Congruence of Triangles

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Discover the concept of congruence of triangles, where two triangles are identical in shape and size. Explore the various congruence rules such as SSS, SAS, ASA, AAS, and RHS, and understand how they are used to determine triangle congruence.

Congruence of Triangles

Congruence of triangles refers to the property of two triangles being identical in shape and size. This means that all three corresponding sides of the triangles are equal in length, and all three corresponding angles have the same measure. Congruent triangles can be obtained by performing various transformations such as slides, rotations, flips, and turns.

To determine whether two triangles are congruent, we can use several congruence rules, which require knowing specific combinations of sides and angles. The four main congruence rules for triangles are:

• SSS (Side-Side-Side)
• SAS (Side-Angle-Side)
• ASA (Angle-Side-Angle)
• AAS (Angle-Angle-Side)
• RHS (Right angle-Hypotenuse-Side)

When two figures or shapes are congruent, it means they can be superimposed or overlapped perfectly onto each other, with their shape and dimensions matching exactly. In geometric figures, congruence implies that line segments have the same length and angles have the same measure.

To summarize, congruence in mathematics refers to the similarity in shape and size between two figures or shapes. In the case of triangles, congruent triangles have equal corresponding sides and angles. There are several conditions, such as SSS, SAS, ASA, AAS, and RHS, that can be used to prove the congruence of triangles.

Congruence of Triangle with Examples

Side-Side-Side (SSS) Congruence:

If the three sides of one triangle are equal to the corresponding three sides of another triangle, then the triangles are congruent.

Example:

Triangle ABC with side lengths AB = 5 cm, BC = 6 cm, and AC = 7 cm is congruent to Triangle DEF with side lengths DE = 5 cm, EF = 6 cm, and DF = 7 cm.

Side-Angle-Side (SAS) Congruence:

If two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of another triangle, then the triangles are congruent.

Example:

Triangle ABC with side lengths AB = 4 cm, BC = 6 cm, and angle ∠BAC = 60 degrees is congruent to Triangle DEF with side lengths DE = 4 cm, EF = 6 cm, and angle ∠EFD = 60 degrees.

Angle-Side-Angle (ASA) Congruence:

If two angles and the included side of one triangle are equal to the corresponding two angles and the included side of another triangle, then the triangles are congruent.

Example:

Triangle ABC with angles ∠BAC = 30 degrees, ∠ABC = 60 degrees, and side length AC = 5 cm is congruent to Triangle DEF with angles ∠EFD = 30 degrees, ∠DEF = 60 degrees, and side length DF = 5 cm.

Angle-Angle-Side (AAS) Congruence:

If two angles and a non-included side of one triangle are equal to the corresponding two angles and the non-included side of another triangle, then the triangles are congruent.

Example:

Triangle ABC with angles ∠BAC = 45 degrees, ∠ABC = 30 degrees, and side length AB = 4 cm is congruent to Triangle DEF with angles ∠EFD = 45 degrees, ∠DEF = 30 degrees, and side length DE = 4 cm.

Hypotenuse-Leg (HL) Congruence (for right triangles only):

If the hypotenuse and one leg of a right triangle are equal to the corresponding hypotenuse and leg of another right triangle, then the triangles are congruent.

Example:

Right triangle ABC with hypotenuse AC = 5 cm and leg AB = 3 cm is congruent to right triangle DEF with hypotenuse DF = 5 cm and leg DE = 3 cm.

Congruent Triangles Rules

The congruence of triangles can be determined using various rules and criteria. Here are the main congruent triangle rules:

1. Side-Side-Side (SSS) Rule: If three sides of one triangle are congruent to the corresponding three sides of another triangle, then the triangles are congruent. This rule states that if all three sides of one triangle are equal in length to the corresponding sides of another triangle, then the two triangles are congruent.
2. Side-Angle-Side (SAS) 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 triangles are congruent. This rule states that if two sides of one triangle are equal in length to the corresponding sides of another triangle, and the included angle between these sides is equal, then the two triangles are congruent.
3. Angle-Side-Angle (ASA) Rule: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. This rule states that if two angles of one triangle are equal to the corresponding angles of another triangle, and the side between these angles is equal in length, then the two triangles are congruent.
4. Angle-Angle-Side (AAS) Rule: If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the triangles are congruent. This rule states that if two angles of one triangle are equal to the corresponding angles of another triangle, and a side not between these angles is equal in length, then the two triangles are congruent.
5. Hypotenuse-Leg (HL) Rule: If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent. This rule applies specifically to right triangles and states that if the hypotenuse and one leg of one right triangle are equal in length to the corresponding parts of another right triangle, then the two triangles are congruent.

These congruence rules are useful in proving various geometric theorems and solving problems involving triangles, as they provide a systematic way to establish the congruence of triangles based on given information.

Congruent Triangle Proofs

Proof 1: Side-Angle-Side (SAS) Congruence

Given: △ABC and △DEF

• AB ≅ DE
• ∠A ≅ ∠D
• BC ≅ EF

To prove: △ABC ≅ △DEF

Proof:

• Construct line segment AC and line segment DF.
• By the given information, AB ≅ DE and BC ≅ EF.
• Also, ∠A ≅ ∠D.
• By the SAS congruence criterion, △ABC ≅ △DEF.

Proof 2: Angle-Side-Angle (ASA) Congruence

Given: △ABC and △DEF

• ∠A ≅ ∠D
• ∠B ≅ ∠E
• AC ≅ DF

To prove: △ABC ≅ △DEF

Proof:

• Draw line segment BC and line segment EF.
• Given that ∠A ≅ ∠D, ∠B ≅ ∠E, and AC ≅ DF.
• By the ASA congruence criterion, △ABC ≅ △DEF.

These are just two examples of congruent triangle proofs using different congruence criteria. There are other criteria such as Angle-Angle-Side (AAS) and Side-Side-Side (SSS) that can also be used to prove the congruence of triangles.

What are the Properties of Congruent Triangles?

Congruent triangles are triangles that have the same shape and size. When two triangles are congruent, it means that all corresponding angles and sides of the triangles are equal. The properties of congruent triangles include:

1. Side-Side-Side (SSS) Congruence: If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
2. Side-Angle-Side (SAS) Congruence: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
3. Angle-Side-Angle (ASA) Congruence: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
4. Angle-Angle-Side (AAS) Congruence: If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.
5. Hypotenuse-Leg (HL) Congruence (applicable for right triangles only): If the hypotenuse and one leg of one right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent.
6. Reflexive Property: Every triangle is congruent to itself. This means that all three sides and angles of a triangle are congruent to themselves.

These properties can be used to prove that two triangles are congruent and establish other relationships between their corresponding parts.

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