GANITA PRAKASH · GRADE 7 · PART II · CHAPTER 1
Study Materials
Think first. Use AI purposefully. Verify always.
- SSS: three equal corresponding sides.
- SAS: two equal corresponding sides and their included angle.
- ASA: two equal corresponding angles and their included side.
- AAS: two equal corresponding angles and a corresponding non-included side.
- RHS: both triangles right, equal hypotenuse and one equal corresponding leg.
AAA does not fix scale. SSA is not sufficient in general. Every criterion requires a consistent correspondence and valid, non-degenerate triangles.
MATCH — Same shape. Same size.
Congruent figures fit exactly after a translation, rotation or reflection. Every part must match. Equal area alone does not establish congruence; a change of scale generally destroys it.
Same shape + same size → congruence
Choose a pair of corresponding points. Move, turn or flip one figure without stretching it. Check the entire boundary, not just one side.
Orientation does not affect congruence. Same shape with different size is not enough.
Textbook connection: Ganita Prakash · Grade 7 · Part II · Chapter 1 — Geometric Twins · printed page 2
Worked example →FIX — Which measurements fix a copy?
Measurements must fix both shape and size. A circle needs its radius. A rectangle needs its two adjacent side lengths. A two-arm symbol also needs the angle between its arms.
Two arm lengths + included angle
Ask what can still move when a measurement is fixed. Add enough information to remove that freedom. Do not confuse a flexible two-arm symbol with a rigid triangle.
Two given sides alone leave the hinge free to move.
Textbook connection: Ganita Prakash · Grade 7 · Part II · Chapter 1 — Geometric Twins · printed page 3
Worked example →TRIPLE — Three sides determine a triangle
SSS fixes a non-degenerate triangle up to a rigid movement or reflection. Three positive lengths must satisfy the triangle inequality; invalid lengths do not create a triangle.
SSS — three corresponding sides
Draw AB = 6 cm. Use circles centred at A and B with radii 4 cm and 8 cm. Join either intersection to A and B. Compare both constructions by reflection in AB.
Two circle intersections do not by themselves mean two non-congruent triangles.
Textbook connection: Ganita Prakash · Grade 7 · Part II · Chapter 1 — Geometric Twins · printed page 5
Worked example →ORDER — Correspondence carries meaning
The order in a congruence statement records a vertex-by-vertex match. For △ABC ≅ △XYZ, A ↔ X, B ↔ Y, C ↔ Z. Corresponding sides and angles follow that same mapping.
List the equal side pairs. Match the shared endpoints. Write the second triangle in the same order as the first, then check every side and angle.
Naming the same two triangles with an arbitrary order can state a false correspondence.
Textbook connection: Ganita Prakash · Grade 7 · Part II · Chapter 1 — Geometric Twins · printed page 7
Worked example →HINGE — Fix the included angle
SAS requires two corresponding sides and the angle included between them. Fixing that hinge fixes the triangle. AAA fixes shape but can leave size different.
SAS — the angle must be included
Identify the vertex shared by both known sides. Check that its angle is given. Draw the sides along the two fixed rays and join the endpoints.
The included angle between AB and AC is ∠A, not ∠B. AAA is not a congruence criterion.
Textbook connection: Ganita Prakash · Grade 7 · Part II · Chapter 1 — Geometric Twins · printed page 10
Worked example →AMBIGUOUS — A non-included angle can leave two answers
SSA is not a general congruence criterion. The source construction PQ = 6 cm, QR = QS = 4 cm and ∠QPR = ∠QPS = 30° gives two distinct triangles with different third-side lengths.
SSA does not guarantee congruence
Draw PQ = 6 cm and a ray at 30° from P. Draw the radius-4 circle from Q. Mark both intersections R and S. Compare PR and PS before deciding congruence.
Do not say SSA never works. Some data have one solution; the right-triangle special case gives RHS.
Textbook connection: Ganita Prakash · Grade 7 · Part II · Chapter 1 — Geometric Twins · printed page 11
Worked example →ANGLE — Two angles and a side
ASA fixes two angles and their included side. AAS also guarantees congruence: the third angle is determined by the 180° sum, converting the data into an ASA determination.
Identify the side and both endpoint angles. If one endpoint angle is missing, compute it from the angle sum. Construct the two rays at the side endpoints and use their intersection.
AAS is valid even though the side is not between the two originally stated angles.
Textbook connection: Ganita Prakash · Grade 7 · Part II · Chapter 1 — Geometric Twins · printed page 15
Worked example →RIGHT — Right angle, hypotenuse, side
RHS applies to two right triangles with equal hypotenuses and one equal corresponding leg. The hypotenuse lies opposite 90°. Do not apply RHS without both right angles.
RHS — right angle + hypotenuse + leg
Draw the given leg. Construct a perpendicular at its endpoint. Intersect it with a circle from the other endpoint whose radius is the hypotenuse. The reflected intersection is a congruent copy.
The longest-looking side is not proof of a hypotenuse. Locate the established right angle.
Textbook connection: Ganita Prakash · Grade 7 · Part II · Chapter 1 — Geometric Twins · printed page 16
Worked example →DEDUCE — Turn congruence into a reason
Construct altitude AD to BC. The triangles ADB and ADC have equal hypotenuses AB and AC, common leg AD and right angles at D. RHS gives equal corresponding base angles. In an equilateral triangle all three angles are therefore 60°.
State a construction before using its properties. Establish congruence with a correctly ordered statement, then use corresponding parts. Finish with an angle sum where needed.
Do not mark a midpoint or angle bisector as given when it still needs to be proved.
Textbook connection: Ganita Prakash · Grade 7 · Part II · Chapter 1 — Geometric Twins · printed page 18
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