Chapter exercises 9 — Polygons

Determine, preferably without using the formula, the sum of the internal angles of a: a) convex pentagon; b) convex hexagon.

Determine the value of $x$ in cases (in and), $\overline{AB} \parallel \overline{ED}$):

In the cases below, determine $x$, knowing that the segments $\overline{AP}$, $\overline{BP}$, $\overline{CP}$ and $\overline{DP}$, in the figures in which they appear, are angle bisectors.

Being $\overline{AP}$ and $\overline{CP}$ angle bisectors from $\hat{A}$ and $\hat{C}$, determine $x$. In a), $\overline{AB} \parallel \overline{PC}$ and $\overline{AP} \parallel \overline{BC}$; in b), $\overline{AB} \parallel \overline{PC}$.

Determine the internal angle and external angle of one: a) equilateral triangle; b) square; c) regular pentagon; d) regular hexagon.

If the triangle $ABP$ is equilateral and $ABCDE$ is a regular pentagon, determine $x$ in the cases:

Determine the values ​​of $x$ and $y$ in the cases: a) regular pentagon and square; b) regular and square hexagon.

Calculate the sum of the interior angles of a nonagon.

Calculate the sum of the interior angles of a decagon.

Calculate the sum of the interior angles of an icosagon.

What is the polygon whose interior angles add up to $1\,800^{\circ}$?

Calculate the number of diagonals in a decagon.

Calculate the number of diagonals of an icosagon.

Find the polygon whose number of diagonals is three times the number of sides.

Determine the polygon whose number of diagonals is four times the number of sides.

Determine the polygon that has 9 distinct diagonals.

Determine the largest angle of a pentagon whose interior angles are in the ratio $3 : 3 : 3 : 4 : 5$.

A regular polygon has, from one of its vertices, as many diagonals as there are diagonals of a hexagon. Find: a) the polygon; b) the total number of diagonals; c) the sum of the internal angles; d) the sum of the external angles; e) the measure of each internal angle and each external angle.

How many diagonals can we draw starting from a vertex of a convex polygon with 20 sides?

Determine the number of sides of a convex polygon, knowing that diagonals depart from one of its vertices 25.

Determine the convex polygon whose sum of internal angles is equal to the number of diagonals multiplied by $180^{\circ}$.

Can the internal and external angles of a regular polygon have equal measurements? In what case does this occur?

Determine the number of diagonals of a regular convex polygon whose exterior angle is $24^{\circ}$.

The ratio of the interior angle to the exterior angle of a regular polygon is 9. Determine the number of sides of the polygon.

The internal angle of a regular polygon is 1,5 times its external angle. Determine the number of sides of the polygon.

The exterior angle of a regular polygon is equal to twice its interior angle. Determine the number of diagonals in this polygon.

The sum of the internal angles and the external angles of a regular polygon is $1\,800^{\circ}$. Determine the number of diagonals in the polygon.

Determine the number of sides of a regular convex polygon whose interior angle is five times the exterior.

Determine the number of sides of a regular polygon $ABCDE\ldots$, knowing that the angle bisectors $\overline{AP}$ and $\overline{CP}$ of the angles $\hat{A}$ and $\hat{C}$ form an angle that is worth $\dfrac{2}{9}$ of its internal angle.

Determine the measure of the angle formed by the extensions of the sides $\overline{AB}$ and $\overline{CD}$ of a regular polygon $ABCD\ldots$ with 20 sides.

The perpendicular bisectors of two consecutive sides of a regular polygon form an angle of $24^{\circ}$. Determine the number of diagonals in this polygon.

By increasing the number of sides of a polygon by 3, its number of diagonals increases by 21. Determine the number of diagonals in this polygon.

In the figure below, determine the sum of the measures of the angles $\hat{a} + \hat{b} + \hat{c} + \hat{d} + \hat{e} + \hat{f}$.

Given two polygons with $n$ and $n + 6$ sides, respectively, calculate $n$, knowing that one of the polygons has 39 more diagonals than the other.

Three convex polygons have $n$, $n + 1$, and $n + 2$ sides, respectively. Since $2\,700^{\circ}$ is the sum of all the internal angles of the three polygons, determine the value of $n$.

The numbers that express the number of sides of three polygons are $n - 3$, $n$ and $n + 3$. Determine the number of diagonals of each of the polygons, knowing that the sum of all their internal angles is $3\,240^{\circ}$.

Three polygons have the number of sides expressed by consecutive integers. Knowing that the total number of diagonals of the three polygons is equal to 28, determine the polygon with the greatest number of diagonals.

Two convex polygons have the number of sides expressed by the numbers $n$ and $n + 4$. Determine the value of $n$, knowing that one of the polygons has 34 more diagonals than the other.

A convex polygon has 5 more sides than another. Knowing that the total number of diagonals is 68, determine the number of diagonals of each polygon.

Given two regular polygons, with $n + 1$ sides and $n$ sides, respectively, determine $n$, knowing that the interior angle of the first polygon exceeds the interior angle of the second by $5^{\circ}$.

A regular polygon has 30 diagonals that do not pass through its center. How long is each internal angle?

What is the regular polygon that has 6 diagonals passing through its center?

A regular polygon has 170 diagonals. How many pass through the center?

The interior angle of a regular polygon measures $140^{\circ}$. How many diagonals pass through the center?