Let's see how to construct a quadrilateral when two adjacent sides and three angles are given.
Construct a quadrilateral \(CAKE\) where \(CA = 7 \ cm\), \(AK = 6.5 \ cm\), \(\angle C = 90^\circ\), \(\angle K = 110^\circ\), \(\angle E = 100^\circ\).
Let us first draw the rough quadrilateral, which will help us to draw a fair quadrilateral.
Step 1: Draw a line segment \(CA = 7 \ cm\).
Step 2: Construct \(\angle ACX = 90^\circ\) using ruler and compass.
Step 3: We know that sum of all the angles of a quadrilateral is \(360^\circ\).
\(\angle C + \angle A + \angle K + \angle E = 360^\circ\)
\(90^\circ + \angle A + 110^\circ + 100^\circ = 360^\circ\)
So, \(\angle A = 60^\circ\)
Construct \(\angle CAY = 60^\circ\) using ruler and compass.
Step 4: With \(A\) as centre and \(6.5 \ cm\) as radius, draw an arc, cutting \(AY\) at \(K\).
Step 5: With \(K\) as centre, make an angle of \(110^\circ\), cutting \(CX\) at \(E\), then join \(EK\).
Thus, the quadrilateral \(CAKE\) is obtained.