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In the given figure, \(UB \parallel AT\) and \(CU \equiv CB\). Prove that \(\triangle CUB \sim \triangle CAT\) and hence \(\triangle CAT\) is isosceles.
Proof:
| Statements | Reasons | |
| 1 | \(\angle CUB = \angle CBU\) | |
| 2 | \(\angle CUB = \angle CAT\) | |
| 3 | \(\angle CBU = \angle CTA\) | |
| 4 | \(\angle UCB = \angle ACT\) | |
| 5 | \(\triangle CUB \sim \triangle CAT\) | |
| 6 | \(CA = CT\) | |
| 7 | \(\triangle CAT\) is an isosceles triangle |
Answer variants:
By AAA Similarity criteria(1,2,4)
Common angle
Since \(\angle CUB = \angle CBU\) and \(\angle CAT = \angle CTA\)
Given that \(UB \parallel AT\), corresponding angles are equal if \(CA\) is transversal.
By AAA Similarity criteria(1,2,3)
Given that in \(\triangle CUB\), \(CU = CB\).
Given that \(UB \parallel AT\), corresponding angles are equal if \(CT\) is transversal.
By AAA Similarity criteria(2,3,4)
Opposite sides of equal angle are equal.
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