What Are the Key Conditions for Triangle Congruence?
Two triangles are said to be congruent when all three corresponding sides are equal and all three corresponding angles are identical in measure. These triangles can be moved around, rotated, flipped, and turned to make them seem the same. They will line up if they are moved. The symbol of congruence is’ ≅’. Congruent triangles have the same sides and angles as each other. There are basically four congruence rules that are rhs sss sas asa used to prove if two triangles are congruent.However, all six dimensions must be discovered. As a result, the congruence of triangles may be determined using only three of the six variables.
In this article we will learn rhs congruent triangles and other congruent triangles sss sas asa rhs along with solved examples.
Conditions of Congruence in Triangles
Two triangles are said to be congruent if both the triangles have similar size and shape. It is not necessary to find all six corresponding elements of both triangles to be equal in order to determine that they are congruent. There are five prerequisites for two triangles to be congruent, according to studies and experiments. They are sss sas asa rhs and aas congruence properties.
Congruent Triangles
Both triangles are said to be congruent if their three angles and three sides are equivalent to the equivalent angles and sides of another triangle. In the Δ PQR and ΔXYZ,We can see that PQ = XY, PR = XZ, and QR = YZ in the PQR and XYZ, and that ∠P = ∠ X, ∠Q = ∠ Y, and also ∠R = ∠Z. Hence we can say that Δ PQR ≅ ΔXYZ.
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To be congruent, the two triangles must be the same size and shape. Both of the given triangles should be superimposed on one another. A triangle's location or appearance appears to change when we rotate, reflect, and/or translate it. In that instance, we must determine the six parts of a triangle as well as the parts of the other triangle that correspond to them. Consider Δ ABC and ΔPQR as shown below.
Identifying the Corresponding Parts
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Thus on identifying the corresponding parts of the given triangles, we say that the Δ ABC ≅ ΔPQR.
CPCT
We come across the word CPCT, when we study about the congruent triangle. CPCT means “Corresponding Parts of Congruent Triangles”. The matching parts of congruent triangles are equal, as we know. We typically utilise the abbreviation cpct in brief phrases instead of the complete version while dealing with triangle principles and solving problems.
We can predict the congruence without actually measuring the sides and angles of a triangle. Different rules of congruency are as follows.
SSS (Side-Side-Side)
SAS (Side-Angle-Side)
ASA (Angle-Side-Angle)
AAS (Angle-Angle-Side)
RHS (Right angle-Hypotenuse-Side)
RHS Congruent Triangles
Here we will discuss congruence rhs sss and other congruence methods of triangle.
According to congruent triangles rhs rule : In two right-angled triangles, when the length of the hypotenuse and corresponding side of one triangle equals the length of the hypotenuse and corresponding side of the second triangle, the two triangles are congruent. It is also known as rhs triangle congruence.
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We can see in above figure, hypotenuse XZ = RT and side YZ=ST
Hence triangle XYZ ≅ triangle RST.
SSS Congruence Rule
According to the theorem: The two triangles are congruent if the three sides of one triangle are equal to the corresponding three sides (SSS) of the other triangle
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We can see in the above-given figure, AB= PQ, QR= BC and AC=PR
Hence Δ ABC ≅ Δ PQR.
ASA ( Angle - Side - Angle ) Congruence Rule
The two triangles are said to be congruent if any two angles and the side included between the angles of one triangle are comparable to the corresponding two angles and side included between the angles of the second triangle. Hence the two triangles are said to be congruent by ASA rule.
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We can see in above figure, ∠ B = ∠ Q, ∠ C = ∠ R and the sides between ∠B =∠C , ∠Q =∠ R i.e. BC= QR. Hence, Δ ABC ≅ Δ PQR.
SAS (Side-Angle-Side)
According to the SAS theorem - The two triangles are said to be congruent if any two sides and the angle included between the sides of one triangle are equivalent to the corresponding two sides and the angle between the sides of the another triangle.
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We can see in above given figure, length of sides AB= PQ, AC=PR and angle between AC and AB equal to angle between PR and PQ i.e. ∠A = ∠P. Hence, Δ ABC ≅ Δ PQR.
Angle - Angle - Side( AAS)
According to the Angle - Angle - Side rule : Two triangles are congruent if their corresponding two angles and one non-included side are equal.
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Given that:
∠ BAC = ∠ QPR, ∠ ACB = ∠ RQP and AB=QR,
So the triangle ABC and PQR are congruent to each other (△ABC ≅△ PQR).
Solved Examples:
1. In the Figure Given Below, AB = BC and AD = CD. Show that BD Bisects AC at Right Angles.
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Sol: Consider ∆ABD and ∆CBD,
Given AB = BC
AD = CD
Common Side BD = BD
By SSS congruency Therefore, ∆ABD ≅ ∆CBD
(By CPCT)
∠ABD = ∠CBD
Now, consider ∆ABE and ∆CBE,
(Given) AB = BC
∠ABD = ∠CBD (Proved above)
Common Side BE = BE
Therefore, ∆ABE≅ ∆CBE (By SAS congruency)
∠BEA = ∠BEC (CPCTC)
According to Linear pair
∠BEA +∠BEC = 180°
2∠BEA = 180° (∠BEA = ∠BEC)
∠BEA = 180°/2 = 90° = ∠BEC
By CPCT AE = EC
Hence, BD is a perpendicular bisector of AC.
