Question: $AB$, $BC$ and $CD$ are adjacent sides of a regular polygon, and $\angle CAB = 10°$. Find the exterior angle of the polygon, the number of sides $n$, and $\angle ACD$.

[diagram in original sheet: three adjacent sides of the polygon with diagonal AC drawn]

Step 1. $AB = BC$ (sides of a regular polygon), so $\triangle ABC$ is isosceles:

$\angle ABC$ is an interior angle of the polygon.

Step 2. Interior and exterior angles sit on a straight line:

Step 3. All $n$ exterior angles of any polygon sum to $360°$, and in a regular polygon they are equal:

Step 4. $\angle BCD$ is also an interior angle ($160°$), and $\angle BCA = 10°$ lies inside it:

⚠ Watch out: To find $n$, always convert to the exterior angle first and divide into $360°$ — solving $\frac{(n-2)180}{n} = 160$ works too but takes three times as long. The isosceles step needs regular: equal sides is what makes the two base angles $10°$.

Question: $ABCDE$ is a pentagon with interior angles $x°$, $2x°$, $3x°$, $3x°$ and $99°$. Find the value of $x$.

Step 1. The angle sum of any pentagon (regular or not) comes from $(n-2) \times 180°$:

Step 2. Form an equation — all five interior angles together make $540°$:

Step 3. Collect the $x$ terms:

Step 4. Solve:

Check. The five angles are $49°$, $98°$, $147°$, $147°$ and $99°$.

Sum: $49 + 98 + 147 + 147 + 99 = 540$ ✓ — and each one is less than $180°$, as interior angles of an ordinary (convex) pentagon should be.

The same equation-building works for any polygon: count the sides, write the sum, add the angles.

⚠ Watch out: The formula is $(n-2) \times 180°$, not $n \times 180°$ — a pentagon sums to $540°$, not $900°$. This sum works for any pentagon; only the $\frac{540°}{5}$ = equal-angles step would need it to be regular, and this pentagon is not.

Question: $ABCDE$ is a regular pentagon and $DEF$ is a straight line. Find the interior angle and the exterior angle of the pentagon.

[diagram in original sheet: regular pentagon with side DE extended to F]

Step 1. Sum of interior angles of an $n$-sided polygon is $(n-2) \times 180°$. A pentagon has $n = 5$:

Step 2. Regular means all interior angles are equal, so each one is

Step 3. The exterior angle $\angle AEF$ sits on the straight line $DEF$ with the interior angle:

Step 4. Cross-check with the exterior-angle shortcut — the exterior angles of any polygon sum to $360°$, so each exterior angle of a regular pentagon is

Both routes agree; use whichever the question makes shorter (for angles, the $360°/n$ route is usually faster).

⚠ Watch out: $(n-2) \times 180°$ is the sum — dividing by $n$ needs the polygon to be regular; an irregular pentagon still sums to $540°$ but its individual angles differ. And exterior angles always total $360°$, no matter how many sides.