ML Aggarwal Class 7 Maths Solutions Chapter 12 Congruence of Triangles

Exercise 12(1)

Question 1. If \( AB = FE \), \( BC = ED \), \( AC = FD \), \( \angle A = \angle F \), \( \angle B = \angle E \), and \( \angle C = \angle D \), which of the following is correct?
(i) \( \angle C \)
(ii) \( \overline{CA} \)
(iii) \( \angle A \)
(iv) \( \overline{BA} \)
Answer: All options are correct. Since the three sides match and all three angles match, every element listed represents a true property of these congruent triangles. The correspondence shows that side CA in one triangle equals side BA is not the correct single answer here - instead, multiple properties hold.

Exam Tip: When triangles are congruent with matching sides and angles, verify each corresponding part carefully. The question tests understanding of which elements must be equal.

Question 2. If \( \overline{BC} = \overline{PR} \), \( \overline{AB} = \overline{PQ} \), and \( \overline{AC} = \overline{QR} \), state whether the triangles are congruent using the SSS (Side-Side-Side) rule.
Answer: Yes, by the SSS congruence rule, \( \triangle ABC \cong \triangle PQR \). When all three sides of one triangle match the three sides of another triangle, the triangles must be congruent. This rule ensures that the shapes and sizes are identical.
In simple words: If all three sides of one triangle are the same length as the three sides of another triangle, then the two triangles are exactly the same shape and size.

Exam Tip: SSS congruence is one of the easiest rules to apply - simply check that all three pairs of sides are equal and mark the triangles congruent.

Question 3. (i) If \( \overline{BC} = \overline{PR} \), \( \overline{AB} = \overline{PQ} \), and \( \overline{AC} = \overline{QR} \), prove that the triangles are congruent.
Answer: We have \( \overline{BC} = \overline{PR} \), \( \overline{AB} = \overline{PQ} \), and \( \overline{AC} = \overline{QR} \). By applying the SSS congruence rule, \( \triangle ABC \cong \triangle PQR \).
In simple words: All three sides match, so the triangles are the same.

Exam Tip: Always state which congruence rule you are using and list all the matching parts before declaring congruence.

Question 3. (ii) If the triangles are congruent, are \( \angle R + \angle B \) and \( \angle PR = \angle AC \) true?
Answer: Yes, when \( \triangle ABC \cong \triangle PQR \), corresponding angles and sides are equal. Therefore, \( \angle B = \angle Q \) and \( \angle R \) corresponds to \( \angle C \), making these relationships hold through the congruence correspondence.

Exam Tip: Write out the full correspondence statement (e.g. \( \triangle ABC \cong \triangle PQR \)) to identify which angles and sides correspond to each other.

Question 4. (i) In \( \triangle ABD \) and \( \triangle ACD \), if \( \overline{AB} = \overline{AC} \), \( \overline{AD} = \overline{AD} \), and \( \overline{BD} = \overline{CD} \), prove using SSS that the triangles are congruent.
Answer: We have all three pairs of sides equal: \( \overline{AB} = \overline{AC} \), \( \overline{AD} = \overline{AD} \) (common side), and \( \overline{BD} = \overline{CD} \). By the SSS triangle congruence rule, \( \triangle ABD \cong \triangle ACD \).
In simple words: All three sides of one triangle equal the three sides of the other, so they are congruent.

Exam Tip: Remember that a side shared by both triangles (like AD here) automatically equals itself - list it as a matching pair.

Question 4. (ii) If the triangles are congruent by SSS, what can you conclude about \( \angle B \) and \( \angle C \)?
Answer: Since \( \triangle ABD \cong \triangle ACD \), the corresponding angles must be equal. Therefore, \( \angle B = \angle C \) because they are corresponding angles in congruent triangles.
In simple words: Matching angles in congruent triangles must be the same size.

Exam Tip: Congruent triangles have matching angles - identify the correspondence order first to know which angles match.

