Question

Please help!!!
Given that lines marked with arrows in figure 10.36 are parallel, determine the sum of the angles a+b+c+d+e without measuring the angles. Explain your reasoning

Answer 1

Answer:

a + b + c + d + e = 180.

Step-by-step explanation:

We can prove that a = angle CBA (vertically opposite angles are equal).

Now, since lines DE and CA are parallel, then it follows that c = angle ACE (alternate angles are equal). Similarly, b = angle CAD.

But notice that what we've just done is found the angles of the triangle CBA.

So the sum of the angles (a + b + c + d + e) is the sum of the angles of a triangle.

We then conclude that a + b + c + d + e = 180 degrees.

Answer 2

You can use properties like angle made by a line intersecting parallel lines are equal(corresponding angles). And another properties that alternate interior angles are same.

The sum of angles a + b + c + d + e is 180 degrees.

What are vertically opposite angles?

Vertically opposite angles are angles made between two intersecting lines  at the point of intersection(located oppositely and not adjacent). They're of equal measure.

What are alternate interior angles?

In given diagram, the lines AC and DE are parallel. The angle made by CE on AC and DE has two pairs of alternate interior angles.

$\angle ACE, \angle CED\\ and\\ \angle DCE, \angle CEA$

These angle pairs are called alternate interior angles.

Finding the sum:

In triangle ABC, we have $\angle ABC = a \text{\: (Since they're alternate interior angles)}$

Also, we have following pairs as alternate interior angles:

$\angle ACE = \angle DEC = c\\ \angle ADE = \angle CAD = b$

Thus, we have:

$\angle DCE + \angle ECA + \angle CAD + \angle DAE = d + c + b + e\\ or\\ \angle DCA + \angle CAE = d + c + b + e$

Since sum of interior angles of a triangle is 180 degrees, thus we have:

$\angle CBD + \angle BCA + \angle CAE = 180^\circ\\ a + b + c + d + e = 180^\circ$

Thus, we have: $a + b + c + d + e = 180^\circ$

Learn more about angles made by a line intersecting parallel lines here:

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