What is Side Angle Side Formula?

In geometry, two figures or objects are considered congruent if they have the same shape and size, or if one of them has the same shape and size as the mirror image of the other. More formally, a set of two points is said to be congruent only if one can be transformed into the other using the isometric method. In other words, you can move and mirror any object to exactly match another object, but you cannot resize it. So, if you can cut two different planar shapes on a sheet of paper and match them perfectly, they’re congruent. In this sense, the congruence of two planar figures means that their respective properties are “congruent” or “equal”, including their respective sides and angles as well as their respective diagonals, perimeters, and areas.

Congruence of Triangles

The congruence of two or more triangles depends on the measure of the sides and angles. The three sides of a triangle determine the size and the three angles of the triangle determine the shape. Two triangles are said to be congruent if their corresponding pairs of sides and corresponding angles are equal. They are the same shape and size. These triangles can be slid, rotated, flipped, and flipped to look the same. If you rearrange them, they match each other. The sign of congruence is “≅”.

In mathematics, congruence means that two figures are similar in shape and size. There are basically four laws of congruence that prove the congruence of two triangles. However, you need to find all six dimensions. Therefore, the congruence of a triangle can be estimated by knowing only three out of six values. 

Congruence in Triangles

Two triangles are said to be congruent if three angles and three sides of a triangle are equal to the corresponding angles and sides of another triangle. From Δ PQR and ΔXYZ, we observe that  PQ = XY, PR = XZ, QR = YZ, and  ∠P = ∠X, ∠Q = ∠Y, and ∠R = ∠Z. Then we can say ΔPQR ≅ ΔXYZ. Two triangles must be the same size and shape to be congruent. Two triangles to be considered must overlap each other, when you rotate, reflect, or move a triangle, its position or shape will not change.                  

Conditions of Congruence in Triangles

Two triangles are said to be congruent if three angles and three sides of a triangle are equal to the corresponding angles and sides of another triangle. It is not necessary to find all six corresponding elements of two triangles to determine congruence. Studies and experiments have shown that there are five conditions under which two triangles are congruent.

• SSS (Side – Side – Side) Rule      

Two triangles are said to be congruent by the SSS law if three sides of one triangle are equal to the corresponding three sides of the second triangle. In the given figure, AB = PQ, BC = QR, AC = PR,             

Therefore, ΔABC ≅ ΔPQR

• SAS (Side – Angle – Side) Rule

Two triangles are said to be SAS congruent if the angle between two sides of a triangle is equal to the angle between two sides of the second triangle. In the given figure, the sides are AB = PQ, AC = PR, and the angle between AC and AB is equal to the angle between PR and PQ. That is, ∠A = ∠P. So, ∆ABC ≅ ∆PQR. 

• ASA (Angle – Side – Angle) Rule

According to the ASA criteria, two triangles are congruent if two angles of one triangle and the side between them are equal to the corresponding angles of another triangle and the side between them. 

In the above figure, ∠ B = ∠ Q, ∠ C = ∠ R, and the sides between ∠ B and ∠ C, ∠ Q and ∠ R are equal. That is, BC = QR. So ∆ABC ≅ ∆PQR.

• AAS (Angle – Angle – Side) Rule

According to the AAS criterion, two triangles are congruent if two angles of a triangle and a side not between them are equal to the corresponding angle of the other triangle and the sides that are not between them are equal. According to the AAS criterion, if ΔABC ≅ ΔXYZ, then the third angle (∠ABC) and the other two sides (AC and BC) ΔABC must be equal to that angle (∠XYZ) and sides (XZ and YZ) ΔXYZ.                                                                                                                 

• RHS (Right angle- Hypotenuse-Side) Rule 

Two right triangles are RHS congruent if the hypotenuse and side of a right triangle are equal to the hypotenuse and side of the second right triangle. In the figure above, the hypotenuse is XZ = RT and the side is YZ = ST, so ∆XYZ ≅ ∆RST.                                

Side Angle Side

SAS congruence is a term also known as lateral angle congruence used to describe the relationship between two congruent shapes. Let’s take a closer look at SAS triangle congruence to understand what SAS means.                                                             

These two triangles are the same size and shape. So we can say that they match. They can be seen as examples of congruent triangles. We can represent this in mathematical form using the congruent triangle (≅) symbol, (ΔDEF ≅ ΔPQR). That is, D corresponds to P, E corresponds to Q, and F corresponds to R. ED corresponds to PQ, EF to QR, and DF to PR. Therefore, we can conclude that the corresponding parts of the congruent triangle are equal.