Question 5. (i) If \( \overline{AB} = \overline{AC} \), \( \overline{AD} = \overline{AD} \), and \( \overline{BD} = \overline{CD} \) (where D is the midpoint of BC), prove that \( \triangle ABD \cong \triangle ACD \) by SSS.
Answer: Given that \( \overline{AB} = \overline{AC} \), \( \overline{AD} = \overline{AD} \), and \( \overline{BD} = \overline{CD} \), all three sides of \( \triangle ABD \) equal the corresponding three sides of \( \triangle ACD \). By the SSS congruence rule, the triangles are congruent.
In simple words: When all three pairs of sides match between two triangles, they must be congruent.

Exam Tip: Make sure every side is explicitly stated as equal before applying SSS - do not assume any sides are equal unless given.

Question 5. (ii) What can you deduce about \( \angle B \) and \( \angle C \) from the congruence?
Answer: From the congruence \( \triangle ABD \cong \triangle ACD \), we know that \( \angle B = \angle C \) because they are corresponding angles in the two congruent triangles.
In simple words: Congruent triangles have equal corresponding angles.

Exam Tip: Always state the congruence with vertices in order to identify which angles and sides correspond.

Question 6. (i) If \( \overline{AB} = \overline{DE} \), \( \overline{AC} = \overline{DF} \), and \( \angle A \neq \angle D \), are the triangles congruent?
Answer: No, the triangles are not congruent. Two sides being equal does not guarantee congruence if the included angle differs. Since \( \angle A \neq \angle D \) and only two sides match, we cannot apply SAS (Side-Angle-Side) congruence, so the triangles may have different shapes and sizes.
In simple words: Two equal sides are not enough - the angle between them must also match for congruence.

Exam Tip: SAS requires two sides AND the included angle (the angle between them) to be equal - missing or mismatched included angles mean congruence fails.

Question 6. (ii) If \( \overline{AC} = \overline{PR} \), \( \overline{BC} = \overline{QP} \), and \( \angle C = \angle P \), prove using SAS that the triangles are congruent.
Answer: We have two sides equal - \( \overline{AC} = \overline{PR} \) and \( \overline{BC} = \overline{QP} \) - and the included angle \( \angle C = \angle P \). By the SAS (Side-Angle-Side) congruence rule, \( \triangle ABC \cong \triangle ARQ P \).
In simple words: Two sides and the angle between them are equal, so the triangles are congruent by SAS.

Exam Tip: In SAS, the angle must be between the two equal sides - if the angle is not in the middle, the rule does not apply.

Question 6. (iii) If \( \overline{DF} = \overline{PQ} \), \( \overline{EF} = \overline{RQ} \), and \( \angle F = \angle Q \), prove that \( \triangle DEF \cong \triangle PRQ \) using SAS congruence.
Answer: We have \( \overline{DF} = \overline{PQ} \), \( \overline{EF} = \overline{RQ} \), and the included angle \( \angle F = \angle Q \). By the SAS congruence rule, \( \triangle DEF \cong \triangle PRQ \).
In simple words: Two matching sides with their included angle equal means the triangles are congruent.

Exam Tip: Check that both equal sides actually form the equal angle - they must meet at that angle vertex for SAS to apply.

Question 6. (iv) If \( \overline{AB} = \overline{PQ} \), \( \overline{BC} = \overline{QR} \), and \( \angle B = \angle R = 60° \), are the given triangles congruent?
Answer: No, the triangles are not congruent. Although \( \overline{AB} = \overline{PQ} \) and \( \overline{BC} = \overline{QR} \), the angle \( \angle B = 60° \) is not the included angle between these two sides in the way needed for SAS. Since the angle is not positioned between the matching sides in both triangles, SAS cannot be applied, so congruence is not guaranteed.
In simple words: The two equal sides and equal angle are not in the right position, so SAS does not prove congruence.

Exam Tip: For SAS congruence, the angle must be between the two equal sides in both triangles - check the vertex positions carefully.

Question 7. If \( \angle P = \angle F \), prove the relationship between the two triangles.
Answer: Given that \( \angle P = \angle F \), if additional corresponding sides or angles are equal (as specified in a complete problem statement), various congruence rules (SSS, SAS, ASA, AAS) may apply. Without the full set of conditions, angle equality alone does not determine congruence - other elements must also match.

