CBSE Class 7 Mathematics Congruence Of Triangles Worksheet Set 01

Question. If m ∠ABC m ∠LMN then∠ ABC ______________ ∠LMN.
Answer : ∠ABC ≅ ∠LMN.

Question. Congruent figures are equal in size.
Answer : False, congruent figures are equal in shape and size.

Question. When are two geometric figures said to be congruent?
Answer : Two geometric figures are said to be congruent if they have the same shape and size.

Question. ΔABC is an equilateral triangle. Is ΔABC ≅ ΔACB ? If yes, state the three criteria of congruence.
Answer : Yes, S.A.S., S.S.S., A.S.A. or A.A.S.

Question. If ΔLMN ≅ ΔRST, name the corresponding sides and the corresponding angles
Answer : LM = RS, MN = ST, LN = RT, ∠L = ∠R, ∠M = ∠S, ∠N = ∠T.

Question. When are two line segments said to be congruent?
Answer : Two line segments are congruent if they have the same length

Question. Two rectangles are congruent if _____________________.
Answer : They have the same length and breadth.

Question. Two circles are congruent if _________________________.
Answer : They have the same radii

Question. If two sides and the included angle of one triangle are respectively equal to any two sides and the included angle of the other then the two triangles are congruent.
Answer : True

Question. When are two triangles said to be congruent?
Answer : Two triangles are congruent if and only if one of then can be made to superimpose on the other, so as to cover the other exactly

Question. When are two angles said to be congruent?
Answer : Two angles are said to be congruent if they have the same measure and shape

Question. In the figure measure of sides and angles of PQR and LMN are given. Which of the statements are true?
Answer : (ii) ΔLMN ≅ ΔQPR.

Question. In the figure of Q. 25 which criterion is suitable to prove the congruence of the two triangles?
Answer : RHS

Question. When are two squares said to be congruent?
Answer :  Two squares are said to be congruent if they have the same side length

Question. If ΔPQR ≅ ΔMNL
(i) Which side of PQR is equal to the side of LM of ΔMNL ?
(ii) Which angle of ΔMNR is equal to R in the ∠PQR.
Answer : (i) RP

Question. Congruent figures are same in shape.
Answer : False, congruent figures are equal in shape and size.

Question. In Q. 26 in the congruence of triangles, which of the sides and angles are equal?
Answer : ∠R = ∠P = 90°
RS = PQ   SQ = QS

Question. In Q. 28 if the triangles are congruent then name them according to the correspondence. Also name the pairs of corresponding angles
Answer : ΔPRQ ≅ ΔLMN, ∠P = 105° = ∠L, ∠R = ∠M = 35

Question. If ΔABC ≅ ΔLMN, name the corresponding sides and the corresponding angles.
Answer : AB = LM, BC = MN, AC = LN, ∠A = ∠L, = ∠B = ∠M, ∠C = ∠N

Question. If ΔPQR ≅ ΔXYZ, name the corresponding sides and the corresponding angles.
Answer : PQ = XY, QR = YQ, PR = XZ, ∠P = ∠X, ∠Q = ∠Y, ∠R = ∠Z.

Question. Two triangles are congruent if three sides of one triangle are respectively equal to the three sides of the other trignale.
Answer : True

Question. Congruent figures cover each other completely?
Answer : True

Q.1. If triangle ABC and triangle DEF are congruent under the correspondence: ABC \(\leftrightarrow\) FED. Write the parts of triangle ABC that corresponds to:
a) DE
b) Angle E
c) FD
Answer:
Given correspondence is \(\text{ABC} \leftrightarrow \text{FED}\). This means vertex A corresponds to F, vertex B corresponds to E, and vertex C corresponds to D.
a) Side DE corresponds to side AB.
b) Angle E corresponds to angle B.
c) Side FD corresponds to side AC.
In simple words: Corresponding parts of congruent triangles match in the exact order given by the letter correspondence.

Exam Tip: Always align the letters of the two triangles according to the given correspondence before identifying corresponding sides or angles.

Q.2. Which congruence criterion will you use in the following. Write the congruence in symbolic form:
Answer:
Based on the given figures (referencing the standard triangle congruence figures with side lengths and marked angles):
a) By SAS (Side-Angle-Side) congruence criterion, the triangles are congruent, and written in symbolic form as \(\triangle ABD \cong \triangle ACD\) (or equivalent based on the figure labels).
b) By ASA (Angle-Side-Angle) or SAS criterion depending on the given marked sides and included angles, written symbolically as \(\triangle PQR \cong \triangle SRQ\).
In simple words: Identify which sides and angles are equal in both triangles to choose between SSS, SAS, ASA, or RHS criteria.

Exam Tip: Make sure the angle is strictly between the two given equal sides when applying the SAS criterion.

Q.3. In the given figure BD and CE are the altitudes of triangle ABC such that BD = CE
a) Prove that \(\triangle CBD \cong \triangle BCE\)
b) Is angle DCB = angle EBC
Give reasons
Answer:
a) In right-angled triangles \(\triangle CBD\) and \(\triangle BCE\):
Hypotenuse \(BC = BC\) (Common side)
Side \(BD = CE\) (Given altitudes are equal)
\(\angle BDC = \angle CEB = 90^\circ\) (Given altitudes)
Therefore, by RHS congruence criterion, \(\triangle CBD \cong \triangle BCE\).
b) Yes, \(\angle DCB = \angle EBC\).
Reason: Since \(\triangle CBD \cong \triangle BCE\), their corresponding parts are equal (CPCT), which gives \(\angle DCB = \angle EBC\).
In simple words: We use the RHS rule because both triangles have a right angle, a common hypotenuse, and one equal side.

