SSA Congruence Rule
SSA congruence rule is also known as side-side-angle congruence rule refers to the congruence of two triangles. Two triangles are said to be congruent when it one of these five conditions are met, SSS, SAS, ASA, AAS, and RHS criteria. SSA congruence rule states that if two sides and an angle not included between them are respectively equal to two sides and an angle of the other then the two triangles are equal. However, this congruence or criterion is not valid. Let us see why it is not and find proof for it.
| 1. | What is the SSA Congruence Rule? |
| 2. | Proof of SSA Congruence Rule |
| 3. | FAQs on SSA Congruence Rule |
What is the SSA Congruence Rule?
The SSA congruence rule states that if two sides and an angle not included between them are respectively equal to two sides and an angle of the other then the two triangles are equal. However, with this congruence rule, two triangles are not said to be congruent since the sides of the two triangles may not be on the same corresponding sides. Both the triangles might end up having different shapes and sizes from each other. Thus, the SSA congruence rule is not valid.
Proof of SSA Congruence Rule
As we already learned that this congruence rule is not valid and triangles cannot be congruent, let us see the reasons as to why SSA will not work. Consider the following figure:
In the two triangles ∆ABC and ∆DEF, we have AB = DE, BC = EF, and ∠C = ∠F (non-included angles). We see that even though two pairs of sides and a pair of angles are (correspondingly) equal, the two triangles are not congruent. The two triangles do not have the same shape and size. Therefore, the SSA congruence rule is not valid.
Let us consider an example to understand this better. Suppose that there is a triangle two of whose sides have lengths 4cm and 3cm, and a non-included angle is 30°. Let’s try to geometrically construct such a triangle. If during our construction process, we find that we can construct only one (unique) such triangle, then SSA congruence would be valid, but on the other hand, if we find that we can construct more than one such triangle, then SSA congruence would be invalid because then two different triangles can have the same two lengths and a non-included angle.
Here’s a step-by-step construction:
Step 1: Construct BC = 4cm.
Step 2: Through C, draw a ray CX such that ∠BCX = 30°
Step 3: Take a point A on the ray CX such that AB = 3cm. How many locations of A are possible? Configure the compass such that the distance between its tip and the pencil’s tip is 3cm. Place the tip of the compass on B such that two points can be marked off on CX as shown in the image below.
Thus, two different triangles have been successfully constructed i.e. ∆A1BC and ∆A2BC with a pair of sides and a non-included angle. This means two things:
- A pair of sides and a non-included angle will not uniquely determine a triangle. In other words, congruence through SSA is invalid.
- A pair of sides and the included angle will uniquely determine a triangle. In other words, congruence through SAS is valid.
Therefore, it is proved that the SSA congruence rule is not valid.
Related Topics
Listed below are a few topics related to the SSA congruence rule, take a look.
Examples on SSA Congruence Rule
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Example 1: Triangle ABC is an isosceles triangle and the line segment AD is the angle bisector of the angle A. Can you prove that ΔADB is congruent to the ΔADC 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 ΔADB and ΔADC. As the line segment AD is the angle bisector of the angle A then it divides the ∠A into two equal parts. Therefore, ∠BAD=∠CAD. Now according to the SAS rule, the two triangles are congruent. Hence, ΔADB≅ΔADC.
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Example 2: Prove that △ACF≅△AEB, if ∠C=∠E and AC=AE.
Solution:
Here,
∠C=∠E (given)
AC=AE (given)
∠A=∠A (common)
Therefore, Δ ABD ≅ Δ ACD (ASA rule)
FAQs on SSA Congruence Rule
What is Meant by SSA Congruence Rule?
SSA congruence rule states that if two sides and an angle not included between them are respectively equal to two sides and an angle of the other then the two triangles are equal. However, this congruence or criterion is not valid.
Why is SSA Congruence Rule not Possible?
The SSA congruence rule is not possible since the sides could be located in two different parts of the triangles and not corresponding sides of two triangles. The size and shape would be different for both triangles and for triangles to be congruent, the triangles need to be of the same length, size, and shape.
When Can SSA Prove Triangles are Congruent?
SSA congruence rule can prove if triangles are congruent in two scenarios:
- If three sides of a triangle are congruent to three sides of another triangle, the triangles are considered congruent. (SSS Congruence Rule).
- If two sides and the included angle of one triangle are congruent to the corresponding parts of the other triangle, the triangles are considered congruent. (SAS Congruence Rule).
Is SSA a Criterion for Congruence of Triangles?
No, the SSA congruence rule is not a valid criterion that proves if two triangles are congruent to each other.
Does SSA Work in Right Angle Triangle?
One of the cases the SSA congruence rule might work is in right angle triangles where the angles are at right angles. The condition for this to be possibly valid is if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the triangles are congruent.
What is SSS, SAS, ASA, and AAS?
The 4 different triangle congruence theorems are:
- SSS(Side-Side-Side): Where three sides of two triangles are equal to each other.
- SAS(Side-Angle-Side): Where two sides and an angle included in between the sides of two triangles are equal to each other.
- ASA(Angle-Side-Angle): Where two angles along with a side included in between the angles of any two triangles are equal to each other.
- AAS(Angle-Angle-Side): Where two angles of any two triangles along with a side that is not included in between the angles, are equal to each other.
Why SSA is Not a Postulate?
SSA is not a postulate because two sides and a non-included angle do not guarantee the triangles to be congruent. The sides could be of any length and at different locations.