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 $45^\circ$.
the measure of angle x is $36^\circ$.
the measure of the vertex angle is $72^\circ$.
the perpendicular bisector of $\overline{xz}$ creates two smaller isosceles triangles.

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 $45^\circ$.
  the measure of angle x is $36^\circ$.
  the measure of the vertex angle is $72^\circ$.
  the perpendicular bisector of $\overline{xz}$ creates two smaller isosceles triangles.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Perpendicular Bisector Theorem" ], "new_concepts": [ "Isosceles Triangle Properties", "Triangle Angle Sum Theorem" ], "current_concepts": [ "Isosceles Triangle Properties", "Triangle Angle Sum Theorem", "Perpendicular Bisector Theorem" ] } ## Set up the angle equations $$ \begin{aligned} &\text{Let the base angles be } m\angle X = m\angle Z = x \ &\text{The vertex angle is } m\angle Y = 2x \ &x + x + 2x = 180^\circ \ &4x = 180^\circ \ &x = 45^\circ \end{aligned} $$ ## Calculate the angle measures $$ \begin{aligned} &m\angle X = 45^\circ \ &m\angle Z = 45^\circ \ &m\angle Y = 2(45^\circ) = 90^\circ \end{aligned} $$ ## Analyze the perpendicular bisector of the base $$ \begin{aligned} &\text{The perpendicular bisector of base } \overline{XZ} \text{ passes through the vertex } Y. \ &\text{It divides } \triangle XYZ \text{ into two right triangles with angles } 45^\circ, 45^\circ, \text{ and } 90^\circ. \ &\text{These two smaller triangles are also isosceles.} \end{aligned} $$

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.

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QUESTION IMAGE

Question

Explanation:

Set up the angle equations

Calculate the angle measures

Analyze the perpendicular bisector of the base

Answer: