Question

In \[\Delta ABC\], \[AB = BC\], \[AD \bot BC\], \[CE \bot AB\]. Prove that \[AD = CE\].

Answer 1

Hint: Here, we are required to prove that in the given \[\Delta ABC\], \[AD = CE\]. First of all, we will draw a \[\Delta ABC\] satisfying the given conditions. Then we will observe that the sides AD and CE lie on which triangles which can be proved congruent. After proving those two triangles congruent, we will prove that \[AD = CE\] by simply using the CPCT rule.

Complete step-by-step answer:
First of all, we will draw a triangle \[\Delta ABC\] satisfying all the given conditions:

Now, from the above figure,
 In \[\Delta ABC\], \[AB = BC\], \[AD \bot BC\], \[CE \bot AB\].
Now, we will prove the congruence of two triangles.
Two triangles are said to be congruent if they completely cover each other, i.e. three sides of one triangle are equal to three sides of the other triangle and their angles are also equal in any orientation.
Now, in \[\Delta ABD\] and \[\Delta CBE\],
1.It is given that \[AB = BC\].
2.It is given that \[AD \bot BC\]and \[CE \bot AB\], so we can say that
\[\angle ADB = \angle CEB\]
3.In both the triangle, \[\angle B\] is common, so
\[\angle B = \angle B\]
Therefore, \[\Delta ABD \cong \Delta CBE\] by Side-Angle-Side or SAS Congruence.
Now, when two triangles are congruent, their corresponding sides and angles are equal, this is known as CPCT Rule which stands for Corresponding Parts of Congruent Triangles.
As, \[\Delta ABD \cong \Delta CBE\], then by CPCT rule
\[AD = CE\]
Hence, proved.

Note: As we have discussed, two triangles are said to be congruent if they have the same sides and the same angles. Now, there are 5 ways to prove that two triangles are congruent:
1.Side Side Side or SSS Rule: According to this, all the three sides of the two triangles should be equal to each other respectively.
2.Side Angle Side or SAS Rule: In this rule, if two sides and the angle between them of two triangles are equal then they are congruent.
3.Angle Side Angle or ASA Rule: According to this, if two angles and a side between them of two triangles are equal, then they are congruent to each other.
4.Angle Angle Side or AAS Rule: According to this, if two angles and then, the side in two triangles are equal, then they are congruent.
5.Right Hypotenuse Side or RHS Rule: If the right angle, the hypotenuse and any one side in two right angles are equal, then, they both are congruent.
Hence, these are all the rules which we can apply to prove the congruence of any two triangles (whichever is possible).