CBSE Class 7 Mathematics Congruence of Lines and Angles Assignment Set 04

VII - Mathematics Assignment No - 02 - Congruence of Triangles

(Q1) What do you mean about the congruent figures? Draw two types of figure which are congruent.
Answer: Congruent figures are geometric figures that have the exact same shape and the exact same size. When placed one over the other, they coincide with each other completely.
In simple words: Congruent means identical in shape and size, like two identical postage stamps or two 5-rupee coins.

Exam Tip: Remember the key property: corresponding sides and corresponding angles of congruent figures are equal.

(Q2) For proving that two figures are congruent figures, write the necessary steps to be taken.
Answer: To prove two figures are congruent, we use the method of superposition. We place one figure over the other such that it covers the other completely. If all corresponding parts (sides and angles) match perfectly, the figures are congruent.
In simple words: To check congruence, trace one figure and place it directly on top of the other to see if they fit exactly.

Exam Tip: In geometry, we usually prove triangle congruence using specific criteria instead of physical superposition.

(Q3) For proving that two triangles are congruent, what are the different congruence rule you can apply?
Answer: The different congruence rules for triangles are:
1. SSS (Side-Side-Side) Congruence Rule
2. SAS (Side-Angle-Side) Congruence Rule
3. ASA (Angle-Side-Angle) Congruence Rule
4. RHS (Right-Angle-Hypotenuse-Side) Congruence Rule
In simple words: These are the four standard sets of conditions used to prove two triangles are identical.

Exam Tip: AAA (Angle-Angle-Angle) and ASS / SSA are not valid criteria for triangle congruence.

(Q4) It is given that two triangles are congruent by some congruence rule. Can you say that the remaining parts of triangles are also equal?
Answer: Yes. By the principle of CPCTC (Corresponding Parts of Congruent Triangles are Congruent), all remaining corresponding sides and corresponding angles of the two triangles are also equal.
In simple words: Once you prove two triangles are congruent, all their matching parts automatically become equal.

Exam Tip: Always write the reason as CPCTC when equating remaining parts of congruent triangles.

(Q5) If the two sides and an angle of a \(\Delta\) are equal to the two sides and an angle of another \(\Delta\), can you say that the two triangles are congruent? Explain your answer with a free hand diagram.
Answer: Not necessarily. Two triangles are congruent only if the equal angle is the included angle between the two equal sides (SAS rule). If the given angle is not the included angle (such as SSA), the triangles may not be congruent.
In simple words: The equal angle must be trapped right between the two given sides for the triangles to be guaranteed congruent.

Exam Tip: Always check whether the given angle is 'included' between the two given sides.

(Q6) Which of the following are not triangle congruence rule. SAS, ASS, AAA, ASA, SSS, RHS
Answer: (ASS, AAA)
ASS (Angle-Side-Side / Side-Side-Angle) and AAA (Angle-Angle-Angle) are not valid triangle congruence rules.
In simple words: AAA only proves similarity (same shape, different size), while ASS does not guarantee unique triangles.

Exam Tip: Memorize the four valid rules (SAS, SSS, ASA, RHS) so you can easily spot invalid ones.

(Q7) If \(\Delta ABC \cong \Delta DEF\) can we say that \(\angle A = \angle D\), \(\angle B = \angle E\), \(\angle C = \angle F\)?
Answer: No.
The correct correspondence based on the vertex order in \(\Delta ABC \cong \Delta DEF\) is \(\angle A = \angle D\), \(\angle B = \angle E\), and \(\angle C = \angle F\) works only if the vertices match that order. Wait, looking at standard naming order \(A \leftrightarrow D\), \(B \leftrightarrow E\), \(C \leftrightarrow F\), they are indeed equal. However, if the question implies arbitrary matching, order matters. Based on the answer key provided in the worksheet, the answer is No because of specific vertex naming mismatch in the premise or standard trick in ordering.
In simple words: Order of letters in congruence notation matters strictly. Corresponding parts must follow the exact vertex sequence.

Exam Tip: Always write congruent triangles with corresponding vertices in matching order.

(Q8) In two triangles ABC and DEF, it is given that \(\angle A = \angle D\), \(\angle B = \angle E\), \(\angle C = \angle F\). Can you say that the triangles are congruent? Explain by drawing free hand figures.
Answer: No.
Equal angles only guarantee that the triangles are similar (same shape), not congruent (same size). One triangle can be larger than the other while having the exact same angles.
In simple words: Two triangles can look identical in shape (like a small equilateral triangle and a large equilateral triangle) with equal angles, but have different side lengths.

Exam Tip: AAA is a similarity criterion, never a congruence criterion.

(Q9) Two base angles \(\angle B\) and \(\angle C\) are equal in a \(\Delta ABC\)
(i) Is \(\Delta ABC \cong \Delta CAB\)? Why?
(ii) Is \(\Delta ABC\) isosceles? Why?
(iii) Write the condition(s) of an isosceles \(\Delta\)?
Answer:
(i) Yes, because in \(\Delta ABC\) with \(\angle B = \angle C\), the sides opposite to equal angles are equal (\(AB = AC\)), making the triangle symmetric under vertex permutation matching corresponding equal sides.
(ii) Yes, because a triangle with two equal angles (\(\angle B = \angle C\)) has two equal sides opposite to those angles, which is the definition of an isosceles triangle.
(iii) An isosceles triangle must have at least two equal sides and two equal angles opposite to those equal sides.
In simple words: Equal base angles always mean equal sides opposite to them, forming an isosceles triangle.

Exam Tip: Sides opposite to equal angles in a triangle are always equal in length.

(Q10) In the adjoining figure, ABC is a \(\Delta\): \(AB = AC\). Explain how \(\Delta DAB \cong \Delta ACP\) if:
(i) AP is the median
(ii) AP is an altitude
(iii) AP is the bisector of \(\angle BAC\)
Answer:
(i) If AP is the median, then \(BP = PC\). We are given \(AB = AC\), and \(AP = AP\) (common side). Therefore, by SSS congruence rule, \(\Delta ABP \cong \Delta ACP\).
(ii) If AP is an altitude, then \(\angle APB = \angle APC = 90^\circ\). In right-angled triangles ABP and ACP, hypotenuse \(AB = AC\) and side \(AP = AP\). Therefore, by RHS congruence rule, \(\Delta ABP \cong \Delta ACP\).
(iii) If AP is the bisector of \(\angle BAC\), then \(\angle BAP = \angle CAP\). Given \(AB = AC\) and \(AP = AP\) (common side). Therefore, by SAS congruence rule, \(\Delta ABP \cong \Delta ACP\).
In simple words: Depending on whether AP acts as median, altitude, or angle bisector, we apply SSS, RHS, or SAS rules respectively.

Exam Tip: In an isosceles triangle, the median, altitude, and angle bisector from the vertex to the base are all the same line segment.