Chapter 4.3, Problem 12WE

To determine

To find: The relation between $\angle 1,\angle 2,\angle 3$ and $\angle 4.$

$\angle 1=\angle 2$ and $\angle 3=\angle 4$

Given information

$\begin{array}{l}\overline{\text{WX}}\perp \overline{\text{UV}}\\ \overline{\text{WX}}\perp \overline{\text{YZ}}\\ \overline{\text{WU}}\cong \overline{\text{WV}}\end{array}$

Calculations:

As, given, $\overline{\text{WX}}\perp \overline{\text{UV}}$ as well as $\overline{\text{WX}}\perp \overline{\text{YZ}}$

So, $\overline{\text{UV}}$ must be parallel to $\overline{\text{YZ}}$

⇒$\overline{\text{UV}}\left|\right|\overline{\text{YZ}}$.....(i)

In, $\Delta \text{UWX}$ and $\Delta \text{WXV}$

$\overline{\text{UW}}\cong \overline{\text{WV}}$   (given)

$\overline{\text{WX}}\cong \overline{\text{XW}}$   (Reflexive property of congruence)

$\angle \text{UWX}\cong \angle \text{VWX}=90°$ (as $\overline{\text{WX}}\perp \overline{\text{UV}}$ )

So, by SAS congruency,

$\Delta \text{UWX}\cong \Delta \text{WVX}$

Therefore, by congruent part of congruent triangles,

we have,

$\angle 3=\angle 4$.....(ii)

and,

$\angle \text{WUX}\cong \angle \text{WVX}$.....(iii)

Since,

$\overline{\text{UV}}\left|\right|\overline{\text{YZ}}$ [from (i)]

⇒$\angle \text{WUX}\cong \angle \text{1}$ (alternate interior angles)

and,

$\angle \text{WVX}\cong \angle \text{2}$ (alternate interior angles)

Therefore, $\angle 1=\angle 2$.....(iv)

Conclusion

$\angle 1=\angle 2$ and $\angle 3=\angle 4$

Answer to Problem 12WE

$\angle 1=\angle 2$ and $\angle 3=\angle 4$

Explanation of Solution

Given information

$\begin{array}{l}\overline{\text{WX}}\perp \overline{\text{UV}}\\ \overline{\text{WX}}\perp \overline{\text{YZ}}\\ \overline{\text{WU}}\cong \overline{\text{WV}}\end{array}$

Calculations:

As, given, $\overline{\text{WX}}\perp \overline{\text{UV}}$ as well as $\overline{\text{WX}}\perp \overline{\text{YZ}}$

So, $\overline{\text{UV}}$ must be parallel to $\overline{\text{YZ}}$

⇒$\overline{\text{UV}}\left|\right|\overline{\text{YZ}}$.....(i)

In, $\Delta \text{UWX}$ and $\Delta \text{WXV}$

$\overline{\text{UW}}\cong \overline{\text{WV}}$   (given)

$\overline{\text{WX}}\cong \overline{\text{XW}}$   (Reflexive property of congruence)

$\angle \text{UWX}\cong \angle \text{VWX}=90°$ (as $\overline{\text{WX}}\perp \overline{\text{UV}}$ )

So, by SAS congruency,

$\Delta \text{UWX}\cong \Delta \text{WVX}$

Therefore, by congruent part of congruent triangles,

we have,

$\angle 3=\angle 4$.....(ii)

and,

$\angle \text{WUX}\cong \angle \text{WVX}$.....(iii)

Since,

$\overline{\text{UV}}\left|\right|\overline{\text{YZ}}$ [from (i)]

⇒$\angle \text{WUX}\cong \angle \text{1}$ (alternate interior angles)

and,

$\angle \text{WVX}\cong \angle \text{2}$ (alternate interior angles)

Therefore, $\angle 1=\angle 2$.....(iv)

Conclusion

$\angle 1=\angle 2$ and $\angle 3=\angle 4$