Question

triangle xyz is isosceles. the measure of the vertex angle, y, is twice the measure of a base angle. what is true about triangle xyz? select three options.
☐ angle y is a right angle.
☐ the measure of angle z is 45°.
☐ the measure of angle x is 36°.
☐ the measure of the vertex angle is 72°.
☐ the perpendicular bisector of \\(\overline{xz}\\) creates two smaller isosceles triangles.

Explanation:

Step1: Define variables for angles

Let the measure of each base angle (∠X and ∠Z) be \( x \). Then the vertex angle ∠Y is \( 2x \).

Step2: Use triangle angle sum property

The sum of angles in a triangle is \( 180^\circ \). So, \( x + x + 2x = 180^\circ \).
Simplify: \( 4x = 180^\circ \), then \( x = \frac{180^\circ}{4} = 45^\circ \)? Wait, no, wait: Wait, \( x + x + 2x = 4x \)? Wait, no, \( x + x + 2x = 4x \)? Wait, no, \( x + x + 2x = 4x \)? Wait, no, \( x + x + 2x = 4x \)? Wait, no, \( x + x + 2x = 4x \)? Wait, no, let's recalculate. \( x + x + 2x = 4x \)? Wait, no, \( x + x = 2x \), plus \( 2x \) is \( 4x \)? Wait, no, \( x + x + 2x = 4x \). Then \( 4x = 180^\circ \), so \( x = 45^\circ \)? Wait, that can't be. Wait, no, maybe I made a mistake. Wait, the vertex angle is twice a base angle. So base angles are equal (isosceles), so let base angles be \( x \), vertex angle \( 2x \). Then sum is \( x + x + 2x = 4x = 180^\circ \), so \( x = 45^\circ \), vertex angle \( 90^\circ \). Wait, but then let's check the options.

Wait, let's re-express:

Let base angles be \( x \) (∠X and ∠Z, since Y is vertex). Then ∠Y = \( 2x \).

Sum: \( x + x + 2x = 4x = 180^\circ \) ⇒ \( x = 45^\circ \), so ∠Y = \( 90^\circ \), ∠X = ∠Z = \( 45^\circ \)? Wait, but that contradicts some options. Wait, maybe I mixed up vertex and base. Wait, in isosceles triangle, the vertex angle is the angle between the two equal sides. So if Y is the vertex, then the equal sides are XY and YZ? Wait, no, in triangle XYZ, vertices are X, Y, Z. So if Y is the vertex, then the base is XZ, and the equal sides are XY and YZ? Wait, no, the legs are XY and YZ, base is XZ. So base angles are ∠X and ∠Z, vertex angle is ∠Y. So sum: ∠X + ∠Z + ∠Y = 180. ∠X = ∠Z = x, ∠Y = 2x. So 2x + x + x = 4x = 180 ⇒ x = 45, ∠Y = 90. So ∠Y is right angle (90°), ∠Z is 45°, and the perpendicular bisector of XZ: since triangle XYZ is isosceles with XY = YZ (wait, no, if Y is vertex, then XY = YZ? Wait, no, in isosceles triangle, the two equal sides are the legs, so if vertex is Y, then XY = YZ, so the base is XZ. Then the perpendicular bisector of XZ would pass through Y, creating two triangles: XYW and ZYW, where W is the midpoint of XZ. Then XY = YZ, XW = WZ, YW is common. So triangles XYW and ZYW are congruent, and each has angles: ∠X = 45°, ∠XWY = 90°, so ∠XYW = 45°, so they are isosceles (45-45-90). Wait, but let's check the options:

Option 1: Angle Y is a right angle. ∠Y = 90°, so yes.

Option 2: The measure of angle Z is 45°. ∠Z = x = 45°, yes.

Option 3: The measure of angle X is 36°. No, ∠X is 45°, so no.

Option 4: The measure of the vertex angle is 72°. No, vertex angle is 90°, so no.

Option 5: The perpendicular bisector of XZ creates two smaller isosceles triangles. Let's see: perpendicular bisector of XZ (base) in isosceles triangle XYZ (with Y as vertex) will pass through Y (since in isosceles triangle, the altitude, median, and angle bisector from vertex to base coincide). So the perpendicular bisector of XZ is YW, where W is midpoint of XZ. Then triangles XYW and ZYW: XY = YZ (given, isosceles), XW = WZ (bisector), YW is common. So triangles are congruent. Angles: ∠X = 45°, ∠XWY = 90°, so ∠XYW = 45°, so triangle XYW has angles 45, 45, 90, which is isosceles (two angles equal). Similarly for ZYW. So yes, this option is correct.

Wait, but earlier calculation: 4x = 180 ⇒ x = 45, so ∠Y = 90°, ∠X = ∠Z = 45°. So correct options are:

1. Angle Y is a right angle.

2. The measure of angle Z is 45°.

5. The perpendicular bisector of XZ creates two smaller isosceles triangles.

Wait, but let's recheck the angle sum. If vertex angle is twice base angle, base angles are x, vertex 2x. Sum: x + x + 2x = 4x = 180 ⇒ x = 45, 2x = 90. So that's correct. So:

- Angle Y (vertex) is 90°, so right angle: correct.

- Angle Z is 45°: correct (since base angles are equal, ∠X = ∠Z = 45°).

- Perpendicular bisector of XZ: in isosceles triangle, the perpendicular bisector of the base passes through the vertex, so it splits the triangle into two congruent right triangles, each with angles 45, 45, 90, which are isosceles: correct.

The other options:

- Angle X is 36°: no, it's 45°.

- Vertex angle 72°: no, it's 90°.

So the three correct options are:

- Angle Y is a right angle.

- The measure of angle Z is 45°.

- The perpendicular bisector of \( \overline{XZ} \) creates two smaller isosceles triangles.

Answer:

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