Congruence Criteria- SSS And RHS - Free MCQ Practice Test with solutions,

∠LQ = ∠MNR,

∠Q = ∠R

QM = MR,

Hence, ΔQLM ≅ ΔMNR (by AAS)

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

• In a parallelogram, if the diagonals are equal, then the parallelogram is a rectangle because only rectangles have equal diagonals.
• Since ABCD is a rectangle, its diagonals bisect each other and are equal in length, meaning each half of the diagonal is also equal.

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

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.

We are asked to prove that triangle ABD is congruent to triangle PQM.

Step 1: Compare given information

• AB = PQ (given)
• BC = QR (given)
• AD = PM (given, both are medians)

Step 2: Use property of medians

• Since AD is the median of triangle ABC, point D is the midpoint of BC.
  So BD = DC = half of BC.

• Since PM is the median of triangle PQR, point M is the midpoint of QR.
  So QM = MR = half of QR.

Step 3: Use the equality of sides

• From given, BC = QR.
• Therefore, half of BC = half of QR.
• This means BD = QM.

Step 4: Now check corresponding sides of triangles ABD and PQM

• In triangle ABD and triangle PQM:

1. AB = PQ (given)
2. BD = QM (proved using medians and BC = QR)
3. AD = PM (given)

So, all three sides of triangle ABD are equal to the corresponding three sides of triangle PQM.

Step 5: Congruence criterion

• When three sides of one triangle are equal to the three sides of another triangle, the two triangles are congruent by the SSS (Side-Side-Side) criterion.

Final Answer: The correct option is b) SSS.

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.

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 B

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