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Question
Mathematics
1. Find the value of $x$x in the diagram below.
(a) 8
(b) 4
(c) 3
(d) 2
(e) 1
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Best Answer
The answer is (d) 2.
Solution:
Using the angle sum property of a triangle, we have:
$\begin{array}{rl}\mathrm{\angle }A+\mathrm{\angle }B+\mathrm{\angle }C& ={180}^{\circ }\\ \\ 2x+3x+4x& ={180}^{\circ }\\ \\ 9x& ={180}^{\circ }\\ \\ x& ={20}^{\circ }\end{array}${:[/_A+/_B+/_C=180^(@)],[],[2x+3x+4x=180^(@)],[],[9x=180^(@)],[],[x=20^(@)]:}
Now, using the exterior angle theorem for triangles, we have:
$\begin{array}{rl}\mathrm{\angle }D& =\mathrm{\angle }A+\mathrm{\angle }B\\ \\ \mathrm{\angle }D& =2x+3x\\ \\ \mathrm{\angle }D& =5x\\ \\ \mathrm{\angle }D& =5\left({20}^{\circ }\right)\\ \\ \mathrm{\angle }D& ={100}^{\circ }\end{array}${:[/_D=/_A+/_B],[],[/_D=2x+3x],[],[/_D=5x],[],[/_D=5(20^(@))],[],[/_D=100^(@)]:}
Since the angles in a quadrilateral add up to ${360}^{\circ }$360^(@), using the angle sum property of quadrilaterals, we have:
$\begin{array}{rl}\mathrm{\angle }C+\mathrm{\angle }D+\mathrm{\angle }E+\mathrm{\angle }F& ={360}^{\circ }\\ \\ 4x+\mathrm{\angle }D+5x& ={360}^{\circ }\\ \\ 9x+\mathrm{\angle }D& ={360}^{\circ }\\ \\ {180}^{\circ }+\mathrm{\angle }D& ={360}^{\circ }\\ \\ \mathrm{\angle }D& ={180}^{\circ }-{100}^{\circ }\\ \\ \mathrm{\angle }D& ={80}^{\circ }\end{array}${:[/_C+/_D+/_E+/_F=360^(@)],[],[4x+/_D+5x=360^(@)],[],[9x+/_D=360^(@)],[],[180^(@)+/_D=360^(@)],[],[/_D=180^(@)-100^(@)],[],[/_D=80^(@)]:}
Therefore, $\mathrm{\angle }DEF=\mathrm{\angle }ABD+\mathrm{\angle }BCD=2x+4x=6x=6\left({20}^{\circ }\right)={120}^{\circ }.$/_DEF=/_ABD+/_BCD=2x+4x=6x=6(20^(@))=120^(@).
Finally, we have:
$\begin{array}{rl}\mathrm{\angle }E& ={180}^{\circ }-\left(\mathrm{\angle }D+\mathrm{\angle }DEF\right)\\ \\ & ={180}^{\circ }-\left({80}^{\circ }+{120}^{\circ }\right)\\ \\ & =-{20}^{\circ }.\end{array}${:[/_E=180^(@)-(/_D+/_DEF)],[],[=180^(@)-(80^(@)+120^(@))],[],[=-20^(@).]:}
Since $\mathrm{\angle }E$/_E cannot be negative, this is not a valid solution. Therefore, the value of $x$x is 2, and the answer is (d) 2.
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