GSEB Class 7 Maths Solutions Chapter 7 Congruence of Triangles Exercise 7.2

 

Question 1. Which congruence criterion do you use in the following?
(a) Given:
AC = DF
AB = DE
BC = EF
So, ΔABC ≅ ΔDEF

(b) Given:
ZX = RP
RQ = ZY
So, ΔPQR ≅ ΔXYZ

(c) Given: ∠MLN = ∠FGH
∠NML = ∠GFH
ML = FG
So, ΔLMN ≅ ΔGFH

(d) Given: EB = DB
AE = BC
∠A = ∠C = 90°
So, ΔABE ≅ ΔCDB

Answer:
(a) SSS congruence criterion: We use the SSS (Side-Side-Side) rule because all three sides of one triangle are found to be equal to the corresponding three sides of the other triangle.
(b) SSS congruence criterion: This is another SSS (Side-Side-Side) rule case, as all three sides of the first triangle match the three corresponding sides of the second triangle.
(c) ASA congruence criterion: We use the ASA (Angle-Side-Angle) rule here since two angles and the included side of one triangle are equivalent to the two corresponding angles and included side of the other triangle.
(d) RHS congruence criterion: This employs the RHS (Right angle-Hypotenuse-Side) rule because both triangles have a right angle, their hypotenuses are equal, and one pair of corresponding sides are also equal.
In simple words: Match the parts that are given as equal to figure out which congruence rule applies. For example, if all sides are equal, it's SSS. If two angles and the side between them are equal, it's ASA.

Exam Tip: Carefully observe the given conditions and the figures. Each congruence criterion has specific requirements for sides and angles; know them well to identify the correct one.

 

Question 2. You want to show that ΔART ≅ ΔPEN,
(a) If you have to use SSS criterion, then you need to show
(i) AR =
(ii) RT =
(iii) AT =

(b) If it is given that ∠T = ∠N and you are to use SAS criterion, you need to have

(c) If it is given that AT = PN and you are to use ASA criterion, you need to have

(i) ∠ATR =
(ii) ∠TAR =
Answer:
Here ΔART ≅ ΔPEN means that the corresponding vertices are: A ↔ P, R ↔ E and T ↔ N.
(a) If you must use the SSS (Side-Side-Side) criterion, then you need to demonstrate the equality of the following corresponding sides:
(i) AR = PE
(ii) RT = EN
(iii) AT = PN
(b) If it is stated that ∠T = ∠N, and you are instructed to use the SAS (Side-Angle-Side) criterion, then you need to show the equality of the sides that include these angles:
(i) RT = EN
(ii) AT = PN
(c) If it is stated that AT = PN, and you are instructed to use the ASA (Angle-Side-Angle) criterion, then you need to demonstrate the equality of the angles that include these sides:
(i) ∠ATR = ∠PNE
(ii) ∠TAR = ∠NPE
In simple words: For triangles to be exactly the same (congruent), specific parts must match. For SSS, all three sides must be equal. For SAS, two sides and the angle between them must be equal. For ASA, two angles and the side between them must be equal.

Exam Tip: When dealing with congruence, correctly identifying corresponding vertices is crucial. This mapping helps you match corresponding sides and angles accurately for any given criterion.

 

Question 3. You have to show that ΔAMP ≅ ΔAMQ. In the following proof, supply the missing reasons.

Answer:
Here is the completed proof with the necessary reasons supplied:
In simple words: We showed these triangles are identical because we found two matching sides (PM=QM and AM=AM) and the angle between those sides (∠PMA=∠QMA) was also the same for both. This uses the SAS rule.

Exam Tip: For proofs, always list your reasons clearly for each step. Common reasons include "Given," "Common side/angle," "Vertically opposite angles," or specific congruence criteria like SAS, SSS, ASA, RHS.

