Given quadrilateral ABCD where AB = BC = CD, ∠ ABC = 70°, ∠ BCD = 170°,
find ∠ BAD without using a calculator.
The following solution selected as the Best Answer by the asker was worked out by my son, Kuntal, a networking professional with Cisco Systems despite being out of touch with the subject. I posted it in my account as he is not registered with YA!
Originally, Mr. Duke was the first to post an elegant solution to this problem of geometry. His ability to think of unique solutions to the problems of math is amazing! He drew my attention to this problem and inspired me to attempt. I too posted my solution which was followed by a stream of good solutions from Kuntal, rozeta53, Rakesh Dubey and Half-Blood Prince, who are my other valuable contacts of YA!
The three answers of Half-Blood Prince can be read in comments.
ABCD is the given quadrilateral. Draw equilateral Δ BEC. E may lie inside or outside the quadrilateral ABCD or on side AD. We assume that it is inside ABCD and prove that this is not possible. With the given quadrilateral and the construction, different angles are as shown in the figure.
=> Sum of all angles of the quadrilateral
= α° + 85° + 70° + 170° + 35° + β°
= 360° + α° + β°
=> α° + β° = 0.
This can be proved even if E were outside ABCD.
=> E is on AD so that α° + β° = 0.
=> ∠ BAD = 85°.
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