Question

By the definition of supplementary angles, the base angles of the top isosceles triangle have a measure of °. Therefore, the measure of the vertex angle is ° by the Triangle Sum Theorem. The base angles of the bottom isosceles triangle will also have a measure of ° by the Vertical Angles Theorem. Thus, x°, which is the bottom isosceles triangle's vertex angle, will equal

Answer 1

1) Lesser than 180º 2)180º-2x 3) the same 4)
`x=90-(y)/(2)`

Explanation:

Let's start by defining some concepts.

Supplementary Angles= Two angles whose sum is equal to 180º

Triangle Sum Theorem= The sum of all of the inner three angles of any triangle, is equal to 180º.

Vertical Angles Theorem= If two angles are vertical, i.e. opposed by the vertex they are congruent. They share the same measure.

Isosceles Triangle = A triangle that has two congruent legs, and two congruent angles out of three.

--

1)By the definition of supplementary angles, the base angles of the top isosceles triangle have a measure of?

The base angles of a triangle are not Supplementary, because they do not sum 180º. According to the Triangle Sum Theorem, we need the third angle of the triangle to sum 180º.

2)Therefore, the measure of the vertex angle is 180º-2x by the Triangle Sum Theorem.

Check the process of obtaining it:

In any Isosceles, The Triangle Sum Theorem says:

y +x+x=180º

y=180º-2x

3)The base angles of the bottom isosceles triangle will also have the same measure, of the base angle of the Top Isosceles triangle by Vertical AnglesTheorem.

Since in the Vertical Angles Theorem angles opposed by the Vertex angle have the same measure. We have two isosceles triangles then. In the picture below, an example.

4) Thus, x°, which is the bottom isosceles triangle's vertex angle, will be equal to:

2x=180º-y

x=90º-y/2

x=90º-y/2

(For x and y in degrees)