Question

in diagram 1, klm pu is a regular pentagon and mpq is a straight line.
find the value of x.

Explanation:

Step1: Find interior angle of pentagon

First, calculate the interior angle of a regular pentagon. The formula for the interior angle of an n-sided polygon is $\frac{(n-2)\times180^\circ}{n}$. For $n=5$:
$$\frac{(5-2)\times180^\circ}{5}=108^\circ$$
So $\angle UPM = 108^\circ$.

Step2: Calculate reflex angle at U

We know $\angle TUP = 120^\circ$, so the reflex angle at U is:
$$360^\circ - 120^\circ = 240^\circ$$

Step3: Sum angles of polygon RTUPQ

We use the polygon angle sum formula for a 5-sided polygon RTUPQ: $\text{Total interior angles}=(5-2)\times180^\circ=540^\circ$. We also know $\angle T=40^\circ$, $\angle R=70^\circ$, reflex $\angle U=240^\circ$, $\angle UPQ=180^\circ - 108^\circ=72^\circ$. Let $\angle X$ be the exterior angle, so the interior angle at Q is $180^\circ - x^\circ$. Set up the equation:
$$40^\circ + 70^\circ + 240^\circ + 72^\circ + (180^\circ - x^\circ) = 540^\circ$$

Step4: Solve for x

Simplify the left side:
$$40+70+240+72+180 - x = 540$$
$$602 - x = 540$$
$$x = 602 - 540 = 62$$

Answer:

$62^\circ$