Question

In the figure above, $\overline{AB},\overline{CD}$ and $\overline{EF}$ intersect at $P$.
If $r={90}^o,s={50}^o,t={60}^o,u={45}^o$ and $w={50}^o$, what is the value of $x$ ?

• A. ${45}^o$
• B. ${50}^o$
• C. ${65}^o$
• D. ${75}^o$
• E. It cannot be determined from the information given

Answer 1

The correct option is C ${65}^o$

From the given figure,

Sum of the angles in a $△$$PBF$ $=$ $180$

$\angle BPF$ $+$ $s$ $+$ $t$ $=$ $180$

Given that $s$ $=$ $50$ and $t$ $=$ $60$,

$\angle BPF$ $+$ $50$ $+$ $60$ $=$ $180$

$\angle BPF$ $=$ $180$ $-$ $110$

$\angle BPF$ $=$ $70$

Sum of the angles in a $△$$PAD$ $=$ $180$

$\angle APD$ $+$ $w$ $+$ $u$ $=$ $180$

Given that $w$ $=$ $50$ and $u$ $=$ $45$,

$\angle APD$ $+$ $50$ $+$ $45$ $=$ $180$

$\angle APD$ $=$ $180$ $-$ $95$

$\angle APD$ $=$ $85$

Sum of the angles in a straight line $=$ $180$

$\angle BPF$ $+$ $\angle APD$ $+$ $\angle DPF$ $=$ $180$

$70$ $+$ $85$ $+$ $\angle DPF$ $=$ $180$

$\angle DPF$ $=$ $180$ $-$ $155$

$\angle DPF$ $=$ $25$

Vertically opposite angles are equal.

Here, $\angle DPF$ and $\angle EPC$ are vertically opposite angles.

$\angle DPF$ $=$ $\angle EPC$ $=$ $25$

Sum of the angles in a $△$$EPC$ $=$ $180$

$\angle EPC$ $+$ $r$ $+$ $x$ $=$ $180$

Given that $r$ $=$ $90$,

$25$ $+$ $90$ $+$ $x$ $=$ $180$

$x$ $=$ $180$ $-$ $115$

$x$ $=$ $65$

Therefore, the value of ${}^{\prime }{x}^{\prime }$ is ${}^{\prime }{65}^{\prime }$.