APPLY — Propositions in Number Theory and Geometry

02 · UNDERSTAND

Propositions in Number Theory and Geometry

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.

03 · KEY IDEAS

Keep the conditions clear.

• 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.

04 · VISUALISE

Follow the direction.

06 · ORIGINAL AIMS WORKED EXAMPLE

Follow each step.

Two right triangles have perpendicular legs 3, 8 and 4, 6 respectively. Compare area and congruence.

1. Given: the two pairs of perpendicular legs. To decide: whether equal area forces congruence.
2. Both areas are 12 square units, from one half of base times height.
3. Their side-length sets differ: 3, 8, $\sqrt{73}$ and 4, 6, $\sqrt{52}$. They are not congruent.
4. Result: equal area does not imply congruence. Verify: congruent triangles still always have equal area.

07 · AIMS METHOD

Your turn to reason.

1. AAttempt independently
2. IInvestigate the idea
3. MMake connections
4. SScrutinise your reasoning

Does $x^2$ = $y^2$ imply x = y for real numbers?

08 · PRACTICE SPRINT

Use the idea in a new situation.

Ten original AIMS questions: 3 Warm-up, 4 Build, 2 Think and 1 Challenge.

Check & Feedback: incorrect answers receive a conceptual hint and another attempt; correct answers include reasoning.

TEXTBOOK CONNECTION

Return to the source. Bring your reasoning.

Ganita Manjari · Grade 9 · Part II · Chapter 9. Use your textbook for the complete questions. These are topic references and original AIMS guidance, not reproduced exercises.