GANITA PRAKASH · GRADE 7 · PART II · CHAPTER 1
AMBIGUOUS — A non-included angle can leave two answers
Think first. Use AI purposefully. Verify always.
DISCOVER
Start with a sketch and a prediction.
A 30° ray from P meets a circle centred at Q. Could the circle cut the ray twice?
Try on paper first. Use the model to test your idea before accepting the explanation.
Understand
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.
Key Ideas
SSA does not guarantee congruence
Do not say SSA never works. Some data have one solution; the right-triangle special case gives RHS.
Visualise
One ray. Two possible third vertices.
PQ = 6 cm; QR = QS = 4 cm; ∠P = 30°.
Measured model lengths: PR ≈ 2.550 cm; PS ≈ 7.842 cm. They are different, so the triangles are not congruent.
The dashed circle is clipped to the viewing window. Both intersections are shown in the combined view.
Method
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.
Worked Example
Given
PQ = 6 cm; QR = QS = 4 cm; R and S lie on the same 30° ray from P.
To Find
Whether △PQR and △PQS must be congruent
Property
The two circle–ray intersections satisfy the same SSA data.
Solution
- Both share PQ and the angle at P.
- Both Q-to-intersection lengths are 4 cm.
- The intersections are different distances along the ray, so PR ≠ PS.
Result
The two triangles are non-congruent. SSA is insufficient in general.
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
Why are two intersections in this construction different from the two SSS intersections?
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 11