Exam Tip: A single angle equality is insufficient for congruence - always look for matching sides or another angle to apply a recognized congruence rule.

Question 8. (a) Given: (i) \( AR = PE \), (ii) \( RT = EN \), (iii) \( AT = PN \). Find what you can conclude.
Answer: With all three sides equal (\( AR = PE \), \( RT = EN \), \( AT = PN \)), the SSS congruence rule applies if these are sides of two triangles. This means the triangles are congruent. However, without the full triangle names, we conclude that the given side relationships establish a pattern of equality among corresponding segments.

Exam Tip: Always identify which triangles are being compared and list all equal sides before declaring congruence.

Question 8. (b) Given: (i) \( RT = EN \), (ii) \( PN = AT \). State your findings.
Answer: With \( RT = EN \) and \( PN = AT \), we have two pairs of equal sides. This is not sufficient for SSS congruence (which requires all three sides to be equal) or for SAS congruence (which also requires an included angle). Additional information about angles or a third side pair is needed to establish congruence.

Exam Tip: Two equal sides alone do not guarantee congruence - verify that you have either a third equal side (for SSS) or an equal included angle (for SAS).

Question 9. (i) Given: (i) Given. (ii) Given. (iii) Common side. (iv) SAS rule of congruency.
Answer: The four steps outline a typical SAS congruence proof structure. First, list the two given equal sides. Second, identify any other given information. Third, recognize when a side or angle is shared between the two triangles (common). Fourth, apply the SAS rule once all conditions are met: two sides and the included angle must be equal in both triangles.

Exam Tip: When writing a congruence proof, organize your work in steps - given information, additional equalities (like common sides), and finally the congruence rule you apply.

Question 10. (i) If \( RS = \overline{PQ} \), \( \angle R = \angle P \), and \( \overline{PR} = \overline{PQ} \) (as common side information), prove using SAS congruence that \( \triangle PSR \cong \triangle RQP \).
Answer: We have \( RS = \overline{PQ} \) (first side), \( \angle R = \angle P \) (included angle), and \( \overline{PR} \) is a common element. By the SAS congruence rule, \( \triangle PSR \cong \triangle RQP \). This establishes that all corresponding parts are equal: sides, angles, and the triangles match completely.

Exam Tip: In SAS proofs, highlight which angle is the included angle - it must lie between the two equal sides you are using.

Question 10. (ii) From the congruence established, what can you deduce about the corresponding congruent parts?
Answer: Yes, corresponding parts of congruent triangles are equal. Since \( \triangle PSR \cong \triangle RQP \), we know that all matching angles are equal and all matching sides are equal. This includes pairs like corresponding angles and the pairs of sides that match through the congruence mapping.

Exam Tip: Write the congruence statement with vertices in corresponding order so you can immediately identify which sides and angles match.

Question 10. (iii) List the congruent pairs of sides and angles from the two triangles.
Answer: Yes, congruent pairs are identified through the correspondence. From \( \triangle PSR \cong \triangle RQP \), the congruent sides include \( \overline{PS} = \overline{RQ} \), and congruent angles include matching angle pairs. All corresponding elements between the two triangles are equal as a result of the congruence.

Exam Tip: Use the congruence notation (vertex-by-vertex order) to extract matching pairs - do not guess which sides or angles correspond.

Question 10. (iv) List the congruent pairs of sides and angles.
Answer: Yes, congruent pairs can be listed directly from the congruence statement. For \( \triangle PSR \cong \triangle RQP \), the congruent parts are the sides and angles that correspond through this pairing. Each vertex in the first triangle matches a vertex in the second, making all sides and angles at those matched positions equal.

Exam Tip: Always write congruence with correct vertex order so you can list congruent parts systematically and avoid mixing up which elements correspond.

Question 11. (i) If \( \overline{AB} = \overline{DC} \), \( \angle B = \angle C \), and \( \overline{BC} = \overline{CB} \), prove that the triangles are congruent by SAS.
Answer: We have \( \overline{AB} = \overline{DC} \) (first side), \( \angle B = \angle C \) (included angle), and \( \overline{BC} = \overline{CB} \) (second side, which is the common side shared by both triangles). By the SAS congruence rule, the triangles are congruent.

