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? choose three correct answers. the perpendicular bisector of xz creates two smaller isosceles triangles. angle y is a right angle. the measure of angle x is 36°. the measure of the vertex angle is 72°. the measure of angle z is 45°.

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

The sum of angles in a triangle is \( 180^\circ \). So, \( x + x + 2x = 180^\circ \).
Simplify: \( 4x = 180^\circ \), so \( x = 45^\circ \)? Wait, no, wait—wait, the problem says vertex angle Y is twice a base angle. Wait, maybe I misread. Wait, let's re - express. Let base angle be \( x \), vertex angle \( Y = 2x \). Then \( x + x + 2x=180 \), \( 4x = 180 \), \( x = 45 \)? But that contradicts some options. Wait, maybe the vertex angle is twice one base angle, but maybe the triangle has two equal base angles. Wait, no, isosceles triangle has two equal angles (base angles) and vertex angle. So let's re - check. Wait, maybe the problem is that the vertex angle Y is twice the measure of a base angle. So let base angles be \( x \), vertex angle \( 2x \). Then \( x + x+2x = 180 \), \( 4x = 180 \), \( x = 45 \), vertex angle \( 90 \)? But one option says Angle Y is a right angle (90 degrees). But also, let's check the perpendicular bisector of XZ: in an isosceles triangle, the perpendicular bisector of the base (if Y is vertex, then XZ is base) will bisect the vertex angle and create two smaller isosceles triangles. Also, if vertex angle is 72, then base angles are 54? Wait, maybe I made a mistake. Wait, maybe the vertex angle is twice a base angle, but let's suppose base angle is \( x \), vertex angle \( 2x \). Then \( x + x+2x = 180 \), \( 4x = 180 \), \( x = 45 \), vertex angle 90 (right angle). Then the perpendicular bisector of XZ (base) will create two smaller isosceles triangles. Also, if angle X is 36, then vertex angle would be 72 (since 36*2 = 72, and 36+36 + 72=144? No, that's wrong. Wait, maybe the problem is that the vertex angle is twice a base angle, but the triangle has vertex angle Y, and base angles X and Z. So let's re - solve: Let \( \angle X=\angle Z = x \), \( \angle Y = 2x \). Then \( x + x+2x=180 \), \( 4x = 180 \), \( x = 45 \), \( \angle Y=90 \) (right angle). So Angle Y is a right angle is correct. The perpendicular bisector of XZ (base) will create two smaller isosceles triangles (since it's perpendicular bisector, so the two triangles formed will have two equal sides and angles). Now, if we consider another case: maybe the vertex angle is 72, then base angles are 54, but that doesn't fit. Wait, maybe the problem has a typo, but based on the options:

1. "The perpendicular bisector of \( \overline{XZ} \) creates two smaller isosceles triangles." - In an isosceles triangle, the perpendicular bisector of the base (XZ, if Y is vertex) will bisect the vertex angle and the base, creating two congruent (and thus isosceles) triangles. So this is correct.

2. "Angle Y is a right angle." - If \( \angle Y = 2x \), \( \angle X=\angle Z=x \), and \( x + x+2x = 180 \), \( 4x = 180 \), \( x = 45 \), \( \angle Y = 90 \), so this is correct.

3. Wait, another option: "The measure of angle X is 36°" - If \( \angle Y = 2x \), \( \angle X=\angle Z=x \), and \( x = 36 \), then \( \angle Y=72 \), and \( 36 + 36+72 = 144 \), which is wrong. Wait, maybe the vertex angle is \( x \), and base angles are \( 2x \)? No, the problem says "the measure of the vertex angle, Y, is twice the measure of a base angle". So vertex angle is twice base angle. So base angle is \( x \), vertex \( 2x \). So sum is \( x + x+2x = 180 \), \( 4x = 180 \), \( x = 45 \), vertex \( 90 \). So angle X is 45, not 36. Wait, maybe the problem is different. Wait, maybe the triangle is isosceles with \( XY = YZ \), so vertex angle at Y, and base XZ. Then the perpendicular bisector of XZ will pass through Y, bisecting XZ and angle Y. Then the two triangles formed are isosceles. Also, if angle Y is 90, then it's a right isosceles triangle. Also, maybe the "The measure of the vertex angle is 72°" - if vertex angle is 72, then base angles are (180 - 72)/2 = 54. But that doesn't match the "twice" condition. Wait, maybe the problem has a mistake, but based on the options, the correct ones are:

- The perpendicular bisector of \( \overline{XZ} \) creates two smaller isosceles triangles. (Because in an isosceles triangle, the perpendicular bisector of the base is also the angle bisector and median of the vertex angle, creating two congruent isosceles triangles.)

- Angle Y is a right angle. (If vertex angle is twice base angle, and base angles are 45, vertex is 90.)

- Wait, another option: "The measure of angle X is 36°" - no, if vertex is 72, base angles are 54. Wait, maybe the problem means that the base angle is twice the vertex angle? Let's try that. Let vertex angle \( x \), base angles \( 2x \). Then \( x + 2x+2x = 180 \), \( 5x = 180 \), \( x = 36 \), so base angles are 72. Then vertex angle is 36, base angles 72. Then angle X is 72? No. Wait, this is confusing. But based on the options, the three correct answers are:

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

2. Angle Y is a right angle.

3. Wait, maybe "The measure of the vertex angle is 72°" is wrong. Wait, let's re - examine the options. The options are:

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

- Angle Y is a right angle.

- The measure of angle X is 36°.

- The measure of the vertex angle is 72°.

- The measure of angle Z is 45°.

Wait, if vertex angle Y is 90 (right angle), then base angles X and Z are 45 each. So "The measure of angle Z is 45°" is correct. And "The perpendicular bisector of \( \overline{XZ} \) creates two smaller isosceles triangles" is correct. And "Angle Y is a right angle" is correct.

So:

1. The perpendicular bisector of \( \overline{XZ} \) creates two smaller isosceles triangles. (Correct, because in isosceles triangle, perpendicular bisector of base creates two congruent isosceles triangles.)

2. Angle Y is a right angle. (Correct, since \( \angle Y = 90^\circ \) when base angles are 45°.)

3. The measure of angle Z is 45°. (Correct, since base angles are equal and \( (180 - 90)/2=45 \).)

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°.