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

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

Answer 2

`m\angle DEC=60^(o)`

Explanation:

We have been given a diagram of two triangles, in which line AB is parallel to CD and BC is parallel to DE.

We can see that
`\angle DCE` is corresponding to
`\angle BAC` as both angles are formed on the same side of parallel lines AB and CD and transversal AE, therefore,
`m\angle BAC=m\angle DCE`.

We can see that
`\angle EDC` is corresponding to
`\angle CBA` as both angles are formed on the same side of parallel lines BC and DE and transversal AE, therefore,
`m\angle CBA=m\angle EDC`.

Upon Substituting our given angle measures we will get,

`64^(o)=m\angle DCE`

`56^(o)=m\angle EDC`

Since the sum of all the angles of a triangle is always 180 degrees, therefore, we can find measure of angle DEC by equating the sum of angles of our given triangle with 180.

`m\angle EDC+m\angle DCE+m\angle DEC=180^(o)`

`56^(o)+64^(o)+m\angle DEC=180^(o)`

`120^(o)+m\angle DEC=180^(o)`

`m\angle DEC=180^(o)-120^(o)`

`m\angle DEC=60^(o)`

Therefore, the measure of angle DEC is 60 degrees.