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Congruence of Triangles - Congruence of Plane Figures and Line Segments

Grade 7CBSE

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

🔑Concepts

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Congruence of Plane Figures: Two plane figures are congruent if they have the same shape and size. If you place one figure over the other and they cover each other completely (superposition), they are congruent. The symbol used for congruence is ≅\cong.

Two identical rectangles labeled F1 and F2 showing congruence.
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Congruence of Line Segments: Two line segments are congruent if and only if they have equal lengths. If AB≅CDAB \cong CD, then the length of AB=CDAB = CD. Conversely, if two segments have the same length, they are congruent.

Two parallel line segments AB and CD both 5cm long.
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Congruence of Angles: Two angles are congruent if they have the same measure. The length of the arms of the angle does not affect its congruence. For example, ∠ABC≅∠PQR\angle ABC \cong \angle PQR if m∠ABC=m∠PQRm\angle ABC = m\angle PQR.

Two angles both measuring 60 degrees.
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Superposition Method: This is a mental or physical test for congruence where you 'cut out' one figure and place it over another. If they match exactly, the figures are congruent.

📐Formulae

AB≅CD  ⟺  length(AB)=length(CD)AB \cong CD \iff length(AB) = length(CD)

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

Square S1≅S2  ⟺  side1=side2\text{Square } S_1 \cong S_2 \iff side_1 = side_2

Circle C1≅C2  ⟺  radius1=radius2\text{Circle } C_1 \cong C_2 \iff radius_1 = radius_2

d=2×r (where d is diameter and r is radius)d = 2 \times r \text{ (where } d \text{ is diameter and } r \text{ is radius)}

💡Examples

Problem 1:

Given two line segments, PQPQ of length 8.58.5 cm and RSRS of length 8.58.5 cm. Are they congruent? If the length of PQPQ is increased by 0.50.5 cm, will they remain congruent?

Solution:

Step 1: Compare the initial lengths. PQ=8.5PQ = 8.5 cm and RS=8.5RS = 8.5 cm. Since PQ=RSPQ = RS, the line segments are congruent: PQ≅RSPQ \cong RS. Step 2: Calculate the new length of PQPQ. New PQ=8.5+0.5=9.0PQ = 8.5 + 0.5 = 9.0 cm. Step 3: Compare the new length of PQPQ with RSRS. Now PQ=9.0PQ = 9.0 cm and RS=8.5RS = 8.5 cm. Since 9.0≠8.59.0 \neq 8.5, PQ≆RSPQ \ncong RS.

Explanation:

Congruence of line segments depends strictly on their lengths being equal. Once the lengths differ, they can no longer be congruent.

Problem 2:

Consider two circles. Circle AA has a radius of 44 cm. Circle BB has a diameter of 88 cm. Are these two circles congruent? Justify your answer.

Solution:

Step 1: Identify the radius of Circle AA, rA=4r_A = 4 cm. Step 2: Find the radius of Circle BB using the diameter formula r=d2r = \frac{d}{2}. Given dB=8d_B = 8 cm, then rB=82=4r_B = \frac{8}{2} = 4 cm. Step 3: Compare the radii. Since rA=4r_A = 4 cm and rB=4r_B = 4 cm, their radii are equal. Step 4: Conclusion. Since the radii are equal, the circles are congruent: Circle A≅Circle B\text{Circle } A \cong \text{Circle } B.

Explanation:

Two circles are congruent if their radii are the same. Since the diameter of Circle B is twice its radius, we calculate the radius to show it matches Circle A.

Problem 3:

Identify which of the following line segments are congruent: AB=4.2AB = 4.2 cm, CD=4CD = 4 cm, EF=4.2EF = 4.2 cm, and GH=5GH = 5 cm.

Comparison of four line segments of different lengths.

Solution:

  1. Compare the lengths of the segments.
  2. AB=4.2AB = 4.2 cm and EF=4.2EF = 4.2 cm.
  3. Since their lengths are equal, AB≅EFAB \cong EF.

Explanation:

According to the rule of congruence for line segments, two segments are congruent if they have the same length. Here, ABAB and EFEF both measure 4.24.2 cm, while CDCD and GHGH have different measures.

Problem 4:

Given two squares, Square XX with side 55 cm and Square YY with side 55 cm. Are they congruent? What if Square YY is rotated by 45∘45^\circ?

Two squares with side 5cm, one oriented normally and one rotated 45 degrees.

Solution:

  1. Side of Square XX = 55 cm.
  2. Side of Square YY = 55 cm.
  3. Since the sides are equal, Square X≅X \cong Square YY.
  4. Rotation does not change the size or shape, so they remain congruent.

Explanation:

Two squares are congruent if their sides are equal. The orientation (rotation) of the figure does not affect its congruence as the shape and size remain identical.

Congruence of Plane Figures and Line Segments Class 7 Notes & Examples