 

Question 4. In ΔABC, ∠A = 30°, ∠B = 40° and ∠C = 110°. In ΔPQR, ∠P = 30°, ∠Q = 40° and ∠R = 110°. A student says that ΔABC ≅ ΔPQR by AAA congruence criterion. Is he justified? Why or why not?
Answer: No, the student is not justified in saying that ΔABC ≅ ΔPQR by the AAA (Angle-Angle-Angle) congruence criterion. This is because AAA is not a valid congruence rule. While having all three angles equal means the triangles are similar (they have the same shape), it does not guarantee that they are congruent (meaning they also have the same size). Two triangles can have the same angles but different side lengths, making them similar but not congruent.
In simple words: Even if all the angles of two triangles are the same, the triangles might not be the exact same size, so AAA isn't a rule for congruence. It only means they have the same shape.

Exam Tip: Remember the four main congruence criteria: SSS, SAS, ASA, and RHS. AAA is a criterion for similarity, not congruence. A common mistake is to confuse similarity with congruence.

 

Question 5. In the figure, the two triangles are congruent. The corresponding parts are marked. We can write ΔRAT ≅ ?

Answer:
By carefully observing the marked corresponding parts, we can establish the following vertex correspondence:
R ↔ W
A ↔ O
T ↔ N
Therefore, if ΔRAT is congruent to another triangle, that triangle would be ΔWON.
In simple words: When triangles are the same, their matching corners and sides align. Looking at the markings, corner R matches W, A matches O, and T matches N, so ΔRAT is the same as ΔWON.

Exam Tip: When writing a congruence statement (e.g., ΔABC ≅ ΔXYZ), the order of the vertices is crucial. It indicates which vertices, sides, and angles correspond to each other.

 

Question 6. Complete the congruence statement:
(i) ΔBCA = ?

(ii) ΔQRS = ?
Answer:
(i) We have the following correspondences based on the provided figure and markings:
A ↔ A
B ↔ T
C ↔ B
Therefore, the congruence statement is: ΔBCA ≅ ΔTBA.
(ii) We have the following correspondences based on the provided figure and markings:
R ↔ P
Q ↔ T
S ↔ Q
Therefore, the congruence statement is: ΔQRS ≅ ΔTPQ.
In simple words: Look at the matching sides and angles to see how the corners of one triangle line up with the corners of the other. The order of letters matters for congruent triangles.

Exam Tip: Matching angles and sides carefully is key. The vertex at the right angle, for instance, should correspond to the right-angle vertex in the other triangle.

 

Question 7. In a squared sheet, draw two triangles of equal areas such that
(i) the triangles are congruent.
(ii) the triangles are not congruent.
What can you say about their perimeters?

Answer:
(i) For congruent triangles with equal areas, consider two right-angled triangles ΔABC and ΔCDE, drawn on a squared sheet. Let both have a base of 3 cm and a height of 4 cm. The hypotenuse for both will be 5 cm.
Area of ΔABC = \( \frac { 1 }{ 2 } \) x base x height = \( \frac { 1 }{ 2 } \) x 4 x 3 = 6 sq. cm.
Area of ΔCDE = \( \frac { 1 }{ 2 } \) x base x height = \( \frac { 1 }{ 2 } \) x 4 x 3 = 6 sq. cm.
Perimeter of ΔABC = 3 + 4 + 5 = 12 cm.
Perimeter of ΔCDE = 3 + 4 + 5 = 12 cm.
Since both triangles have identical side lengths (3, 4, 5 cm), they are congruent, and their perimeters are equal.
(ii) For non-congruent triangles with equal areas, consider ΔPQR and ΔPRS, drawn on a squared sheet. Let ΔPQR be a right-angled triangle with a base of 3 cm and a height of 4 cm. Let ΔPRS be another triangle with a base of 3.5 cm and heights chosen to give the same area.
Area of ΔPQR = \( \frac { 1 }{ 2 } \) x base x height = \( \frac { 1 }{ 2 } \) x 4 x 3 = 6 sq. cm.
Area of ΔPRS (with height 4 cm and base 3 cm, but different sides for the actual drawing provided) = 6 sq. cm.
Perimeter of ΔPQR = 3 + 4 + 5 = 12 cm.
Perimeter of ΔPRS = 4 + 3.5 + 4 = 11.5 cm.
The two triangles have equal areas but are not congruent because their shapes and side lengths are different. Consequently, their perimeters are also not equal.
In simple words: Two triangles can have the same amount of space inside them (area) but be different shapes and sizes (not congruent). If they are congruent, their areas and perimeters are both the same. If they are not congruent, their areas can be the same, but their perimeters might be different.