Exam Tip: Mentioning CPCT (Corresponding Parts of Congruent Triangles) is mandatory when stating equality of parts after proving congruence.

Q.4. In the given figure AB and CD bisect each other at O. Prove that the \(\triangle AOC \cong \triangle BOD\)
Answer:
In \(\triangle AOC\) and \(\triangle BOD\):
\(AO = BO\) (Since AB and CD bisect each other at O)
\(CO = DO\) (Since AB and CD bisect each other at O)
\(\angle AOC = \angle BOD\) (Vertically opposite angles)
Therefore, by SAS congruence criterion, \(\triangle AOC \cong \triangle BOD\).
In simple words: Two pairs of sides are equal because O is the midpoint of both lines, and the opposite angles at the center are equal.

Exam Tip: Always identify vertically opposite angles clearly when two lines intersect.

Q.5. In the given figure ray AZ bisects angle BAD and angle DCB:
a) Prove that the \(\triangle BAC \cong \triangle DAC\)
b) Is AB = AD?
c) Is CD = CB?
Give reasons
Answer:
a) In \(\triangle BAC\) and \(\triangle DAC\):
\(\angle BAC = \angle DAC\) (Since ray AZ bisects \(\angle BAD\))
\(\angle BCA = \angle DCA\) (Since ray AZ bisects \(\angle DCB\))
\(AC = AC\) (Common side)
Therefore, by ASA congruence criterion, \(\triangle BAC \cong \triangle DAC\).
b) Yes, \(AB = AD\).
Reason: Corresponding parts of congruent triangles (CPCT) are equal since \(\triangle BAC \cong \triangle DAC\).
c) Yes, \(CD = CB\).
Reason: CPCT from the congruence of \(\triangle BAC\) and \(\triangle DAC\).
In simple words: The shared side and the bisected angles create two identical triangles back-to-back.

Exam Tip: When a line bisects an angle, it divides the angle into two equal parts.

Q.6. In the given figure AB = AC and D is the midpoint of BC.
a) Prove that \(\triangle ADB \cong \triangle ADC\)
b) Is angle B = angle C
Give reasons.
Answer:
a) In \(\triangle ADB\) and \(\triangle ADC\):
\(AB = AC\) (Given)
\(BD = CD\) (Since D is the midpoint of BC)
\(AD = AD\) (Common side)
Therefore, by SSS congruence criterion, \(\triangle ADB \cong \triangle ADC\).
b) Yes, \(\angle B = \angle C\).
Reason: Corresponding parts of congruent triangles (CPCT) are equal. This proves that angles opposite to equal sides of an isosceles triangle are equal.
In simple words: All three sides of triangle ADB are equal to all three sides of triangle ADC, making them congruent.

Exam Tip: State the midpoint condition clearly to show that the two divided segments of the base are equal.

Q.7. If AC = BD, AD = BC which of the following statements is meaningfully written
a) \(\triangle ABC \cong \triangle ABD\)
b) \(\triangle ABC \cong \triangle BAD\)
Answer: (b) \(\triangle ABC \cong \triangle BAD\)
From the given figure and conditions: \(AC = BD\) and \(AD = BC\), and the common base is \(AB = BA\).
Matching the vertices: Vertex A in triangle ABC corresponds to vertex B in triangle BAD, vertex B corresponds to vertex A, and vertex C corresponds to vertex D. Thus, the correct correspondence is \(\triangle ABC \cong \triangle BAD\).
In simple words: The order of letters in triangle congruence must strictly match the equal sides and angles.

Exam Tip: Double-check the vertex mapping by verifying that the first two letters give equal sides (e.g., AC = BD).

Q.8. By applying given congruence rule write what additional information is needed to establish congruence
a) \(\triangle PQR \cong \triangle FAD\) by SAS congruence rule, PQ = FE and RP = DF
b) \(\triangle ABC \cong \triangle RPQ\) by RHS congruence rule, angle B = angle P = 90^\circ and AB = RP
Answer:
a) For SAS congruence with given sides \(PQ = FE\) and \(RP = DF\), the included angle between sides PQ and RP is \(\angle P\), and the included angle between sides FE and DF is \(\angle F\). Therefore, the additional information needed is \(\angle P = \angle F\).
b) For RHS congruence with given right angles \(\angle B = \angle P = 90^\circ\) and side \(AB = RP\), we need the hypotenuse of both triangles to be equal. Therefore, the additional information needed is hypotenuse \(AC = hypotenuse RQ\).
In simple words: Identify what is missing from the specific rule requirements (SAS needs the included angle; RHS needs the hypotenuse and one side/right angle).

Exam Tip: Always check that the angle is strictly between the two given sides for SAS, and opposite to the right angle (hypotenuse) for RHS.

Please refer to attached file for CBSE Class 7 Maths Worksheet - Congruence of Triangles (1)