Question

In figure, \[OA = OD\] and \[\angle 1 = \angle 2\]. Prove that \[\vartriangle OCB\] is an isosceles triangle.

Answer 1

Hint: Here we use the property of supplementary angles to find angles on a straight line along with \[\angle 1,\angle 2\]. And using the given sides equal we prove two triangles \[\vartriangle AOC,\vartriangle DOB\] as congruent triangles which will give us their sides equal which are also sides of triangle\[\vartriangle OCB\].
* An isosceles triangle has two sides equal to each other and angles opposite to opposite sides are also equal.
* Vertically opposite angles are the angles which lie on the exact opposite side of the intersection of two lines.
* Two triangles are said to be congruent if they are associative in any of the forms stated
SSS- Side, side, side
SAS- Side, angle, side
ASA- Angle, side, angle

Complete step-by-step answer:
In this figure we have \[\angle 1 = \angle 2\]
Take straight line \[AB\] , then sum of angles on straight line is equal to \[{180^ \circ }\]
\[
  \angle 1 + \angle OAC = {180^ \circ } \\
  \angle OAC = {180^ \circ } - \angle 1 \\
 \] \[...(i)\]
Similarly, Take straight line \[CD\] , then sum of angles on straight line is equal to \[{180^ \circ }\]
\[
  \angle 2 + \angle ODB = {180^ \circ } \\
  \angle ODB = {180^ \circ } - \angle 2 \\
 \] \[...(ii)\]
Since \[\angle 1 = \angle 2\]
Therefore if we subtract same angle from \[{180^ \circ }\], both angles obtained will be equal
\[
  {180^ \circ } - \angle 1 = {180^ \circ } - \angle 2 \\
  \angle OAC = \angle ODB \\
 \]
Also when two lines intersect then vertically opposite angles are equal.
So, we get \[\angle AOC = \angle DOB\]
Since two sets of angles of two triangles are equal therefore the third angle of triangles will also be equal. So, \[\angle ACO = \angle DBO\]
Now we prove triangles congruent.
Taking two triangles, \[\vartriangle CAO,\vartriangle BDO\]
We have \[\angle OAC = \angle ODB\] , \[\angle AOC = \angle DOB\], \[\angle ACO = \angle DBO\] and sides \[AO = OD\]
Therefore, we can say triangles are congruent i.e. \[\vartriangle CAO \cong \vartriangle BDO\]
Which means the sides of triangles are equal, which gives us \[OC = OB\]
In \[\vartriangle OCB\] two sides are equal to each other, so \[\vartriangle OCB\] is an isosceles triangle.

Note:Students many times make the mistake of taking alternate interior angles equal here but we are not given the set of lines \[AD,BC\] parallel so we don’t use that approach. In these types of questions, always prove congruency using the help of vertically opposite angles, alternate angles (if lines are parallel).