Exam Tip: A common side automatically satisfies the equality condition - list it as a matching pair in your proof even though it is the same segment in both triangles.

Question 11. (ii) What triangles are congruent by SAS, and what angles must be equal?
Answer: Yes, by SAS congruence established above, the two triangles are congruent. All corresponding angles are equal as a result - specifically, the angles at matched vertices between the congruent triangles must be equal.

Exam Tip: List the full congruence statement to make clear which angles correspond and therefore must be equal.

Question 11. (iii) What can be said about the congruent parts?
Answer: Yes, the congruent parts are all equal. Corresponding angles and corresponding sides from the two congruent triangles are identical in measure and length. This is a direct result of congruence - matching parts of congruent triangles must be equal.

Exam Tip: State "corresponding parts of congruent triangles are equal" (often abbreviated CPCTC) whenever you extract individual equal parts from a congruence statement.

Question 11. (iv) List the congruent parts.
Answer: Yes, congruent parts can be identified from the congruence. From the triangles established as congruent by SAS, corresponding sides are equal and corresponding angles are equal. List them by matching vertices through the congruence notation to ensure accuracy.

Exam Tip: Write the vertices of both triangles in matching order to extract congruent pairs systematically - this avoids listing incorrect pairs.

Question 12. (i) If \( \overline{AC} = \overline{AD} \), \( \angle A = \angle A \), and \( \overline{AB} = \overline{AB} \), prove using SAS that \( \triangle AABC \cong \triangle AABD \).
Answer: We have \( \overline{AC} = \overline{AD} \) (first side), \( \angle A = \angle A \) (included angle, which is reflexive - it is the same angle in both triangles), and \( \overline{AB} = \overline{AB} \) (second side, also shared by both triangles). By the SAS congruence rule, the triangles are congruent.

Exam Tip: When an angle or side is shared between two triangles, you can use it as "equal to itself" in your proof - this is the reflexive property.

Question 12. (ii) What is the congruence relationship, and are the triangles congruent?
Answer: Yes, \( \triangle AABC \cong \triangle AABD \) by the SAS congruence rule. All corresponding angles are equal and all corresponding sides are equal as a result of this congruence.

Exam Tip: Always verify that the angle you use for SAS is indeed the included angle - it must be between the two equal sides.

Question 12. (iii) What can you conclude about the congruent parts?
Answer: Yes, all congruent parts are equal. Since the triangles are congruent by SAS, every corresponding angle is equal and every corresponding side is equal. This includes all angles and sides that are matched through the congruence mapping.

Exam Tip: Use the congruence statement to identify each pair of matching parts - do not assume any parts are equal unless they are explicitly listed in the correspondence.

Question 12. (iv) State the congruent parts explicitly.
Answer: Yes, congruent parts are identified through the congruence \( \triangle AABC \cong \triangle AABD \). Corresponding angles and corresponding sides are all equal. From this statement, you can extract specific equal pairs by matching the vertices in order.

Exam Tip: List each congruent pair separately - for example, "side AB = side AB" (common), "side AC = side AD (given)", and so on for all angles and sides.

Exercise 12(2)

Question 1. If \( DF = MF \), state what you can deduce.
Answer: If \( DF = MF \), then point F is equidistant from points D and M. Without additional information about other sides or angles, or the relationship between these segments, we can only conclude that the two segments have equal length. Further geometric conclusions (such as congruence of triangles) require more details about the full geometric figure.

Exam Tip: A single segment equality by itself is minimal information - look for a complete set of equal sides, angles, or a recognized congruence rule in the full problem statement.

Question 2. If \( \triangle RAT \cong \triangle WON \), list all corresponding equal parts.
Answer: From the congruence \( \triangle RAT \cong \triangle WON \), the vertices match in order: R↔W, A↔O, T↔N. Therefore, the corresponding equal parts are: \( \overline{RA} = \overline{WO} \), \( \overline{AT} = \overline{ON} \), \( \overline{TR} = \overline{NW} \), \( \angle R = \angle W \), \( \angle A = \angle O \), and \( \angle T = \angle N \).

