Section A — 1 Mark / MCQs Easy
Q1. In $\triangle ABC \cong \triangle PQR$, if $\angle A =
50°$ and $\angle B = 60°$, find $\angle R$.
Q2. Which is NOT a criterion for congruence: SAS, ASA,
SSA, or SSS?
Q3. In an isosceles $\triangle ABC$ with $AB = AC$, if
$\angle A = 80°$, find $\angle B$.
Q4. If $\triangle DEF \cong \triangle BCA$, what part
corresponds to side $EF$?
Q5. What does CPCT stand for?
Q6. True/False: If two squares have equal areas, they are
congruent.
Q7. To apply RHS, which angle is mandatory?
Q8. In $\triangle ABC$, $AB=AC$. Name the base angles.
Section B — Core Proofs (Short) Medium
Q9. In quadrilateral ACBD, AC = AD and AB bisects $\angle
A$. Prove $\triangle ABC \cong \triangle ABD$.
Q10. ABCD is a parallelogram. Draw diagonal AC. Prove
$\triangle ABC \cong \triangle CDA$.
Q11. L and M are two parallel lines intersected by another
pair of parallel lines P and Q. Prove $\triangle ABC \cong \triangle CDA$.
Q12. Prove that altitude of an isosceles triangle from
vertex to base bisects the base using RHS.
Q13. Show that the angles of an equilateral triangle are
60° each.
Q14. Line-segment AB is parallel to another line-segment
CD. O is the mid-point of AD. Show $\triangle AOB \cong \triangle DOC$.
Q15. In an isosceles triangle with AB=AC, D and E are
points on BC such that BE=CD. Show AD=AE.
Q16. Prove that every point on perpendicular bisector of a
line segment is equidistant from its endpoints.
Section C — Medium / Long Proofs Medium
- Q17. BE and CF are two equal altitudes of a $\triangle
ABC$. Using RHS, prove that $\triangle ABC$ is isosceles.
- Q18. ABC is an isosceles triangle with AB=AC. Draw AP
$\perp$ BC. Show that $\angle B = \angle C$ (Using RHS congruence).
- Q19. ABC and DBC are two isosceles triangles on the same
base BC. Show that $\angle ABD = \angle ACD$.
- Q20. $\triangle ABC$ is right angled at A, and AB=AC. Find
$\angle B$ and $\angle C$.
- Q21. In a right triangle ABC, right angled at C, M is mid
point of hypotenuse AB. C is joined to M and produced to D such that DM=CM. Point D joined to B.
Prove $\triangle AMC \cong \triangle BMD$.
- Q22. In the above figure, show that $\angle DBC$ is a right
angle.
- Q23. Further show $\triangle DBC \cong \triangle ACB$.
- Q24. Finally show $CM = \frac{1}{2}AB$. (This is Q7 from Ex
7.1 NCERT, very important).
Section D — Board Level / Complex Problems Hard
- Q25. In right triangle ABC, AD is median to hypotenuse BC.
Prove that $AD = \frac{1}{2}BC$. (Hint: produce AD to E such that AD=DE, join EC).
- Q26. If the bisector of the vertical angle of a triangle
bisects the base, prove that the triangle is isosceles.
- Q27. If the altitude from one vertex of a triangle bisects
the opposite side, then the triangle is isosceles.
- Q28. Prove that the medians of an equilateral triangle are
equal.
- Q29. Two sides AB and BC and median AM of one $\triangle
ABC$ are respectively equal to sides PQ and QR and median PN of $\triangle PQR$. Show $\triangle
ABM \cong \triangle PQN$.
- Q30. AD and BC are equal perpendiculars to a line segment
AB. Show that CD bisects AB.
- Q31. ABCD is a quadrilateral in which AD = BC and $\angle
DAB = \angle CBA$. Prove $\triangle ABD \cong \triangle BAC$.
- Q32. P is a point equidistant from two lines $l$ and $m$
intersecting at A. Show that line AP bisects the angle between them. (RHS rule).
✅ Quick Tips for Proofs: Always write: (1) Given, (2) To Prove, (3) Construction
(if any), (4) Proof. Use exactly 3 statements for congruence + the rule name (e.g., SAS). Q1:
$\angle C=70°$, so $\angle R=70°$ | Q2: SSA | Q3: 50° | Q4: CA | Q20: $\angle B=45°, \angle C=45°$.
