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Let us learn how to construct a triangle with an example when its base, vertical angle and the altitude from the vertex to the base are given.
Example:
Construct a $$\triangle ABC$$ such that $$AB = 5 \ cm$$, $$\angle C = 45^{\circ}$$, and the altitude from $$C$$ to $$AB$$ is of length $$2 \ cm$$.

Solution:

First, let us draw a rough figure.

Construction:

Step 1: Draw a line segment $$AB$$ of length $$5 \ cm$$.

Step 2: At $$A$$, draw $$AE$$ such that $$\angle BAE = 45^{\circ}$$.

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

Step 4: Draw the perpendicular bisector $$XY$$ to $$AB$$, which intersects $$AD$$ at $$G$$ and $$AB$$ at $$F$$.

Step 5: With $$G$$ as centre and $$GA$$ as radius, draw a circle $$ABH$$.

Step 6: From $$F$$, mark an arc in the line $$XY$$ at $$I$$, such that $$FI = 2 \ cm$$.

Step 7: Draw $$NM$$ through $$I$$, which is parallel to $$AB$$.

Step 8: $$NM$$ meets the circle at $$A$$ and $$J$$.

Step 9: Join $$AB$$ and $$AC$$. Thus, $$ABC$$ is the required triangle.