4. When two adjacent sides and three angles are known

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.

 

Rough diagram:

 

 

Fair diagram:

 

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.