Chapter 4.6, Problem 17WE

To determine

To find: the type of triangle is $\Delta XYZ$ .

Answer to Problem 17WE

  $\Delta XYZ$ is an isosceles triangle.

Explanation of Solution

Given information:

  $A,B,C$ and $D$ are noncoplanar. $\Delta ABC,\Delta ACD$ and $\Delta ABD$ are equilateral. $X$ and $Y$ are midpoints of $\overline{AC}$ and $\overline{AD}$ . $Z$ is a point on $\overline{AB}$ .

  

Calculation:

By side angle-side postulate $\Delta XAZ\cong \Delta YAZ$ ,

Therefore, $\overline{AZ}\cong \overline{AZ},\overline{AY}\cong \overline{AY},\angle XAZ\cong \angle YAZ$

By side angle side postulate $\Delta ABC\cong \Delta AED$

Hence, the corresponding parts $\overline{XY}$ and $\overline{YZ}$ of congruent triangles $\Delta XAZ$ and $\Delta YAZ$ are congruent.

Hence $\Delta XYZ$ is an isosceles triangle.