What can we conclude about the triangles on the line shown below?

the triangles are congruent and therefore have the same slope.

the triangles are similar and therefore have the same slope.

the triangles are similar but do not have the same slope.

the triangles are unrelated to one another.

The triangles are similar and therefore have the same slope.

Step-by-step explanation:

see the attached figure with letters to better understand the problem

we know that

If two figures are similar, then the ratio of its corresponding sides is proportional and its corresponding angles are congruent

In this problem triangles ABC and DEC are similar by AA Similarity Theorem  

Because

∠CAB=∠CDE =90°

∠ACB=∠ECD ----> by vertical angles

so

The ratio of its corresponding sides is proportional

----> equation A

Remember that the slope is the change in the y-value by the change in the x-value

so

The slope BC is equal to

The slope EC is equal to

Rewrite the equation A

therefore

The triangles are similar and therefore have the same slope.

δabc ≅ δdcb by sas because

db ≅ ac . . given (side)

bc ≅ bc . . reflexive property (side)

∠dbc ≅ ∠ acb . . alternate interior angles where transversal bc crosses parallel lines bd and ac (included angle)