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In the diagram, ABCDEF is a regular hexagon and ABGH is a squarea) find angle ABC and angle CBGb) find the angle of triangle ACG |
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Answer» Answer: Given that ABCDEF is a regular hexagon, so all sides of it are equal in length. Sum of the angles of a n− sided polygon =(n−2)×180 Hence, Sum of all angles of a hexagon =(6−2)×180 =720 And, Each angle of a regular hexagon = 6 720
=120.......(1) Quadrilateral ABEF contains half the angle measure of the hexagon. Hence, its interior angle measure =720/2=360 ∠AFE=120.......(ii) [ from (1)] In △AFE, AF=EF [ SINCE, they are a part of regular hexagon].......(iii) ⇒∠FAE=∠FEA [ Angles OPPOSITE to equal sides are equal] In △AFE, ∠FAE+∠FEA+∠AFE=180 [ sum of all angles of a triangle is 180 o ] SUBSTITUTING (ii) & (iii) in above equation, ∠FAE=∠FEA=30 o
∠FAE+∠EAB=120 o [ Angle of a regular hexagon] ∴∠EAB=90 o .......(iv) ∠ABE=1/2×120 o [∵∠ABE bisects ∠ABC] =60 o
In △ABE, ∠AEB+∠ABE+∠EBA=180 o
[ sum of all angles of a triangle] ∠AEB=180 o −60 o −90 o =30 o [ from (iv) & (v) ] ∠ABE=60 o ∠EAB=90 o ∠AEB=30 o |
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