Answer:
![\triangle BOT \cong \triangle IEW](https://tex.z-dn.net/?f=%5Ctriangle%20BOT%20%5Ccong%20%5Ctriangle%20IEW)
Step-by-step explanation:
We are given the following in the question:
![\angle BTO \cong \angle IWE\\WI \cong BT\\OT \cong EW](https://tex.z-dn.net/?f=%5Cangle%20BTO%20%5Ccong%20%5Cangle%20IWE%5C%5CWI%20%5Ccong%20BT%5C%5COT%20%5Ccong%20EW)
We have to prove:
![\triangle BOT \cong \triangle IEW](https://tex.z-dn.net/?f=%5Ctriangle%20BOT%20%5Ccong%20%5Ctriangle%20IEW)
Proof:
![\text{In }\triangle BOT \text{ and } \triangle IEW](https://tex.z-dn.net/?f=%5Ctext%7BIn%20%7D%5Ctriangle%20BOT%20%5Ctext%7B%20and%20%7D%20%5Ctriangle%20IEW)
we can write:
![\angle BTO = \angle IWE\text{ (Given)}\\WI \cong BT\text{ (Given)}\\OT \cong EW\text{ (Given)}](https://tex.z-dn.net/?f=%5Cangle%20BTO%20%3D%20%5Cangle%20IWE%5Ctext%7B%20%28Given%29%7D%5C%5CWI%20%5Ccong%20BT%5Ctext%7B%20%28Given%29%7D%5C%5COT%20%5Ccong%20EW%5Ctext%7B%20%28Given%29%7D)
Hence, the two triangle are congruent by SAS congruency rule.
![\triangle BOT \cong \triangle IEW](https://tex.z-dn.net/?f=%5Ctriangle%20BOT%20%5Ccong%20%5Ctriangle%20IEW)
The attached image shows the two triangle.
Answer:
<u>Let's verify is the corresponding sides have same ratio:</u>
- ΔABC, sides: 6, 9, 11
- ΔXYZ, sides: 21, 31.5, 38.5
<u>Ratios:</u>
- 6/21 = 2/7 (divide by 3)
- 9/31.5 = 2/ 7 (divide by 4.5)
- 11/38.5 = 2/7 (divide by 5.5)
The ratios are same, so the triangles <u>are similar</u>
- Yes, the sides are all proportional
The scale factor (ΔABC to ΔXYZ) is <u>2/7</u> (or ΔXYZ to ΔABC is 7/2 = <u>3.5</u>)