SOLUTION: ABCD is a cyclic quadrilateral, angle DAB =80 and angle ACB =50 . Prove that AB=AD.

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Question 1043931: ABCD is a cyclic quadrilateral, angle DAB =80 and angle ACB =50 .
Prove that AB=AD.
Found 2 solutions by rothauserc, ikleyn:
Answer by rothauserc(4718) About Me  (Show Source):
You can put this solution on YOUR website!
a cyclic quadrilateral is a quadrilateral with all four of its vertices on a circle
:
labeling vertices can be done clockwise or counterclockwise
:
angle ACB does not make sense using the above conventions
:

Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
ABCD is a cyclic quadrilateral, angle DAB =80 and angle ACB =50 .
Prove that AB=AD.
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Let me remind you that a cyclic quadrilateral is a quadrilateral inscribed into the circle.

So, ABSD is inscribed into a circle. 
Then the measure of the angle DAB plus the measure of the opposite angle BCD is 180°.


    (If a quadrilateral is inscribed into a circle, then the sum of the measures of its opposite angles is 180°.  

    See the lesson Quadrilateral inscribed in a circle in this site).


So,  mLDAB + mLBCD = 180°.

But  mLBCD = mLACB + mLACD.

Therefore,  mLDAB + mLACB + mLACD = 180°.

But  mLDAB + mLACB = 80° + 50° = 130°.

Therefore,  mLACD = 180° - 130° = 50°.

Thus the inscribed angles  LACB  and  LACD  have equal measures.

Hence, they lean equal arcs: the length of the arc AB is the same as the length of the AD.

It implies that the chords AB and AD have equal lengths.   QED.

Solved.

See the lessons associated with it:

    - Quadrilateral inscribed in a circle

and

    - A circle, its chords, tangent and secant lines - the major definitions
    - The longer is the chord the larger its central angle is
    - An inscribed angle in a circle
in this site.