Question

triangle xyz is isoceles. the measure of the vertex angle, y, is twice the measure of a base angle. what is true about triangle xyz? choose three correct answers. the perpendicular bisector of (overline{xz}) creates two smaller isosceles triangles. angle y is a right angle. the measure of angle z is (45^{circ}). the measure of the vertex angle is (72^{circ}). the measure of angle x is (36^{circ}).

Explanation:

Step1: Let the measure of base angle

Let the measure of base angle \(X = Z=x\). Then the measure of vertex angle \(Y = 2x\).

Step2: Use the angle - sum property of a triangle

By the angle - sum property of a triangle \(X + Y+Z=180^{\circ}\). Substitute \(X = Z=x\) and \(Y = 2x\) into the equation: \(x + 2x+x=180^{\circ}\).

Step3: Solve for \(x\)

Combine like terms: \(4x = 180^{\circ}\), so \(x = 45^{\circ}\). Then \(Y=2x = 90^{\circ}\).

Step4: Analyze the perpendicular bisector of \(\overline{XZ}\)

Let \(M\) be the mid - point of \(XZ\). In isosceles right - triangle \(XYZ\) with \(XY = YZ\) (since \(X = Z = 45^{\circ}\)), the perpendicular bisector of \(XZ\) (which is also the median and the altitude from \(Y\) to \(XZ\)) divides \(\triangle XYZ\) into two smaller isosceles right - triangles \(\triangle XYM\) and \(\triangle ZYM\) (because \(\angle X=\angle Z = 45^{\circ}\), \(\angle XMY=\angle ZMY = 90^{\circ}\), and \(\angle XYM=\angle ZYM = 45^{\circ}\)).

Answer:

The perpendicular bisector of \(\overline{XZ}\) creates two smaller isosceles triangles, Angle \(Y\) is a right angle, The measure of angle \(Z\) is \(45^{\circ}\)