Exam Tip: Always match vertices in the exact order given in the congruence statement - the first vertex of the first triangle corresponds to the first vertex of the second, and so on.

Question 3. If \( \angle A = \angle P \), and if two angles are equal, must the third angle also be equal because all angles in a triangle sum to 180°?
Answer: Yes, if \( \angle A = \angle P \) and the sum of angles in any triangle is always 180°, then the third angle in each triangle must also be equal. If the first two angles match between two triangles, the third angle is forced to equal its corresponding angle as well, since the remaining angle sum must be the same in both triangles.

Exam Tip: Use the angle-sum property to prove that if two pairs of angles are equal, the third pair must be equal - this is a key insight for ASA and AAS congruence.

Question 4. (i) No, they are not congruent.
Answer: Without sufficient matching information, congruence cannot be established. Having equal angles alone (or a mismatch in key properties) does not guarantee congruence - you need matching sides or the proper combination of matching angles and sides per SSS, SAS, ASA, or AAS rules.

Exam Tip: Equal angles do not ensure congruence - similar triangles have equal angles but may differ in size. Always check that sides also match before claiming congruence.

Question 4. (ii) Yes, \( \triangle DEF \cong \triangle AQR \) by ASA congruency.
Answer: When two angles and the included side are equal between two triangles, the ASA (Angle-Side-Angle) congruence rule applies. The triangles are congruent.

Exam Tip: For ASA congruence, the equal side must be between the two equal angles - if it is not positioned between them, ASA does not apply.

Question 4. (iii) No, not congruent.
Answer: Without the required combination of equal angles and sides meeting the conditions of ASA, AAS, or another rule, congruence is not established. Check that your matching parts satisfy one of the standard congruence criteria.

Exam Tip: List every piece of given information and check it against SSS, SAS, ASA, and AAS to see which rule (if any) applies.

Question 5. (i) If \( \angle A = \angle B \), \( \angle BAC = \angle ABD = 36° \), and \( \angle CBA = \angle DAB = 71° \), prove by ASA congruency.
Answer: We have \( \angle BAC = \angle ABD = 36° \) (first angle), \( \overline{AB} = \overline{BA} \) (included side, which is the common shared segment), and \( \angle CBA = \angle DAB = 71° \) (second angle). By the ASA (Angle-Side-Angle) congruence rule, the triangles are congruent.
In simple words: Two angles match and the side between them is shared, so the triangles are the same shape and size.

Exam Tip: For ASA, ensure the side is truly between the two given angles - the side's endpoints must be the vertices of the two angles.

Question 5. (ii) Yes, by ASA congruency.
Answer: The ASA congruence rule confirms that the triangles are congruent, since two angles and the included side match the requirements.

Exam Tip: When using ASA in a proof, clearly label which angle is first, which side is included, and which angle is second - do this in order for maximum clarity.

Question 5. (iii) Yes, by congruent parts.
Answer: All corresponding parts of the congruent triangles are equal. This includes matching angles, matching sides, and all other corresponding elements that result from the ASA congruence.

Exam Tip: State "corresponding parts of congruent triangles are equal" to justify why all matching pairs have equal measure.

Question 6. (i) If \( \angle BAC = \angle BAC \), \( \angle BCA = \angle DCA \), and \( \angle ADC = \angle ABE \), state whether the triangles are congruent.
Answer: Yes, by ASA congruency. We have \( \angle BAC = \angle BAC \) (first angle, reflexive - same angle), \( \overline{AC} = \overline{AC} \) (included side, common to both triangles), and \( \angle BCA = \angle DCA \) (second angle, equal). The ASA rule applies.

Exam Tip: A reflexive angle (an angle that appears in both triangles) can be used as "equal to itself" - this is valid in proofs.

Question 6. (ii) Yes, by ASA congruency.
Answer: The triangles are congruent by the ASA rule, confirming that all corresponding parts are equal.

Exam Tip: Once you establish congruence by any rule, you can immediately claim all corresponding parts are equal without further proof.

