GANITA PRAKASH · GRADE 7 · PART II · CHAPTER 1
DEDUCE — Turn congruence into a reason
Think first. Use AI purposefully. Verify always.
DISCOVER
Start with a sketch and a prediction.
An isosceles triangle has equal sides AB and AC. How could splitting it into two right triangles prove that its base angles agree?
Try on paper first. Use the model to test your idea before accepting the explanation.
Understand
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°.
Key Ideas
Do not mark a midpoint or angle bisector as given when it still needs to be proved.
Visualise
Prove the equal base angles
Given only: AB = AC and ∠A = 80°. Do not assume a midpoint.
Equal sides lead to equal angles
AB = BC = CA. What can the isosceles result tell us?
Method
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.
Worked Example
Given
AB = AC and ∠BAC = 80°.
To Find
The measures of ∠ABC and ∠ACB
Property
Construction: draw AD perpendicular to BC. Use RHS, then the triangle angle sum.
Solution
- AB = AC, AD = AD and ∠ADB = ∠ADC = 90°.
- △ADB ≅ △ADC by RHS, so ∠ABC = ∠ACB.
- Their sum is 180° − 80° = 100°; each is 50°.
Result
∠ABC = ∠ACB = 50°.
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
If A is the centre of a circle, B and C are on it, and ∠BAC = 120°, find the two base angles.
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 18