GANITA MANJARI · GRADE 9 · PART II · CHAPTER 9
APPLY — Concept Note
Think first. Use AI purposefully. Verify always.
Concept
The same reasoning works with numbers, equations and figures. Record all conditions, keep the direction clear and use a result only when its conditions apply. A sketch helps you reason but is not a proof.
Key Ideas
- Adding the same number to equal numbers preserves equality; subtracting it justifies the converse.
- Equal squares of real numbers can come from opposite numbers.
- Congruent triangles have equal area, but equal area alone does not give congruence.
- For two lines cut by a transversal, corresponding-angle equality and parallelism imply each other.
Rule / Property
List the conditions. Choose a known property. Test the reverse using a numerical example or a labelled figure.
Visual Recall
Mini Example
Two right triangles have perpendicular legs 3, 8 and 4, 6 respectively. Compare area and congruence.
Result: equal area does not imply congruence. Verify: congruent triangles still always have equal area.
Common Mistake
Treating a diagram that looks correct as sufficient evidence, or forgetting the real-number domain when comparing squares.
Quick Recall
Does = imply x = y for real numbers?
Check your reasoning
No. For example, x = 6 and y = −6 give equal squares but unequal numbers.