Exam Tip: Equal area does not automatically mean congruence. Congruence implies equal area and perimeter, but equal area only sometimes implies equal perimeter.

 

Question 8. Draw a rough sketch of two triangles such that they have five pairs of congruent parts but still the triangles are not congruent.
Answer: A pair of triangles that have three equal angles and two equal sides, but are still non-congruent, can be depicted as follows:

In this case:
∠A = ∠D (All three angles are equal)
∠C = ∠F
∠B = ∠E
And two sides are equal:
AC = EF = 12
AB = DF = 8
However, ΔABC and ΔDEF are not congruent because the corresponding included angle for the sides (AC and AB for ΔABC, and EF and DF for ΔDEF) is not equal. Specifically, the side BC (in ΔABC) and side DE (in ΔDEF) are not equal, which prevents congruence even with five matching parts.
In simple words: You can have two triangles that look somewhat alike, even with many matching parts (like all angles and two sides), but still not be exactly the same size and shape (not congruent). This often happens when the matching sides don't include the matching angle in the right way.

Exam Tip: This scenario highlights why the position of equal parts is crucial for congruence. For SAS, the angle *must* be between the two sides. For ASA, the side *must* be between the two angles. Simply having five equal parts is not enough.

 

Question 9. If ΔABC and ΔPQR are to be congruent, name one additional pair of corresponding parts. What criterion did you use?

Answer:
Given the figures, we can see that two angles are already marked as equal: ∠B = ∠Q (one arc) and ∠C = ∠R (two arcs).
For ΔABC and ΔPQR to be congruent, we need one additional pair of corresponding parts. Since we already have two angles, the most appropriate criterion to use would be ASA (Angle-Side-Angle) or AAS (Angle-Angle-Side).
If we use the ASA congruence criterion, the additional corresponding part required is the side included between these two angles. This would mean:
BC = QR
So, by providing that BC = QR, we can prove ΔABC ≅ ΔPQR using the ASA congruence criterion.
In simple words: Since we already know two angles are the same (∠B matches ∠Q, and ∠C matches ∠R), to make the triangles congruent, we just need to confirm that the side between those angles (BC and QR) is also the same length. This would use the ASA rule.

Exam Tip: When asked to find an "additional part," always look at the existing given information (angles or sides) and then determine which congruence criterion (SSS, SAS, ASA, AAS, RHS) is easiest to satisfy with one more piece of information.

 

Question 10. Explain, why ΔABC ≅ ΔFED.

Answer:
To explain why ΔABC ≅ ΔFED, we need to examine the corresponding parts that are given as equal:
1. ∠B = ∠E = 90° (Given, as indicated by the right-angle markings).
2. BC = ED (Given, as indicated by the single dash mark on the sides).
3. ∠C = ∠D (Given, as indicated by the double arc marks on the angles).
Since two angles (∠B and ∠C) and the side included between them (BC) of ΔABC are equal to the two corresponding angles (∠E and ∠D) and the included side (ED) of ΔFED, the triangles are congruent by the ASA (Angle-Side-Angle) congruence criterion.
In simple words: The triangles are congruent because two of their angles (B and C in the first, E and D in the second) are the same, and the side that connects these two angles (BC and ED) is also the same length. This is called the ASA rule.

Exam Tip: For ASA, remember that the "S" (side) must be *included* between the two "A"s (angles). Visually confirm that the marked side is indeed the one connecting the two marked angles.