NCERT Question | Class 9 | Exercise 8.1 | Question No. 7
Автор: Unique Tutorials (Maths by Anil Sir)
Загружено: 2025-11-23
Просмотров: 25
ABCD is a trapezium in which AB ∥ CD and AD = BC. Show that
(i) ∠A = ∠B
(ii) ∠C = ∠D
(iii) ΔABC ≅ ΔBAD
(iv) diagonal AC = diagonal BD
Solution (outline)
Setup & diagram — Draw trapezium ABCD with AB ∥ CD and mark AD = BC. Draw diagonals AC and BD.
Use parallel-line angle relations — Since AB ∥ CD, corresponding/alternate interior angles made by a diagonal (or a side) with AB and CD are equal. This gives angle equalities needed to compare triangles.
Congruence by SAS — Compare triangles ABC and BAD:
• AD = BC (given)
• AB = AB (common)
• The included angles formed by AB and the respective sides are equal because AB ∥ CD (so corresponding/alternate interior angles match).
Hence ΔABC ≅ ΔBAD (SAS). (From this congruence we get (i) and (ii).)
Equal diagonals — From congruent triangles we get AC = BD.
Conclude: ∠A = ∠B, ∠C = ∠D, ΔABC ≅ ΔBAD and AC = BD.
Why this matters
This is the standard set of properties for an isosceles trapezium (a trapezium whose non-parallel sides are equal). Understanding this proof helps for many geometry problems and board-exam questions.
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