In Fig. 5.10, if AC = BD, then prove that AB = CD.
In Fig. 5.10, if AC = BD, then prove that AB = CD.
Solution and Explanation
From the given figure, we can observe the following relationships:
- \( AC = AB + BC \)
- \( BD = BC + CD \)
We are given that:
\[ AC = BD \quad \text{(Equation 1)} \]
Now, according to Euclid’s axiom, when equals are subtracted from equals, the remainders are also equal. This axiom allows us to subtract the same quantity from both sides of the equation. Let's subtract \( BC \) from both sides of equation (1):
\[ (AB + BC) - BC = (BC + CD) - BC \]
On simplifying both sides:
\[ AB = CD \]
Thus, we have proven that \( AB = CD \) based on the given relations and Euclid's axiom.
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