Question

6 numeric 1 point find the missing value. x = answer previous next

Explanation:

Step1: Find the base angle of the isosceles triangle

In an isosceles triangle, the base angles are equal. The sum of angles in a triangle is \(180^{\circ}\). Let the base angle be \(y\). So, \(y=\frac{180 - 54}{2}=63^{\circ}\)

Step2: Use the exterior - angle property

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. Here, \(x = 180-(180 - 63)=27^{\circ}\) (Another way: The angle adjacent to \(y\) in the larger triangle is \(180 - 63=117^{\circ}\). Then, using the sum of angles in the larger triangle (where the third side is equal to the side of the smaller isosceles triangle), let the third angle in the larger triangle be \(x\). We know that in a triangle with two equal sides (the side of the smaller isosceles triangle and the side marked equal to it in the larger triangle part), and using the angle sum property \(x=\frac{180 - 126}{2}=27^{\circ}\) (since \(180-(117 + 36)\) is wrong approach. Correct: The angle adjacent to \(y\) is \(180 - 63\). Then in the triangle where \(x\) is located, the other angle (from the isosceles triangle property) is \(180-(180 - 63)\). Wait, better: The triangle with \(x\) has an angle which is supplementary to the base angle of the first isosceles triangle. The base angle of the first isosceles triangle (\(y = 63^{\circ}\)), its supplementary angle is \(180 - 63=117^{\circ}\). The triangle with \(x\) has two equal sides (the side of the smaller isosceles triangle and the side marked equal). Let the third angle be \(x\). Using the angle sum property of a triangle \(x=\frac{180-(180 - 63)}{2}=27^{\circ}\) (No, correct: The triangle where \(x\) is. The angle adjacent to \(y\) (from the first isosceles triangle) is \(180 - 63\). But since the two sides (the side of the first isosceles triangle and the side marked equal) are equal, the triangle where \(x\) is has angles: one angle is \(180-(180 - 63)\) (wrong). Correct: The first isosceles triangle (with \(54^{\circ}\) angle) has base angles \(y=\frac{180 - 54}{2}=63^{\circ}\). The angle adjacent to \(y\) is \(180 - 63 = 117^{\circ}\). Now, the triangle with \(x\) has two equal sides (the side of the first isosceles triangle and the side marked equal). Let the angles of this triangle be \(x\), \(x\) (because of equal sides) and \(180-(180 - 63)\) (No. Wait, the sum of angles in a triangle is \(180^{\circ}\). The angle which is adjacent to \(y\) (from the first isosceles triangle) is \(180 - 63\). But in the triangle where \(x\) is, using the property that angles opposite to equal sides are equal. The sum of angles: \(x+x+(180 - 63)=180\). \(2x=63\), \(x = 27^{\circ}\))

Answer:

\(27\)