Answer:
a) ![\angle G = 125^\circ](https://tex.z-dn.net/?f=%5Cangle%20G%20%3D%20125%5E%5Ccirc)
b) AG = 15 units
c) Perimeter of polygon SUBAG = 78 units
Step-by-step explanation:
Given:
Polygon ROTFL ~ Polygon SUBAG
Similar polygons mean they have similar angles and the ratio of corresponding sides and ratio of their perimeter are equal.
Part A:
Given that
![m\angle R = 100^\circ\\m\angle O = 120^\circ\\m\angle T = 75^\circ\\M\angle L = 60^\circ](https://tex.z-dn.net/?f=m%5Cangle%20R%20%3D%20100%5E%5Ccirc%5C%5Cm%5Cangle%20O%20%3D%20120%5E%5Ccirc%5C%5Cm%5Cangle%20T%20%3D%2075%5E%5Ccirc%5C%5CM%5Cangle%20L%20%3D%2060%5E%5Ccirc)
![m\angle L+M\angle L = 180^\circ\\\Rightarrow m\angle L=180-60=120^\circ](https://tex.z-dn.net/?f=m%5Cangle%20L%2BM%5Cangle%20L%20%3D%20180%5E%5Ccirc%5C%5C%5CRightarrow%20m%5Cangle%20L%3D180-60%3D120%5E%5Ccirc)
Sum of all interior angles of a pentagon is ![540^\circ](https://tex.z-dn.net/?f=540%5E%5Ccirc)
![m\angle R+m\angle O+m\angle T+m\angle F+m\angle L=540^\circ\\\Rightarrow 100+120+75+m\angle F+120=540^\circ\\\Rightarrow m\angle F=125^\circ\\](https://tex.z-dn.net/?f=m%5Cangle%20R%2Bm%5Cangle%20O%2Bm%5Cangle%20T%2Bm%5Cangle%20F%2Bm%5Cangle%20L%3D540%5E%5Ccirc%5C%5C%5CRightarrow%20100%2B120%2B75%2Bm%5Cangle%20F%2B120%3D540%5E%5Ccirc%5C%5C%5CRightarrow%20m%5Cangle%20F%3D125%5E%5Ccirc%5C%5C)
Due to similarity property of the two pentagons, ![\angle F =\angle G = 125^\circ](https://tex.z-dn.net/?f=%5Cangle%20F%20%3D%5Cangle%20G%20%3D%20125%5E%5Ccirc)
Part B:
Ratio of corresponding sides is equal.
Given the sides SU = 12, RO = 8 and FL = 10 units respectively.
![\dfrac{SU}{RO}=\dfrac{AG}{FL}\\\dfrac{12}{8}=\dfrac{AG}{10}\\\Rightarrow AG = 15\ units](https://tex.z-dn.net/?f=%5Cdfrac%7BSU%7D%7BRO%7D%3D%5Cdfrac%7BAG%7D%7BFL%7D%5C%5C%5Cdfrac%7B12%7D%7B8%7D%3D%5Cdfrac%7BAG%7D%7B10%7D%5C%5C%5CRightarrow%20AG%20%3D%2015%5C%20units)
Part C:
Ratio of corresponding sides must be equal to ratio of perimeter of the two polygons:
![\dfrac{RO}{SU} = \dfrac{\text{perimeter of ROTFL}}{\text{perimeter of SUBAG}}\\\Rightarrow \dfrac{8}{12} = \dfrac{52}{\text{perimeter of SUBAG}}\\\Rightarrow \text{perimeter of SUBAG} = 52 \times 1.5\\\Rightarrow \text{perimeter of SUBAG} = 78\ units](https://tex.z-dn.net/?f=%5Cdfrac%7BRO%7D%7BSU%7D%20%3D%20%5Cdfrac%7B%5Ctext%7Bperimeter%20of%20ROTFL%7D%7D%7B%5Ctext%7Bperimeter%20of%20SUBAG%7D%7D%5C%5C%5CRightarrow%20%5Cdfrac%7B8%7D%7B12%7D%20%3D%20%5Cdfrac%7B52%7D%7B%5Ctext%7Bperimeter%20of%20SUBAG%7D%7D%5C%5C%5CRightarrow%20%5Ctext%7Bperimeter%20of%20SUBAG%7D%20%3D%2052%20%5Ctimes%201.5%5C%5C%5CRightarrow%20%5Ctext%7Bperimeter%20of%20SUBAG%7D%20%3D%2078%5C%20units)
So, the answers are:
a) ![\angle G = 125^\circ](https://tex.z-dn.net/?f=%5Cangle%20G%20%3D%20125%5E%5Ccirc)
b) AG = 15 units
c) Perimeter of polygon SUBAG = 78 units