Math / Geometry

QuestionFind the value of $x$ . a) the sum of interior quadrilateral is 360. $x=$ $\qquad$

Studdy Solution

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 $x$ : $2x + 180 = 360$ $2x = 360 - 180$ $2x = 180$ $x = \frac{180}{2}$ $x = 90$

STEP 11

The value of $x$ is $90$ .

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