Question

in the figure above, ab, bc, cd, de, and ea are line - segments. what is the value of x + y? the table above shows the contents of two types of assortment packs that a hardware store sells. sam and ali want to buy a number of the assortment packs so that they can split the contents according to each of their needs. sam needs at least 540 bolts and ali needs at least 450 washers. if they buy 10 regular assortment packs, what is the least number of deluxe assortment packs they can buy to satisfy their requirements? lianna plays a card game in which she gets 5 points for every red card she draws but loses 3 points for every black card she draws. by the end of the game, lianna drew 7 times as many red cards as black cards for a final total of 75 points. how many red cards did she draw during the game?

Explanation:

Step1: Find the sum of interior angles of the pentagon in the star - like figure.

The sum of interior angles of a pentagon is $(5 - 2)\times180^{\circ}=540^{\circ}$.

Step2: Calculate the sum of the known angles in the pentagon.

The known angles are $40^{\circ},100^{\circ},30^{\circ}$. Let the other two angles be $a$ and $b$. So $40 + 100+30+a + b=540$. Then $a + b=540-(40 + 100+30)=370^{\circ}$.

Step3: Use the property of linear - pair angles.

We know that $x$ and $y$ are related to the angles of the pentagon. The non - star angles of the pentagon and the angles $x$ and $y$ form linear pairs. If we consider the angles around the intersection points, we can use the fact that the sum of angles around a point is $360^{\circ}$. But an easier way is to use the angle - sum property of the pentagon.
The sum of the angles of the pentagon can also be thought of in terms of the angles of the star.
We know that the sum of the angles at the vertices of a star (formed by intersecting lines) can be found as follows:
The sum of the angles at the vertices of a star formed by a pentagram is $180^{\circ}$.
Let's use another approach.
We know that for the given figure, we can use the property of angles formed by intersecting lines.
We know that the sum of angles in a triangle formed by the lines of the star:
Let's consider one of the triangles formed by the lines of the star.
We know that the sum of angles in a triangle is $180^{\circ}$.
We can find that $x + y=180-(40 + 30)=110^{\circ}$.

Answer:

$110$