Question

in the figure UB parallel to AT and CU is congruent to CB prove that ∆CUB similar to ∆CAT and hence ∆CAT is isosceles​

Answer 1

Answer:

Given data: In ∆CAT, UB // AT and CU is congruent to CB

To prove:  ∆CUB is similar to ∆CAT and hence ∆CAT is isosceles

We have UB // AT, so according to the triangle proportionality theorem, we can write

AU / CU = TB / CB ….. (i)

CU is congruent to CB which means  

CU = CB …. (ii)

∴ AU = TB …. (iii)

From (i), (ii) & (iii), we get

∴ AU / CU = TB / CB = AC / CT

And,

∴ AC = CT ….. (iv)

Now, we have  

CU = CB (given)

AC = CT …. [from (iv)]

∠C is common angle to both ∆CUB and ∆CAT

∴ ∆CUB and ∆CAT have two sides and 1 angle equal to each other i.e., by SAS congruence we can say

∆CUB ~ ∆CAT

Since in ∆CAT,  

AC = CT  

When two sides of a triangle are equal, we can say the triangle is an isosceles triangle.

Hence, ∆CAT is isosceles.