Chapter 6.2, Problem 19PSC

To determine

To prove: The triangle SZW is isosceles

Explanation of Solution

Given information:

Given line is perpendicular to both the given planes, i.e., $\overleftrightarrow{\text{SX}}\perp \text{m}$ and $\overleftrightarrow{\text{SX}}\perp \text{n}$

Two given line segments are congruent, i.e., $\overline{\text{TP}}\cong \overline{\text{TR}}$

Formula used:

Congruency means same shape and same size

Congruent Parts of Congruent Triangles are Congruent Reflexive property

If a line is perpendicular to plane than all line passing through its foot will also perpendicular to it

Perpendicular lines always form right angles

Proof:

  $\overleftrightarrow{\text{SX}}\perp \text{m}$ and $\overleftrightarrow{\text{SX}}\perp \text{n}$ ........... given

  $\overline{\text{SX}}\perp \overline{\text{SZ}}\text{ \& }\overline{\text{SX}}\perp \overline{\text{SW}}$ ............ If a line is perpendicular to plane than all line passing through its foot will also perpendicular to it

  $\angle \text{SXZ}=\angle \text{SWZ}=90°$ ........... Perpendicular lines from right angles

  $\overline{\text{XZ}}\cong \overline{\text{XW}}$ ........... reflexive property

  $\Delta \text{SXZ}\cong \Delta \text{SWZ}$ ............ SAS criteria

  $\overline{\text{SZ}}\cong \overline{\text{SW}}$ ............ Congruent Parts of Congruent Triangles are Congruent Reflexive property

  $\Delta \text{SZW is a isosceles }$

Hence proved