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? Check all that apply. 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 creates two smaller isosceles triangles.

Answer 1

Your answer is A, B, and D.

Explanation:

A- Angle Y is a right angle

B- The measure of angle Z is 46°

D- The perpendicular bisector creates two smaller isosceles triangles.

All three of these statements are true!

Answer 2

Step 1

Find the measure of the vertex angle and the base angle

we know that

An isosceles triangle has two equal sides and two equal angles

The two equal angles are called the base angle

The third angle is called the vertex angle

Let

Y------> the vertex angle

X-----> the base angle

Z----> the base angle

`Z=X` ------> equation A

Remember that

The sum of the internal angles of a triangle is equal to
`180\°`

so

`X+Y+Z=180\°` -----> equation B

In this problem

The measure of the vertex angle, Y, is twice the measure of a base angle

`Y=2X` -----> equation C

Substitute equation A and equation C in equation B

`X+[2x]+[x]=180\°`

Solve for X

`4X=180\°`

`X=45\°`

so

`Z=X=45\°`

Find the value of Y

`Y=2X`

`Y=90\°`

Step 2

Verify the statements

case A) The measure of angle Y is aright angle

The statement is True

Because, the measure of the vertex angle Y is
`90\°`

case B) The measure of angle Z is
`45\°`

The statement is True

See the procedure in the Step
`1`

case C) The measure of angle X is
`36\°`

The statement is False

Because, the measure of angle X is
`45\°`

case D) The measure of the vertex angle is
`72\°`

The statement is False

Because, the measure of the vertex angle Y is
`90\°`

case E)The perpendicular bisector of creates two smaller isosceles triangles

The statement is True

Because, the perpendicular bisector creates two smaller
`45\°-90\°-45\°` triangles, and we know that a
`45\°-90\°-45\°` triangle is an isosceles triangle

the answer is

The measure of angle Y is aright angle

The measure of angle Z is
`45\°`

The perpendicular bisector of creates two smaller isosceles triangles