SAS (Side Angle Side) Formula

The side angle formula, also known as the SAS formula, is used to calculate the area of ​​a triangle using the rules of trigonometry. This formula, based on the side angle theorem, helps you calculate the area of ​​a triangle. As the name suggests, the side angle side refers to two sides and the angle between them. Let’s learn more about the angle formula to find the area of ​​a triangle.

Derivation of SAS (Side Angle Side) Formula

We know that the area of ​​a triangle is given as area = 1/2 × base × height. So let us consider the given triangle to understand the derivation of the SAS formula using the steps given below. 

Step 1: Two sides “a” and “b”, the angle between them “c” are given.

Step 2: Draw ‘p’ perpendicular from X to side YZ and use a trigonometric relationship to write the value of p as p = a × sin c. Assume that p is the height and apply the formula. Sin c = p/a.

Step 3: Because we know that the area of ​​the triangle = 1/2 × base × height. Substituting the value of the base into b and the height of p, the area of ​​the triangle is = 1/2 × b × p.

Step 4: Since p = a × sin c, applying the formula for the area of ​​the triangle is = 1/2 × b × a × sin c.

So, SAS Formula, area = 1/2 × a × b × sin c.

Sample Questions

Question 1: What is SAS Principle?

Solution:

The SAS theorem says that two triangles are congruent when the two sides and the angles between them are equal. This requirement does not make the triangles similar but congruent.

Question 2: What is the side angle formula used for? 

Answer: 

If you know two sides and the angle between them, you can use the angle of change formula to find the area of ​​​​a triangle. Another use of the formula is that you can use the trigonometric law of cosines to find the hypotenuse, which is the unknown side of a right triangle. We can use the law of sines to find the smaller angle, and then, knowing the two angles, we can calculate the third angle of the triangle.

Question 3: What is the area of ​​a triangle when three sides and an angle are given?

Answer: 

Given the sides of a triangle along with the angles between them, the area of ​​the triangle can be calculated by the formula

Area = (ab × sin C)/2, where “a” and “b” are the two given sides and C is the angle between them. This method is also known as the side angle method. For example, if two sides of a triangle are 9 and 11 units, and the angle between the two sides is 60°, then area = (11× 9 × sin 60°)/2 = 42.86 units².

Question 4: How to find the perimeter of the SAS triangle?

Answer: 

The perimeter of a triangle is defined as the total length of the triangle boundaries. That is, perimeter = sum of all lengths of the sides of a triangle.   

Given a  triangle ABC with two known sides AB and AC, and an angle ∠BAC  between the two sides. Let b be the length of AB, b be the length of AC, and c be the length of BC, then the perimeter can be given as perimeter = a + b + c. Since c is unknown, it can be obtained from the cosine formula (the law of cosines or the law of cosines) c² = a² + b² 2ab cos(∠BAC). So the perimeter of the triangle SAS  = a + b + √[a² + b² − 2abcos(∠BAC)] .

Question 5: What is the area of ​​a triangle with sides of 15 cm and 18 cm and the angle between them is 45°?

Solution:

By Side Angle Side Formula,

Area of triangle = 1/2 × a × b × sin c

Given, a = 15cm

b = 18 cm

Area = 1/2 × 15 × 18 × sin 45° = 95.47 cm².

Question 6: What is the area of ​​a triangle with sides of 5 cm and 10 cm and the angle between them is 30°?

Solution:

The formula for one side is,

Triangle area = (1/2) × side₁ × side₂ × sin (including angle)

Given, side₁ = 5cm, side₂ = 10cm, sin(including angle) = sin 30° = 1/2. Substitute values,

Area = (1 /2) × 5 × 10 × sin 30°

= (1/2) × 5 × 10 × (1/2)

= 12.5 cm².

So, the area of ​​​​the triangle is 12.5 cm².

Question 7: If EA = 30 units, ED = 20 units, and ∠AED = 30°, find the area of ​​the quadrilateral ADCB. Also, B divides EA by 1:2 and C is the midpoint of ED.      

Solution:  

It is given that sides,

EA = 40 units and ED = 30 units

Also, angle between EA and ED = ∠AED = 30°

By side angle side formula, Area of ΔAED = 1/2 × EA × ED × sin (30°) =  1/2 × 40 × 30 × sin (30°) = 300 units²

As B divides EA by 1:2 

3EB = AE

EB = AE/3 = 40 / 3 = 13.3 units

Given that C is the midpoint of ED , thus  EC = ED/2 = 30/2 = 15 units

Now area of ΔECB = 1/2 × EB × EC × sin (30°) =1/2 × 13.3 × 15 × sin (30°) = 49.875 units²

Area (ADCB) = Area  (ΔADE) –  Area  (ΔBCE) = 300 – 49.875 = 250.125 unit²

Area (ADCB) = 250.125 unit².