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Let us learn how to construct a triangle if the following are given.

**1**. The base, vertical angle and median on the base.

**2**. The base, vertical angle and altitude on the base.

**3**. The base, vertical angle and the point where the bisector of the vertical angle meets the base.

Let us consider the following construction.

Construction of a segment of a circle on a given line segment containing an angle \(\theta\)

**Construction**:

**Step 1**: Draw a line segment \(\overline{AB}\).

**Step 2**: At \(A\), make \(\angle BAC = \theta\). Draw \(AC\).

**Step 3**: Draw \(AD \perp AC\).

**Step 4**: Draw the perpendicular bisector of \(AB\) meeting \(AD\) at \(M\).

**Step 5**: With \(M\) as centre and \(MA\) as radius, draw a circle \(ABH\).

**Step 6**: Take any point \(N\) on the circle; by the alternate segments theorem, the major arc \(ANB\) is the required segment of the circle containing the angle \(\theta\).