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In quadrilateral ACBD, AC = AD and AB bisects $\angle A$. Show that $\Delta ABC\cong \Delta ABD$. What can you say about BC and BD?

Last updated date: 20th Jun 2024
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Hint: Here in this question, we are given a quadrilateral ACBD which is made up of two triangles i.e. $\Delta ABC$ and $\Delta ABD$. We have to prove that both the triangles are congruent to each other. For proving congruency, we have to apply rules of congruency.

Complete step by step answer:
Let’s solve the question now.
As we know that two triangles are congruent if they satisfy three conditions by applying any rule i.e. SSS rule, SAS, rule, ASA rule, AAS rule or by RHS rule. Here the SAS rule will be applied. SAS rule says that two sides and one angle should be equal if we want to make triangles congruent.
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There is a quadrilateral ACBD which consists of two triangles $\Delta ABC$ and $\Delta ABD$. And it is also given that AC = BD. In the question, it is given that AB bisects $\angle A$ which means AB divides $\angle A$ into two equal parts i.e. $\angle CAB=\angle DAB$. To prove that two triangles are congruent, it is necessary that three conditions should be satisfied.
So, for $\Delta ABC$ and $\Delta ABD$,
$\Rightarrow $AB = AB (common side in both the triangles)
$\Rightarrow $ AC = AD (given)
$\Rightarrow \angle CAB=\angle DAB$ (AB bisects $\angle A$)
By SAS rule,
$\therefore \Delta ABC\cong \Delta ABD$
BC and BD are the sides of the congruent triangles.
$\therefore $BC = BD [ By Corresponding Parts of Congruent Triangles ]

Note: Students should note that while applying the conditions, the reasons for the condition should be written along with them in brackets. Then only marks will be given. Before applying the rule, first check all the necessary conditions satisfying that rule.