2. In the Given Triangle Figures, Prove that Two Triangles are Congruent.
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Sol: To prove two triangles are congruent we need to find value of ∠ABC
Given ∠BAC = 65° and ∠BCA = 55°
In ∆ABC, ∠BAC + ∠ABC + ∠BCA = 180°
⟹ 65° + ∠ABC +55° = 180°
⟹ ∠ABC = 60°.
We can see in both ∆ABC and ∆XYZ,
Gien AB = XZ = 4 cm, BC = YZ = 5 cm
Also we found value of ∠ABC
So, ∠ABC = ∠XZY = 60°.
Therefore, both the triangles are congruent by SAS (Side-Angle-Side) criterion of congruence.
Conclusion:
We have discussed different types of congruence methods of triangles. We have learnt congruence rhs sss and rhs triangle congruence. From above we can conclude that following are the important points that we need to remember about congruence of triangle:
If the six parts of one triangle are equal to the corresponding six parts of the other triangle, the two triangles are congruent..
There are five criteria used to determine triangle congruence. Five conditions are SSS, SAS, ASA, AAS, and RHS criteria.
Two triangles that have equal corresponding angles may not be congruent to each other because one triangle may be an enlarged copy of the another triangle. Hence, there is no AAA Criterion for Congruence.
Triangles with corresponding sides and angles are said to be congruent. Congruence is denoted by the symbol “≅”. Both the triangles have the same area as well as the same perimeter
FAQs on Congruence of Triangles Made Simple
1. What does it mean for two triangles to be congruent?
Two triangles are said to be congruent if they are exact copies of each other. This means they have the same size and the same shape. If you could place one triangle on top of the other, it would cover it perfectly. All corresponding sides and corresponding angles of congruent triangles are equal. The symbol for congruence is ≅.
2. What are the five main criteria for proving triangle congruence?
To prove that two triangles are congruent, you don't need to check all six corresponding parts (3 sides and 3 angles). Instead, you can use any of the five established congruence criteria:
SSS (Side-Side-Side): If all three sides of one triangle are equal to the corresponding three sides of another triangle.
SAS (Side-Angle-Side): If two sides and the included angle (the angle between those two sides) of one triangle are equal to the corresponding parts of another.
ASA (Angle-Side-Angle): If two angles and the included side (the side between those two angles) of one triangle are equal to the corresponding parts of another.
AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are equal to the corresponding parts of another.
RHS (Right angle-Hypotenuse-Side): This special rule applies only to right-angled triangles. It requires the hypotenuse and one other side to be equal.
3. Can you explain the RHS (Right angle-Hypotenuse-Side) congruence rule with an example?
The RHS congruence rule is a special criterion used exclusively for right-angled triangles. For two right-angled triangles to be congruent under RHS, they must satisfy:
R: Both triangles must have a Right angle (90°).
H: The Hypotenuses (the side opposite the right angle) of both triangles must be equal.
S: Any one pair of corresponding Sides must be equal.
For example, if ΔABC is right-angled at B and ΔPQR is right-angled at Q, and we know that hypotenuse AC = PR and side AB = PQ, then we can declare that ΔABC ≅ ΔPQR by the RHS rule.
4. What is the key difference between the ASA and AAS congruence criteria?
The key difference between the ASA (Angle-Side-Angle) and AAS (Angle-Angle-Side) criteria lies in the position of the equal side relative to the equal angles.
In ASA, the side must be the included side, meaning it is located directly between the two angles being considered (e.g., ∠A, side AB, ∠B).
In AAS, the side is a non-included side, meaning it is not located between the two angles being considered (e.g., ∠A, ∠B, side BC).
Although different, both are valid criteria for proving congruence.
5. Why is 'CPCT' so important after proving triangles are congruent?
CPCT stands for 'Corresponding Parts of Congruent Triangles'. Its importance lies in what it allows you to do *after* you have proven two triangles are congruent using one of the rules (like SSS, SAS, etc.). Once congruence is established (ΔABC ≅ ΔPQR), CPCT serves as the reason to state that all other remaining corresponding parts are also equal. For example, you can deduce that ∠C = ∠R or BC = QR. It is the logical step that connects the concept of congruence to solving for unknown angles or sides.
6. Where is the concept of congruence of triangles used in real life?
The concept of congruence is fundamental in many real-world applications, ensuring stability, uniformity, and precision. Key examples include:
Architecture and Engineering: Triangular trusses in bridges and roofs are made of congruent triangles to distribute weight evenly and ensure structural strength.
Manufacturing: Mass production of items like tiles, machine parts, or mobile phone components relies on creating thousands of congruent shapes to ensure they fit together perfectly.
Art and Design: Artists use congruent shapes to create patterns and symmetry in their designs, such as in quilting or mosaic art.
7. Is there an AAA (Angle-Angle-Angle) congruence criterion? Why or why not?
No, there is no AAA congruence criterion in geometry. While knowing that all three angles of one triangle are equal to the corresponding angles of another confirms that the triangles are similar (they have the same shape), it does not guarantee they are congruent (the same size). For instance, an equilateral triangle with 2 cm sides and another with 10 cm sides both have angles of 60°, but they are clearly not congruent. One is just an enlargement of the other.
8. Why isn't SSA (Side-Side-Angle) a valid rule for proving congruence?
SSA (Side-Side-Angle) is not a valid congruence rule because it is ambiguous. Knowing the lengths of two sides and the measure of a non-included angle can lead to two different possible triangles. For a given Side-Side-Angle combination, you can often construct two distinct triangles that satisfy the conditions but are not congruent to each other. Because it doesn't uniquely determine a single triangle, SSA cannot be used as a reliable criterion for proving congruence, with the sole exception of the RHS rule for right-angled triangles.
