Question

MN, OP, and QR are three lines intersecting at the same point A, as shown in the figure below. Find the value of 5a, if x = 3a, y = 5x, and z = 6x.

• A. $5^{\circ }$
• B. ${10}^{\circ }$
• C. ${12}^{\circ }$
• D. ${25}^{\circ }$

Answer 1

The correct option is D ${25}^{\circ }$

The lines QR and MN are intersecting at point A.
So, ∠QAN = ∠MAR = z
[Vertically opposite angles]

Since, OP is a line, so, sum of all angles on the line is ${180}^o.$
So, $\angle MAO+\angle MAR+\angle RAP={180}^o$
Now, according to the question,

$\angle MAO+\angle MAR+\angle RAP={180}^o$
$\Rightarrow y+z+x={180}^o$
$\Rightarrow 5x+6x+3a={180}^o$
[Given: y = 5x, z = 6x, and x = 3a]
$\Rightarrow 5\left(3a\right)+6\left(3a\right)+3a={180}^o$
$\Rightarrow 15a+18a+3a={180}^o$
$\Rightarrow 36a={180}^o$
$\Rightarrow a=5^o$
Hence, $5a=5\times 5^o={25}^o$