ABCD is a quadrilateral, such that AB=DC, and angle ADB and angle DBC are supplementary angles. Show that angle A = angle C, if angle A = ` alpha `, angle D = `theta` , angle C =`gamma` and angle DBC = 180 - `theta` .

These angles can be proven using the Sine Law.

Expert Answers

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We will use the Sine Law: for a triangle `M N P , ` the following expressions are equal: `( M N ) / sin ( M P N ) , ` `( N P ) / sin ( P M N ) , ` `( P M ) / sin ( M N P ) . `

Let's apply the Sine Law to the triangles `A D B ` and `C D B . ` They have a common side `D B ` and equal sides `A B ` and `D C . ` In particular, we obtain from the triangle `A D B :`

`( A B ) / sin ( A D B ) = ( D B ) / sin ( D A B ) ,`

and from the triangle `C D B :`

`( C D ) / sin ( D B C ) = ( D B ) / sin ( D C B ) .`

Because the angles `A D B ` and `D B C ` are supplementary, their sinuses are equal. This way, `sin ( D A B ) = sin ( D C B ) , ` which means they are either equal or supplementary.

It is possible to prove that they are always equal. If the angle `ADB ` is obtuse or right, then `A ` is acute while `AB gt DB . ` Therefore `CD gt DB , ` so `C` cannot be obtuse and `A = C . ` The similar argument may be used if `DBC ` is obtuse or right.

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