Geometry proof and congruence
Class 9 Maths · Triangles · 18 questions with answers
This is the sheet that matters most in Class 9, and it teaches a format rather than a list of theorems. A memorised proof scores on the question it was memorised for. A child who owns the format - figure, given, to prove, then steps each with a reason - can attempt a theorem they have never seen and earn most of the marks.
- Chooses the correct congruence criterion and says why it applies
- Knows why AAA is not a congruence criterion
- Writes a reason beside every statement in a proof
- Uses CPCT correctly and only after congruence has been established
- Sets out a full proof with figure, given and to prove
Exercise 1 of 3 · 11 questions
Which congruence criterion applies? If none does, say so and explain.
Write the answer · Warm-up
Here’s one done for you
AAA is not a congruence criterion, and the reason is worth saying rather than remembering: two equilateral triangles, one with sides of 2 cm and one with sides of 200 cm, have all three angles equal and are obviously not congruent. Angles fix shape; they say nothing about size. This is also the commonest wrong answer in the whole chapter.
- 1)Three pairs of sides are equal.
- 2)Two pairs of sides and the pair of angles between them are equal.
- 3)Two pairs of angles and the pair of sides between them are equal.
- 4)Two pairs of angles and a pair of sides not between them are equal.
- 5)Both triangles are right-angled, with equal hypotenuses and one equal side.
- 6)Three pairs of angles are equal.
- 7)Two pairs of sides and a non-included pair of angles are equal.
- 8)One pair of sides and two pairs of angles, the side lying between the two angles.
- 9)In two right triangles, one pair of legs and the pair of hypotenuses are equal.
- 10)Two pairs of sides are equal and the included angles are both 90 degrees.
- 11)All three pairs of sides are equal but the triangles face opposite ways.
Answers
- 1) SSS.
- 2) SAS. The angle must be the included one - the angle between the two sides.
- 3) ASA. The side must be the one between the two angles.
- 4) AAS. This works because knowing two angles fixes the third, which reduces it to ASA.
- 5) RHS, which applies only to right-angled triangles.
- 6) None. Equal angles make the triangles similar, not congruent - they can be the same shape at completely different sizes.
- 7) No criterion. SSA does not guarantee congruence - two different triangles can fit the same data.
- 8) ASA. The side is included between the two angles, which is what ASA requires.
- 9) RHS. Right angle, hypotenuse and one side.
- 10) SAS. The angle is included between the two known sides, and 90 degrees is still an included angle.
- 11) SSS. Congruence does not depend on orientation - one triangle can be turned or flipped onto the other.
Exercise 2 of 3 · 5 questions
The statements are given. Write the reason for each one.
Write the answer · Guided
Here’s one done for you
This is the proof that angles opposite equal sides of a triangle are equal, and it is five lines long. Notice that every statement has exactly one reason - given, construction, common, a criterion, or CPCT. Those five reasons cover almost every Class 9 proof, and a child who knows they are the only options stops staring at a blank page.
- 1)In triangle ABC, AB = AC, and AD bisects angle A meeting BC at D. Statement 1: AB = AC.
- 2)Statement 2: angle BAD = angle CAD.
- 3)Statement 3: AD = AD.
- 4)Statement 4: triangle ABD is congruent to triangle ACD.
- 5)Statement 5: angle B = angle C.
Answers
- 1) Given.
- 2) By construction, since AD bisects angle A.
- 3) Common side to both triangles.
- 4) By SAS - two sides and the included angle.
- 5) By CPCT - corresponding parts of congruent triangles.
Exercise 3 of 3 · 2 questions
Write the full proof. Figure, given, to prove, then steps with reasons.
Write the answer · Practice
Here’s one done for you
The second proof is worth studying for its last two lines. Congruence gives you that the two angles are equal; it does not by itself give you a right angle. The linear pair does - equal angles adding to 180° must each be 90°. Class 9 proofs frequently end with one extra step like that after CPCT, and children who stop at CPCT lose exactly the mark the question was set for.
- 1)Prove that the diagonals of a parallelogram bisect each other.
- 2)In triangle ABC, AB = AC and AD is the median to BC. Prove that AD is perpendicular to BC.
Answers
- 1) Given: parallelogram ABCD with diagonals AC and BD meeting at O. To prove: AO = OC and BO = OD. Proof: AB = CD (opposite sides of a parallelogram are equal). Angle OAB = angle OCD (alternate angles, since AB is parallel to DC with transversal AC). Angle OBA = angle ODC (alternate angles, with transversal BD). Therefore triangle AOB is congruent to triangle COD (ASA). Therefore AO = OC and BO = OD (CPCT).
- 2) Given: triangle ABC with AB = AC, and D the mid-point of BC. To prove: AD is perpendicular to BC. Proof: AB = AC (given). BD = DC (D is the mid-point). AD = AD (common). Therefore triangle ABD is congruent to triangle ACD (SSS). Therefore angle ADB = angle ADC (CPCT). But these two angles form a linear pair, so they add to 180°. Each is therefore 90°, and AD is perpendicular to BC.
While your child works
- Check the figure first. A proof attempted without a labelled diagram almost always fails, and the diagram itself carries marks.
- Every statement needs one of five reasons: given, construction, common, a congruence criterion, or CPCT. Knowing the list is finite stops the blank page.
- A partial proof with reasons earns partial marks. A conclusion with no reasons earns almost nothing. Say that once and mean it.