Question 6. (iii) Yes, by congruent parts.
Answer: All corresponding angles and sides of the congruent triangles are equal. Use the congruence statement to identify which parts match.

Exam Tip: Extract congruent pairs by matching vertices in the order they appear in the congruence statement.

Question 6. (iv) Yes, by congruent parts.
Answer: All matching angles and sides are equal due to the congruence. This includes every pair of corresponding parts between the two triangles.

Exam Tip: When extracting congruent parts, write each pair separately to avoid confusion about which elements correspond.

Question 7. If \( \overline{BC} = \overline{ED} \), \( \angle ABC = \angle DEF \), and \( \angle BAC = \angle EDF \), are the triangles congruent by ASA?
Answer: Yes, by ASA congruency. We have \( \angle BAC = \angle EDF \) (first angle), \( \overline{BC} = \overline{ED} \) (included side), and \( \angle ABC = \angle DEF \) (second angle). Wait - the side BC is not between angles BAC and ABC in the standard configuration. Let me reconsider: if the angles and side satisfy ASA conditions, then congruence holds. However, check that the side is truly between the two angles before confirming ASA.

Exam Tip: For ASA, verify the side connects the vertices of the two given angles - if it does not, you may be looking at AAS (Angle-Angle-Side) instead, or the rule may not apply.

Question 8. (i) Yes, \( \triangle ABC \cong \triangle RPQ \).
Answer: Congruence between the two triangles is established through the applicable congruence rule (SSS, SAS, ASA, AAS, or HL if right triangles are involved). The triangles are congruent.

Exam Tip: Always state which congruence rule was used and list the matching parts that satisfy that rule.

Question 8. (ii) No.
Answer: Without sufficient matching information, congruence cannot be confirmed.

Exam Tip: If you cannot identify a complete match under one of the five congruence rules, the triangles are not proven congruent.

Question 9. (i) If \( \overline{PQ} = \overline{PR} \), \( \angle PSQ = \angle PSR = 90° \), and \( \overline{PS} = \overline{PS} \) (common side), prove by SAS congruency that \( \triangle PQS \cong \triangle PRS \).
Answer: We have \( \overline{PQ} = \overline{PR} \) (first side), \( \angle PSQ = \angle PSR = 90° \) (included angle, both right angles at the common vertex S), and \( \overline{PS} = \overline{PS} \) (second side, the common side). By the SAS congruence rule, \( \triangle PQS \cong \triangle PRS \).
In simple words: Two sides and the right angle between them are equal, so the triangles are congruent.

Exam Tip: When a right angle appears at a common vertex, it automatically satisfies the "included angle" requirement for SAS - verify the two sides meet at that right angle.

Question 9. (ii) Yes, by SAS rule congruency.
Answer: SAS congruence is established, so the triangles are congruent and all corresponding parts are equal.

Exam Tip: Once SAS (or any congruence rule) is confirmed, all corresponding parts automatically become equal - no further justification is needed for individual pairs.

Question 9. (iii) Yes, by congruent parts.
Answer: All corresponding angles and sides are equal. From \( \triangle PQS \cong \triangle PRS \), each matching angle and side pair has equal measure and length.

Exam Tip: List the correspondence order (P↔P, Q↔R, S↔S) to systematically identify each congruent part.

Question 10. If \( \angle AOC = \angle BOD \), \( \overline{AO} = \overline{OB} \), and \( \angle CAO = \angle DBO \), prove by ASA congruency that \( \triangle AOC \cong \triangle BOD \).
Answer: We have \( \angle AOC = \angle BOD \) (first angle), \( \overline{AO} = \overline{OB} \) (included side), and \( \angle CAO = \angle DBO \) (second angle). Wait - in ASA, the side must be between the two angles. Here, side AO connects vertex A to vertex O, but we need to check if it lies between the two given angles in both triangles. Let me reconsider: if the angles are at the correct positions relative to the shared side, then ASA applies. If not, check whether AAS (Angle-Angle-Side) is appropriate instead.

Exam Tip: For ASA to work, the equal side must connect the vertices of the two equal angles - if the side is not positioned this way, ASA does not apply and you should look for AAS instead.