Question

In the given the figure above, m∠BAC = 64° and m∠CBA = 56°.

Part I: Find the m∠DEC.
Part II: Explain the steps you took to arrive at your answer. Make sure to justify your answer by identifying any theorems, postulates, or definitions used.

Answer 1

`\text{Answer: }\angle DEC=60\textdegree`

Step-by-step explanation:

Since we have given that

m∠BAC = 64° and m∠CBA = 56°

and AB║CD and BC║DE

First, we consider Δ ABC,

As we know two angles of a triangle so we need to find the third angle, for which we'll use the "Angle Sum Property",

Angle Sum Property, states that the sum of three angles of a triangle is
`180\textdegree`

Now, we will apply this,

`\angle BAC+\angle ABC+\angle ACB=180\textdegree\\\\64\textdegree+56\textdegree+\angle ACB=180\textdegree\\\\120\textdegree+\angle ACB=180\textdegree\\\\\angle ACB=180\textdegree-120\textdegree\\\\\angle ACB=60\textdegree`

Now, as we have given that

BC║DE

so,

m∠ACB=m∠DEC

(∵ Corresponding angles are equal for given parallel lines)

So,

`\angle ACB=\angle DEC=60\textdegree`

Answer 2

since the triangles are similar

angle DEC = 60 degrees

3 angles inside a triangle equal 180 degrees

BAC = DCE = 64

CBA = EDC = 56

DEC = 180 -56 -64 = 60 degrees

used angle-angle theorem