Ravi draws a regular hexagon (all sides equal, all angles equal). He draws diagonals from one vertex to every non-adjacent vertex. The diagonal from vertex A to vertex D passes through the centre of the hexagon. What is the angle between two adjacent diagonals drawn from vertex A, given that all three diagonals from A divide the straight angle at A equally?
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Worked Solution
Step 1: In a regular hexagon, each interior angle = (6−2) × 180° ÷ 6 = 720° ÷ 6 = 120°.
Step 2: From vertex A, there are 3 non-adjacent vertices (C, D, E in a hexagon ABCDEF), giving 3 diagonals AC, AD, AE.
Step 3: These 3 diagonals plus the two sides AB and AF divide the 120° interior angle. But the question says "the straight angle at A" is divided by the diagonals.
Step 4: Reconsidering: the two sides AB and AF form 120°. The 3 diagonals from A divide this 120° angle equally into parts: 120° ÷ 2 = 60° between each adjacent diagonal pair (there are 2 gaps between 3 diagonals inside the 120° angle... no, with 3 diagonals inside 120°, they create 4 sections, but adjacent sides AB and AF bound the angle).
Actually: sides AB and AF + 3 diagonals = 5 rays from A, creating 4 equal sections of 120°. Each section = 120° ÷ 4 = 30°. But the question specifies "two adjacent diagonals from A divide the straight angle at A equally" — the 3 diagonals divide the 180° straight angle into 4 equal parts: 180° ÷ 3 = 60° between adjacent diagonals.
Correct: 180° ÷ 3 = 60°.
Correct answer: 60°
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