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Congruence of Triangles - Concept of Congruence

Grade 7CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The Concept of Congruence: Congruence refers to the relation of two objects having the same shape and size. If two figures can be superimposed (placed one on top of the other) so that they cover each other exactly, they are said to be congruent. Visually, imagine two identical 1-rupee coins; they are congruent because they have the exact same dimensions and shape.

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The Congruence Symbol: To denote congruence between two figures, we use the symbol ≅\cong. This symbol conveys that the figures are both equal in size (represented by the ==) and similar in shape (represented by the ∼\sim).

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Congruence of Plane Figures: Two plane figures F1F_1 and F2F_2 are congruent if the trace-copy of F1F_1 fits exactly on F2F_2. For example, two identical square tiles are congruent because one can be laid perfectly over the other to hide it completely.

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Congruence of Line Segments: Two line segments are congruent if and only if they have the same length. If two segments AB‾\overline{AB} and CD‾\overline{CD} are both 5 cm5\text{ cm} long, they are congruent, written as AB‾≅CD‾\overline{AB} \cong \overline{CD}. Visually, if you place one segment on top of the other, their endpoints will coincide.

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Congruence of Angles: Two angles are congruent if they have the same measure. For instance, if ∠ABC=60∘\angle ABC = 60^\circ and ∠PQR=60∘\angle PQR = 60^\circ, then ∠ABC≅∠PQR\angle ABC \cong \angle PQR. This means the degree of 'opening' between the two rays of each angle is identical, regardless of the length of the rays shown.

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Congruence of Triangles: Two triangles are congruent if they are copies of each other and when superposed, they cover each other exactly. This implies that the three sides of one triangle are equal to the corresponding three sides of the other, and the three angles of one are equal to the corresponding three angles of the other. In ΔABC≅ΔPQR\Delta ABC \cong \Delta PQR, vertex AA falls on PP, BB on QQ, and CC on RR.

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Correspondence in Triangles: In congruent triangles, the matching of vertices is called correspondence. For the statement ΔABC≅ΔPQR\Delta ABC \cong \Delta PQR, the correspondence is A↔PA \leftrightarrow P, B↔QB \leftrightarrow Q, and C↔RC \leftrightarrow R. The order of letters in the congruence statement is vital; for example, if AA matches PP, then AA must be in the same position in the name as PP.

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CPCT (Corresponding Parts of Congruent Triangles): This principle states that if two triangles are proved congruent, then all their corresponding parts (sides and angles) are equal. It is a standard justification used in geometric proofs after establishing congruence.

📐Formulae

AB‾≅CD‾  ⟺  AB=CD\overline{AB} \cong \overline{CD} \iff AB = CD

∠A≅∠B  ⟺  m∠A=m∠B\angle A \cong \angle B \iff m\angle A = m\angle B

ΔABC≅ΔPQR  ⟹  AB=PQ,BC=QR,AC=PR\Delta ABC \cong \Delta PQR \implies AB=PQ, BC=QR, AC=PR

ΔABC≅ΔPQR  ⟹  ∠A=∠P,∠B=∠Q,∠C=∠R\Delta ABC \cong \Delta PQR \implies \angle A = \angle P, \angle B = \angle Q, \angle C = \angle R

Correspondence Notation: ABC↔PQR\text{Correspondence Notation: } ABC \leftrightarrow PQR

💡Examples

Problem 1:

If ΔXYZ≅ΔLMN\Delta XYZ \cong \Delta LMN under the correspondence XYZ↔LMNXYZ \leftrightarrow LMN, list all the corresponding congruent parts of the triangles.

Solution:

Step 1: Identify the corresponding vertices from the given statement XYZ↔LMNXYZ \leftrightarrow LMN. The matches are X↔LX \leftrightarrow L, Y↔MY \leftrightarrow M, and Z↔NZ \leftrightarrow N.\nStep 2: List the corresponding angles based on the vertex matches:\n∠X≅∠L\angle X \cong \angle L\n∠Y≅∠M\angle Y \cong \angle M\n∠Z≅∠N\angle Z \cong \angle N\nStep 3: List the corresponding sides based on the pairs of vertices:\nXY‾≅LM‾\overline{XY} \cong \overline{LM}\nYZ‾≅MN‾\overline{YZ} \cong \overline{MN}\nXZ‾≅LN‾\overline{XZ} \cong \overline{LN}

Explanation:

In congruence statements, the order of the letters tells us exactly which parts of one triangle correspond to which parts of the other.

Problem 2:

Two angles, ∠PQR\angle PQR and ∠ABC\angle ABC, are congruent. If the measure of ∠PQR=(2x+10)∘\angle PQR = (2x + 10)^\circ and ∠ABC=70∘\angle ABC = 70^\circ, find the value of xx.

Solution:

Step 1: Use the property of congruence for angles. Since ∠PQR≅∠ABC\angle PQR \cong \angle ABC, their measures must be equal.\nStep 2: Set up the equation:\n(2x+10)∘=70∘(2x + 10)^\circ = 70^\circ\nStep 3: Solve for xx:\n2x=70−102x = 70 - 10\n2x=602x = 60\nx=602x = \frac{60}{2}\nx=30x = 30

Explanation:

Because the angles are congruent, their numerical degree values are identical, allowing us to solve for the unknown variable using basic algebra.