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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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Congruence of plane figures means that two objects are identical in shape and size. If you place one figure over the other, they cover each other exactly. The symbol used for congruence is ≅\cong.

Two identical rectangles showing congruence.
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Congruence of line segments: Two line segments are congruent if they have the same length. For example, if AB=5 cmAB = 5\text{ cm} and CD=5 cmCD = 5\text{ cm}, then AB‾≅CD‾\overline{AB} \cong \overline{CD}.

Two parallel line segments of equal length.
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Congruence of angles: Two angles are congruent if they have the same measure. If m∠ABC=40∘m\angle ABC = 40^\circ and m∠PQR=40∘m\angle PQR = 40^\circ, then ∠ABC≅∠PQR\angle ABC \cong \angle PQR.

An angle of 40 degrees.
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Congruence of Triangles: Two triangles are congruent if their corresponding sides and corresponding angles are equal. This is written as ΔABC≅ΔPQR\Delta ABC \cong \Delta PQR. The order of letters indicates the correspondence, e.g., A↔PA \leftrightarrow P.

📐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: ∠X≅∠L\angle X \cong \angle L ∠Y≅∠M\angle Y \cong \angle M ∠Z≅∠N\angle Z \cong \angle N\nStep 3: List the corresponding sides based on the pairs of vertices: XY‾≅LM‾\overline{XY} \cong \overline{LM} YZ‾≅MN‾\overline{YZ} \cong \overline{MN} XZ‾≅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: (2x+10)∘=70∘(2x + 10)^\circ = 70^\circ\nStep 3: Solve for xx: 2x=70−102x = 70 - 10 2x=602x = 60 x=602x = \frac{60}{2} x=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.

Problem 3:

In the figure provided, check if ΔABC≅ΔFED\Delta ABC \cong \Delta FED given the correspondence ABC↔FEDABC \leftrightarrow FED. If AB=3 cmAB = 3\text{ cm}, BC=4 cmBC = 4\text{ cm}, and AC=5 cmAC = 5\text{ cm}, find the lengths of FEFE, EDED, and FDFD.

Two identical right-angled triangles ABC and FED.

Solution:

Given ΔABC≅ΔFED\Delta ABC \cong \Delta FED. By the property of congruence, corresponding sides must be equal.

  1. FEFE corresponds to ABAB, so FE=AB=3 cmFE = AB = 3\text{ cm}.
  2. EDED corresponds to BCBC, so ED=BC=4 cmED = BC = 4\text{ cm}.
  3. FDFD corresponds to ACAC, so FD=AC=5 cmFD = AC = 5\text{ cm}.

Explanation:

When two triangles are congruent, the parts that 'match' (correspond) are equal in measure. The naming ABC↔FEDABC \leftrightarrow FED tells us exactly which vertices match.

Problem 4:

Two circles are congruent. If the radius of the first circle is r1=7 cmr_1 = 7\text{ cm}, what is the diameter of the second circle?

Two congruent circles with radius 7cm shown in the first circle.

Solution:

  1. Two circles are congruent if they have the same radius.
  2. Since the circles are congruent, r2=r1=7 cmr_2 = r_1 = 7\text{ cm}.
  3. Diameter D=2×radiusD = 2 \times \text{radius}.
  4. D=2×7=14 cmD = 2 \times 7 = 14\text{ cm}.

Explanation:

Congruence in circles depends solely on the radius. If the radii are equal, the circles are identical in size.

Concept of Congruence Class 7 Notes & Examples | CBSE Maths