Chapter 6.6, Problem 1CYP

(a)

To determine

To calculate: the measure of $\angle XWZ$ .

Answer to Problem 1CYP

  $\angle XWZ=45$

Explanation of Solution

Given: WXYZ is an isosceles trapezoid and $m\angle YZW=45$ , $WV=15cm$ and $VY=10cm$

Formula Used:

Isosceles trapezoid is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides.

Calculation:

Consider an isosceles trapezoid WXYZ as follows:

Since WXYZ is an isosceles trapezoid,

Thus, $△XWZ\cong △YZW$

Therefore, $\angle XWZ=\angle YZW$ (since corresponding angles of congruent triangles are equal)

$\angle XWZ=45$

Conclusion:

Hence, $\angle XWZ=45$

(b)

To determine

To calculate: the measure of $\angle WXY$ .

Answer to Problem 1CYP

  $\angle WXY=135$

Explanation of Solution

Given: WXYZ is an isosceles trapezoid and $m\angle YZW=45$ , $WV=15cm$ and $VY=10cm$

Formula Used:

Isosceles trapezoid is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides

Calculation:

Consider an isosceles trapezoid WXYZ as follows:

Since WXYZ is an isosceles trapezoid,

Thus, $XY$ is parallel to $WZ$

Therefore, $\angle XWZ=\angle YZW$ (consecutive interior angles are supplementary)

$\begin{array}{l}\angle XWZ+\angle WXY=180\\ 45+\angle WXY=180\\ \angle WXY=180-45\\ \angle WXY=135\end{array}$

Conclusion:

Hence, $\angle WXY=135°$

(c)

To determine

To calculate: the measure of $XZ$ .

Answer to Problem 1CYP

  $XZ=25cm$

Explanation of Solution

Given: WXYZ is an isosceles trapezoid and $m\angle YWZ=45$ , $WV=15cm$ and $VY=10cm$

Formula Used:

Isosceles trapezoid is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides

Calculation:

Consider an isosceles trapezoid WXYZ as follows:

Since WXYZ is an isosceles trapezoid,

Thus, $△XWZ\cong △YZW$

Therefore, $XZ=YW$ (since corresponding sides of congruent triangles are equal)

$\begin{array}{l}XZ=WV+VY\\ XZ=15+10\\ XZ=25cm\end{array}$

Conclusion:

Hence, $XZ=25cm$ .

(d)

To determine

To calculate: the measure of $XV$ .

Answer to Problem 1CYP

  $XV=10cm$

Explanation of Solution

Given: WXYZ is an isosceles trapezoid and $m\angle YWZ=45$ , $WV=15cm$ and $VY=10cm$

Formula Used:

Isosceles trapezoid is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides

Calculation:

Consider an isosceles trapezoid WXYZ as follows:

Since WXYZ is an isosceles trapezoid,

Thus, $△XZY\cong △YWX$

Therefore, $\angle ZXY=\angle WYX$ (since corresponding angles of congruent triangles are equal)

Now, consider triangle XYV

Since $\angle ZXY=\angle WYX$

Thus, $XV=YV$ (sides opposite to equal angles are equal)

Hence, $XV=10cm$

Conclusion:

Hence, $XV=10cm$