Question

In quadrilateral ACBD, AC = AD and AB bisects ∠ A (see Fig. 7.16). Show that ∆ ABC ≅ ∆ ABD. What can you say about BC and BD?

Answer 1

Proof 

In \( \triangle ABC \) and \( \triangle ABD \), we are given the following conditions:

1. \( AC = AD \) (Given: \( AC \) is equal to \( AD \))
2. \( \angle CAB = \angle DAB \) (Given: \( AB \) bisects \( \angle A \))
3. \( AB = AB \) (Common side: \( AB \) is common to both triangles)

Now, we can apply the SAS Congruence Rule, which states that if two triangles have two sides equal and the included angle between them is also equal, the triangles are congruent.

Applying SAS Congruence Rule:

• Side \( AC \) is equal to side \( AD \).
• Angle \( \angle CAB \) is equal to angle \( \angle DAB \) (since \( AB \) bisects \( \angle A \)).
• Side \( AB \) is common to both triangles.

Thus, by the SAS Congruence Rule, we have:

\[ \triangle ABC \cong \triangle ABD \]

By CPCT (Corresponding Parts of Congruent Triangles), we can deduce that:

\[ BC = BD \]

Conclusion:

Therefore, \( BC \) and \( BD \) are of equal lengths.