Note: Since you have not mentioned the measure of any angle. So, I am assuming one the base angles let say [tex]z^{0} = 45[/tex], , as shown in figure a, to determine the [tex]x^{0}[/tex] and [tex]y^{0}[/tex] spread across the shape.
Answer:
When one of the base angle, let say [tex]z^{0} = 45[/tex] degrees. Then,
[tex]y^{0} =45[/tex] and [tex]x^{0} =90[/tex] degrees.
Step-by-step explanation:
The shape is the combination of three isosceles triangles.
Let us consider the left sided isosceles triangle to determine the measure of the angle [tex]y^{0}[/tex].
According to the Isosceles Triangle Theorem: ''If two sides of a triangle are congruent, then angles opposite those sides are congruent''.
As angle [tex]y^{0}[/tex] is one of the base angles.
Since you have not mentioned the measure of any angle to determine the rest of the angles of isosceles triangles . So, I am suppose the other base angle [tex]z^{0} = 45[/tex] degrees, as shown in figure a.
Let suppose the vertex angle be denoted as [tex]x^{0}[/tex].
As we know that the sum of the angles of a triangle is [tex]180^{0}[/tex].
i.e.
[tex]x^{0}[/tex] + [tex]y^{0}[/tex] + [tex]z^{0}[/tex] = [tex]180^{0}[/tex]
As
[tex]z^{0} = 45[/tex]
Since, it is an isosceles triangle. if one of the base angles i.e. [tex]z^{0} = 45[/tex], it means the other base angle would have the same length. As the other base angle is [tex]y^{0}[/tex]. It means [tex]y^{0} =45[/tex].
Therefore, the the measure of the angle [tex]y^{0} =45[/tex].
Similarly, let us consider the centered isosceles triangle to determine the measure of the angle [tex]x^{0}[/tex].
As we have already determined [tex]y^{0} =45[/tex] when we considered [tex]z^{0} = 45[/tex]. So, the values of [tex]x^{0}[/tex] and [tex]y^{0}[/tex] will remain the same in the current centered isosceles triangle. Means,
As we know that the sum of the angles of a triangle is [tex]180^{0}[/tex].
Because,
[tex]x^{0}[/tex] + [tex]y^{0}[/tex] + [tex]z^{0}[/tex] = [tex]180^{0}[/tex]
Putting [tex]y^{0} =45[/tex] and [tex]z^{0} = 45[/tex] in above equation wold bring the value of [tex]x^{0}[/tex] as 90 degrees. i.e. [tex]x^{0} =90[/tex] degrees
Therefore, the vertex angle [tex]x^{0}[/tex] will be 90 degree i.e. [tex]x^{0} =90[/tex] degrees.
Keywords: isosceles triangle
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