NCERT Exemplar Solutions
Class 6 - Mathematics - Unit 2: Geometry - Problems and Solutions

Question 47

Key idea: To name an angle with three letters, the middle letter is the vertex. Here, the vertex is B. The first and last letters sit on the two arms forming the angle.

• (a) \(\angle 1\):

  Arms used: \(\overrightarrow{BC}\) and \(\overrightarrow{BD}\).

  Start on \(C\), vertex at \(B\), end at \(D\): \(\angle \mathbf{CBD}\).

• (b) \(\angle 2\):

  Arms used: \(\overrightarrow{BD}\) and \(\overrightarrow{BE}\).

  Start on \(D\), vertex at \(B\), end at \(E\): \(\angle \mathbf{DBE}\).

• (c) \(\angle 3\):

  Arms used: \(\overrightarrow{BE}\) and \(\overrightarrow{BA}\).

  Start on \(E\), vertex at \(B\), end at \(A\): \(\angle \mathbf{EBA}\).

• (d) \(\angle 1 + \angle 2\):

  Combine (a) and (b): from \(\overrightarrow{BC}\) to \(\overrightarrow{BE}\) passing through \(\overrightarrow{BD}\).

  Start on \(C\), vertex at \(B\), end at \(E\): \(\angle \mathbf{CBE}\).

• (e) \(\angle 2 + \angle 3\):

  Combine (b) and (c): from \(\overrightarrow{BD}\) to \(\overrightarrow{BA}\) passing through \(\overrightarrow{BE}\).

  Start on \(D\), vertex at \(B\), end at \(A\): \(\angle \mathbf{DBA}\).

• (f) \(\angle 1 + \angle 2 + \angle 3\):

  Combine all three: from \(\overrightarrow{BC}\) to \(\overrightarrow{BA}\).

  Start on \(C\), vertex at \(B\), end at \(A\): \(\angle \mathbf{CBA}\).

• (g) \(\angle CBA - \angle 1\):

  Whole \(\angle CBA\) is from \(\overrightarrow{BC}\) to \(\overrightarrow{BA}\).

  Subtract \(\angle 1\) (from \(\overrightarrow{BC}\) to \(\overrightarrow{BD}\)).

  Remaining part: from \(\overrightarrow{BD}\) to \(\overrightarrow{BA}\) = \(\angle \mathbf{DBA}\).

Notes: The vertex letter B must stay in the middle. Sums of adjacent angles “extend” across the common arm. Subtraction removes the smaller adjacent part from the larger angle.