### Angle between two faces in a pyramid with rectangular base

Question Sample Titled 'Angle between two faces in a pyramid with rectangular base'

In the figure, ${V}{A}{B}{C}{D}$ is a pyramid with rectangular base. ${V}{A}{D}$ and ${V}{B}{C}$ are two isoceles triangles with ${V}{A}={V}{D}$ and ${V}{B}={V}{C}$ . Given that ${V}{D}={72}$ $\text{m}$ , ${V}{B}={60}$ $\text{m}$ , ${B}{C}={42}$ $\text{m}$ and ${C}{D}={57}$ $\text{m}$ . Find the angle between ${V}{A}{D}$ and ${V}{B}{C}$ correct to the nearest ${0.1}^{\circ}$ .

${B}$${C}$${D}$${V}$${A}$${42}$ $\text{m}$${57}$ $\text{m}$${60}$ $\text{m}$${72}$ $\text{m}$
A
${53.1}^{\circ}$
B
${66.4}^{\circ}$
C
${50.2}^{\circ}$
D
${68.1}^{\circ}$

 Add points ${H}$ and ${K}$ on ${A}{D}$ and ${B}{C}$ respectively such that ${V}{H}\bot{A}{D}$ and ${V}{K}\bot{B}{C}$ .

${B}$${C}$${D}$${V}$${A}$${42}$ $\text{m}$${57}$ $\text{m}$${60}$ $\text{m}$${72}$ $\text{m}$${H}$${K}$

 Note that the required angle is $\angle{H}{V}{K}$ . In $\triangle{V}{K}{B}$ , ${V}{B}^{{2}}$ $={V}{K}^{{2}}+{B}{K}^{{2}}$ Pyth. theorem ${V}{K}$ $=\sqrt{{{60}^{{2}}-{\left(\dfrac{{42}}{{2}}\right)}^{{2}}}}$  $={9}\sqrt{{{39}}}$ Similarly, ${V}{H}$ $={3}\sqrt{{{527}}}$ In $\triangle{H}{V}{K}$ , by the cosine formula, ${\cos}\angle{H}{V}{K}$ $=\dfrac{{{V}{H}^{{2}}+{V}{K}^{{2}}-{H}{K}^{{2}}}}{{{2}\cdot{V}{H}\cdot{V}{K}}}$  $=\dfrac{{{\left({9}\sqrt{{{39}}}\right)}^{{2}}+{\left({3}\sqrt{{{527}}}\right)}^{{2}}-{57}^{{2}}}}{{{2}\cdot{9}\sqrt{{{39}}}\cdot{3}\sqrt{{{527}}}}}$ $\angle{H}{V}{K}$ $={53.055749566171045}^{\circ}$

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