Math Snap

STEP 1

What is this asking?
We need to find the value of $x$ in a quadrilateral that includes a triangle with a $60^\circ$ angle, knowing that the sum of interior angles in a quadrilateral is $360^\circ$.
Watch out!
Don't forget that the triangle and the quadrilateral share a side, and that the angles of a triangle add up to $180^\circ$.

STEP 2

1. Find the missing angle in the triangle.
2. Find the total degrees represented by the x angles.
3. Calculate x.

STEP 3

We know two angles in the triangle: $60^\circ$ and $x^\circ$.
Let's call the missing angle $y^\circ$.
Since the sum of angles in a triangle is $180^\circ$, we can write:
$$60 + x + y = 180$$

STEP 4

Notice that the missing angle ($y^\circ$) and the adjacent angle ($x^\circ$) in the quadrilateral form a straight line.
This means they add up to $180^\circ$:
$$x + y = 180$$

STEP 5

Now, we can substitute this relationship into the triangle equation:
$$60 + (x + y) = 180$$ $$60 + 180 = 180$$This doesn't help us directly, so let's re-arrange the equation from 2.1.2. to solve for $y$:
$$y = 180 - x$$

STEP 6

Now, let's substitute this back into the triangle equation from 2.1.1.:
$$60 + x + (180 - x) = 180$$ $$240 = 180$$Oops, that's not right!
Let's try another approach.

STEP 7

Let's think about the quadrilateral.
We know its angles add up to $360^\circ$.
We have three angles represented in terms of $x$, and the remaining angle is adjacent to the triangle's missing angle ($y$).

STEP 8

We know from 2.1.2. that $x + y = 180$.
So, the angles in the quadrilateral are $x$, $x$, $x$, and $y$.
We can write this as:
$$x + x + x + y = 360$$ $$3x + y = 360$$

STEP 9

We also know from 2.1.3. that $y = 180 - x$.
Let's substitute this into our quadrilateral equation:
$$3x + (180 - x) = 360$$ $$2x + 180 = 360$$

STEP 10

Now we can solve for $x$:
$$2x + 180 = 360$$ $$2x = 360 - 180$$$$2x = 180$$$$x = \frac{180}{2}$$$$x = 90$$

The value of $x$ is $90$.