In the figure below, ∠DEC ≅ ∠DCE, ∠B ≅ ∠F, and segment DF is congruent to segment BD . Point C is the point of intersection between segment AG and segment BD while point E is the point of intersection between segment AG and segment DF . Prove ΔABC ≅ ΔGFE. (10 points)

Step-by-step explanation:

Given:

∠DCE ≅ ∠DEC

∠B ≅ ∠F

DF ≅ BD

To prove:

ΔABC ≅ ΔGFE

Solution:

                  Statements                            Reasons

1). ∠DCE ≅ ∠DEC                                1). Given

2). ∠ACB ≅ ∠GEF                               2). Vertically opposite angles to the

                                                                 congruent angles.

3). ∠B ≅ ∠F                                         3). Given

4). DB ≅ DF                                         4). Given

5). DC + CB ≅ DE + EF                       5). Segment addition postulate

6). DC ≅ DE                                        6). Property of isosceles triangle

7). CB ≅ EF                                         7). Transitive property

8). ΔABC ≅ ΔGFE                              8). ASA property of congruence

the revenue of the fitness center, y, is the product of two factors: the number of members and the fee paid by each member. there are 320 members now, but number of members goes down by 10 for each $5 increase in fees, so if x is the number of $5 price increases then members is equal to (320 - 10x). since the fee is currently $45, and the fee increases by $5 for every price bump x, the fee is given by (45 + 5x). so, revenue y is given by y = (320 - 10x)(45 + 5x).