GANITA PRAKASH · GRADE 7 · PART II · CHAPTER 1
HINGE — Fix the included angle
Think first. Use AI purposefully. Verify always.
DISCOVER
Start with a sketch and a prediction.
Hold two side lengths fixed and open the angle between them. Does the third side stay the same?
Try on paper first. Use the model to test your idea before accepting the explanation.
Understand
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.
Key Ideas
SAS — the angle must be included
The included angle between AB and AC is ∠A, not ∠B. AAA is not a congruence criterion.
Visualise
A fixed hinge versus a change of scale
AB = 6 cm, AC = 5 cm. With ∠A fixed, the triangle is fixed. Changing ∠A changes the third side.
Angles stay equal; corresponding lengths have ratio 1. These copies are congruent.
Method
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.
Worked Example
Given
AB = XY = 6 cm, AC = XZ = 5 cm, ∠A = ∠X = 30°.
To Prove
△ABC ≅ △XYZ
Property
SAS: ∠A and ∠X lie between the given side pairs.
Proof
- AB ↔ XY, AC ↔ XZ fix A ↔ X.
- The included angles agree. Match B ↔ Y and C ↔ Z.
Result
△ABC ≅ △XYZ by SAS.
AIMS Method
- A — Attempt independently
Sketch and predict before using a criterion. - I — Investigate the idea
Separate given facts from visual assumptions. - M — Make connections
Match vertices, sides and angles consistently. - S — Scrutinise your reasoning
Check each hypothesis and justify the conclusion.
Think first. Use AI purposefully. Verify always.
Focused Practice
Triangles have angles 30°, 70° and 80°. Is congruence guaranteed?
Check & Feedback
Name the exact given facts, state a valid criterion or explain why none applies, and check the vertex order.
Textbook connection: Ganita Prakash · Grade 7 · Part II · Chapter 1 — Geometric Twins · printed page 10