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.

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.

as QT = TR and QP = PR
then  QT/QP = TR/PR
so PT would be internal bisector of ∠QPR