### Theory:

Learn how to construct a triangle with an example if its base, vertical angle and the point on the base where the bisector of the vertical angle meets the base are given.
Example:
Draw a triangle $$ABC$$ of base $$AB = 7 \ cm$$, $$\angle C = 30^{\circ}$$ and the bisector of $$\angle C$$ meets $$AB$$ at $$C$$ such that $$AD = 5 \ cm$$.

Solution:

First, let us draw a rough figure. Construction: Step 1: Draw a line segment $$AB = 7 \ cm$$.

Step 2: At $$A$$, draw $$AE$$ such that $$\angle EAB = 30^{\circ}$$.

Step 3: At $$A$$, draw $$AF$$ such that $$\angle FAE = 90^{\circ}$$.

Step 4: Draw the perpendicular bisector to $$AB$$, which intersects $$AF$$ at $$O$$ and $$AB$$ at $$P$$.

Step 5: With $$O$$ as centre and $$OA$$ as radius, draw a circle.

Step 6: From $$A$$, mark an arc of $$5 \ cm$$ on $$AB$$ at $$D$$.

Step 7: The perpendicular bisector intersects the circle at $$R$$. Join $$RD$$.

Step 8: $$RD$$ produced meets the circle at $$C$$. Now, join $$AC$$ and $$AB$$.

Thus, $$\triangle ABC$$ is the required triangle.