Test: Congruence Criteria- SSS And RHS - Class 9 MCQ

∠LQ = ∠MNR,

∠Q = ∠R

QM = MR,

Hence, ΔQLM ≅ ΔMNR (by AAS)

Proof of Congruence

1. Given:

• ABCD is a parallelogram.
• Diagonals AC and BD are equal.

2. To Prove:

Triangles △ABD and △ABC are congruent.

3. Explanation:

• In a parallelogram with equal diagonals, the parallelogram is actually a rectangle because, for a general parallelogram, the diagonals are not equal unless it is a rectangle.
• Since ABCD is a rectangle, its diagonals bisect each other and are equal in length. This means that each half of the diagonal is also equal.

4. Congruence Criterion:

• In triangles △ABD and △ABC:
  • AB is common to both triangles.
  • BD = AC, as they are the diagonals of a rectangle and are given to be equal.
  • AD = BC, as opposite sides of a rectangle are equal.
• Thus, by the SSS (Side-Side-Side) criterion, △ABD ≅ △ABC.

Triangles △ABD and △ABC are congruent by the SSS (Side-Side-Side) criterion.

Let's analyze the given information step by step to determine whether triangles ΔABC and ΔPBC are congruent.

1. Given:

   • AB = BP
   • AC = PC
   • BC = BC (common side)

2. Triangles Involved:

   • ΔABC with sides AB, BC, and AC.
   • ΔPBC with sides PB (which is equal to AB), BC, and PC (which is equal to AC).

3. Corresponding Sides:

   • AB corresponds to BP
   • AC corresponds to PC
   • BC corresponds to BC

4. Applying the SSS Congruence Theorem:

   The Side-Side-Side (SSS) Congruence Theorem states that if all three corresponding sides of two triangles are equal in length, then the triangles are congruent.

   • AB = BP (First pair of corresponding sides)
   • AC = PC (Second pair of corresponding sides)
   • BC = BC (Third pair of corresponding sides, common side)

   Since all three pairs of corresponding sides are equal, ΔABC ≅ ΔPBC by the SSS Congruence Theorem.

Given:

• AB = PQ
• BC = QR
• The median AD is equal to the median PM.

To Prove: △ABD is congruent to △PQM by a specific congruence criterion.

Solution:

• In triangles △ABD and △PQM:
  • AB = PQ (given)
  • AD = PM (medians are equal)
  • BD = QM, as D and M are midpoints of BC and QR respectively, making BD = QM (half of equal sides BC and QR)
• Since we have two sides and the included median equal, we can apply the SAS (Side-Angle-Side) criterion for congruence.

Conclusion: △ABD is congruent to △PQM by the SAS (Side-Angle-Side) criterion.

Answer: d) SAS

Explanation: For two equilateral triangles to be congruent, their corresponding sides must be equal in length. In congruent triangles, all corresponding sides and angles are identical. While equilateral triangles always have equal angles (60°), congruence is specifically determined by the equality of sides.

When two triangles are congruent, the corresponding sides and angles are equal based on the order of the vertices in the congruence statement. Given that ΔPQR ≅ ΔABC, the correspondence between the vertices is as follows:

• P corresponds to A
• Q corresponds to B
• R corresponds to C

Using this correspondence:

1. Side PQ corresponds to Side AB
2. Side QR corresponds to Side BC
3. Side PR corresponds to Side AC

Therefore:

• PQ = AB
• QR = BC
• PR = AC

This matches Option B precisely.

Answer: A

Explanation:
Option A is correct because the Hypotenuse-Side (HS) congruence criterion states that if the hypotenuse and one side of one right triangle are equal to the hypotenuse and one side of another right triangle, the two triangles are congruent.

Other options are incorrect:

• B: This statement is partially correct but not universally true for all cases, as the altitude bisecting the opposite side guarantees an isosceles triangle only under specific conditions.
• C: The congruence of two right triangles cannot be guaranteed if just any two sides are equal; the Hypotenuse-Leg or another criterion must be specified.
• D: If the angles are equal, the opposite sides must also be equal, making this statement incorrect.

as QT = TR and QP = PR
then  QT/QP = TR/PR
so PT would be internal bisector of ∠QPR
and as QT = TR, 
hence , T is the bisector of QTR as well.
Therefore, both A and 

In a parallelogram, the diagonals are equal if and only if the parallelogram is a rectangle. This is because equal diagonals imply that the opposite angles are equal, and the only type of parallelogram where this happens is a rectangle.

In a rectangle, all angles are 90^∘. Therefore, the measure of ∠PQR is: 90^∘.

• Option A: Describes the Angle-Angle-Side (AAS) condition, which does guarantee congruence.
• Option B: Describes the Side-Angle-Side (SAS) condition, which does guarantee congruence.
• Option C: Having two sides and a non-included angle equal only guarantees similarity, not necessarily congruence.
• Option D: Describes the Side-Side-Side (SSS) condition, which does guarantee congruence.
• Hence, option C is the answer