Math / Geometry

QuestionCalculate the size of angle $x$ .

Studdy Solution

STEP 1

1. The sum of the interior angles of a triangle is $180^\circ$ .
2. The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

STEP 2

1. Use the exterior angle theorem.
2. Set up an equation for the interior angles.
3. Solve for $x$ .

STEP 3

According to the exterior angle theorem, the exterior angle is equal to the sum of the two opposite interior angles. Here, the exterior angle is $20^\circ$ , and the opposite interior angles are $5x$ and $100^\circ$ . Therefore, we have:
$20^\circ = 5x + 100^\circ$

STEP 4

Now, we need to set up an equation using the sum of the interior angles of the triangle. The angles inside the triangle are $x$ , $5x$ , and $100^\circ$ . Their sum is:
$x + 5x + 100^\circ = 180^\circ$

STEP 5

Solve the equation from Step 2 for $x$ :
$x + 5x + 100^\circ = 180^\circ$ $6x + 100^\circ = 180^\circ$
Subtract $100^\circ$ from both sides:
$6x = 80^\circ$
Divide both sides by 6:
$x = \frac{80^\circ}{6}$ $x = 13.33^\circ$
The size of angle $x$ is:
$\boxed{13.33^\circ}$

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