triangle whose two sides and the angle formed by them is known as a SAS triangle.**

**3. What is the SAS axiom?**

**3. What is the SAS axiom?****SAS axiom is the rule which says that if the two sides of a triangle are equal to the two sides of another triangle, and the angle formed by these sides in the two triangles are equal, then these two triangles are congruent by the SAS criterion.**

**4. How do you prove the SAS similarity theorem? **

**4. How do you prove the SAS similarity theorem?****SAS similarly can be proved by showing that one pair of side lengths of one triangle is proportional to one pair of side lengths of the other triangle and included angles are equal.**

**5. What is the difference between SAS and SSS?**

**5. What is the difference between SAS and SSS?****Both SAS and SSS rules are the triangle congruence rules. The full form of SAS is "Side-Angle-Side" and SSS stands for "Side-Side-Side."**

**In the SAS postulate, two sides and the angle between them in a triangle are equal to the corresponding two sides and the angle between them in another triangle.****In the SSS postulate, all three sides of one triangle are equal to the three corresponding sides of another triangle.**

**6. What are the three triangle similarity theorems?**

**6. What are the three triangle similarity theorems?****The three triangle similarity theorems are:**

**Angle-Angle (AA)****Side-Angle-Side (SAS)****Side-Side-Side (SSS)**

**7. What does SAS mean in math?**

**7. What does SAS mean in math?**

**SAS stands for the Side-Angle-Side theorem in the congruency of triangles. When two sides and the included angle of one triangle are equal to two sides and included angle of another triangle, the SAS rule is used to show that the two triangles are congruent.**