Angles

Polygons Properties of polygons, interior angles of polygons including triangles, quadrilaterals, pentagons, hexagons, heptagons, octagons, nonagons, decagons, 11-gons, dodecagons

Polygons:  Properties of Triangles Sum of the Angles of a Triangle: The sum of the angles of a triangle is 180 degrees. ALWAYS!   Regular Triangles - Equilateral Triangles: The properties of equilateral triangles: All sides are the same length (congruent) and all interior angles are the same size (congruent). To find the measure of the interior angles, we know that the sum of all the angles is 180 degrees (from above)...  And there are three angles... So, the measure of the interior angle of an equilateral triangle is 60 degrees. The measure of the central angles of an equilateral triangle: To find the measure of the central angle of an equilateral triangle, make a circle in the middle...  A circle is 360 degrees around...  Divide that by three angles... So, the measure of the central angle of an equilateral triangle is 120 degrees.

Polygons:  Properties of Quadrilaterals Sum of the Interior Angles of Quadrilaterals: To find the sum of the interior angles of a quadrilaterals, divide it up into triangles... There are two triangles...  Because the sum of the angles of each triangle is 180 degrees...  We get So, the sum of the interior angles of a quadrilateral is 360 degrees. Regular Quadrilaterals - Squares: The properties of squares: All sides are the same length (congruent) and all interior angles are the same size (congruent). To find the measure of the interior angles, we know that the sum of all the angles is 360 degrees (from above)...  And there are four angles... So, the measure of the interior angle of a square is 90 degrees. The measure of the central angles of a square: To find the measure of the central angle of a square, make a circle in the middle...  A circle is 360 degrees around...  Divide that by four angles... So, the measure of the central angle of a square is 90 degrees. Polygons:  Properties of Pentagons Sum of the Interior Angles of a Pentagon: To find the sum of the interior angles of a pentagon, divide it up into triangles... There are three triangles...  Because the sum of the angles of each triangle is 180 degrees...  We get So, the sum of the interior angles of a pentagon is 540 degrees. Regular Pentagons: The properties of regular pentagons: All sides are the same length (congruent) and all interior angles are the same size (congruent). To find the measure of the interior angles, we know that the sum of all the angles is 540 degrees (from above)...  And there are five angles... So, the measure of the interior angle of a regular pentagon is 108 degrees. The measure of the central angles of a regular pentagon: To find the measure of the central angle of a regular pentagon, make a circle in the middle...  A circle is 360 degrees around...  Divide that by five angles... So, the measure of the central angle of a regular pentagon is 72 degrees. Polygons:  Properties of Hexagons Sum of the Interior Angles of a Hexagon: To find the sum of the interior angles of a hexagon, divide it up into triangles... There are four triangles...  Because the sum of the angles of each triangle is 180 degrees...  We get So, the sum of the interior angles of a hexagon is 720 degrees. Regular Hexagons: The properties of regular hexagons: All sides are the same length (congruent) and all interior angles are the same size (congruent). To find the measure of the interior angles, we know that the sum of all the angles is 720 degrees (from above)...  And there are six angles... So, the measure of the interior angle of a regular hexagon is 120 degrees. The measure of the central angles of a regular hexagon: To find the measure of the central angle of a regular hexagon, make a circle in the middle...  A circle is 360 degrees around...  Divide that by six angles... So, the measure of the central angle of a regular hexagon is 60 degrees. Polygons:  Properties of Heptagons Sum of the Interior Angles of a Heptagon: Using the same methods as above (I'll let you do the pictures)... To find the sum of the interior angles of a heptagon, divide it up into triangles... There are five triangles...  Because the sum of the angles of each triangle is 180 degrees...  We get So, the sum of the interior angles of a heptagon is 900 degrees. Regular Heptagons: The properties of regular heptagons: All sides are the same length (congruent) and all interior angles are the same size (congruent). To find the measure of the interior angles, we know that the sum of all the angles is 900 degrees (from above)...  And there are seven angles... So, the measure of the interior angle of a regular heptagon is about 128.57 degrees. The measure of the central angles of a regular heptagon: To find the measure of the central angle of a regular heptagon, make a circle in the middle (I'll let you do the picture)...  A circle is 360 degrees around...  Divide that by seven angles... So, the measure of the central angle of a regular heptagon is about 51.43 degrees. Polygons:  Properties of Octagons Sum of the Interior Angles of an Octagon: Using the same methods as above (I'll let you do the pictures)... To find the sum of the interior angles of an octagon, divide it up into triangles... There are six triangles...  Because the sum of the angles of each triangle is 180 degrees...  We get So, the sum of the interior angles of an octagon is 1080 degrees. Regular Octagons: The properties of regular octagons: All sides are the same length (congruent) and all interior angles are the same size (congruent). To find the measure of the angles, we know that the sum of all the angles is 1080 degrees (from above)...  And there are eight angles... So, the measure of the interior angle of a regular octagon is 135 degrees. The measure of the central angles of a regular octagon: To find the measure of the central angle of a regular octagon, make a circle in the middle (I'll let you do the picture)...  A circle is 360 degrees around...  Divide that by eight angles... So, the measure of the central angle of a regular octagon is 45 degrees. Polygons:  Properties of Nonagons Sum of the Interior Angles of a Nonagon: Using the same methods as above (I'll let you do the pictures)... To find the sum of the interior angles of a nonagon, divide it up into triangles... There are seven triangles...  Because the sum of the angles of each triangle is 180 degrees...  We get So, the sum of the interior angles of a nonagon is 1260 degrees. Regular Nonagons: The properties of regular nonagons: All sides are the same length (congruent) and all interior angles are the same size (congruent). To find the measure of the angles, we know that the sum of all the angles is 1260 degrees (from above)...  And there are nine angles... So, the measure of the angle of a regular nonagon is 140 degrees. The measure of the central angles of a regular nonagon: To find the measure of the central angle of a regular nonagon, make a circle in the middle (I'll let you do the picture)...  A circle is 360 degrees around...  Divide that by nine angles... So, the measure of the central angle of a regular nonagon is 40 degrees. Polygons:  Properties of Decagons Sum of the Interior Angles of a Decagon: Using the same methods as above (I'll let you do the pictures)... To find the sum of the interior angles of a decagon, divide it up into triangles... There are eight triangles...  Because the sum of the angles of each triangle is 180 degrees...  We get So, the sum of the interior angles of a decagon is 1440 degrees. Regular Decagons: The properties of regular decagons: All sides are the same length (congruent) and all interior angles are the same size (congruent). To find the measure of the angles, we know that the sum of all the angles is 1440 degrees (from above)...  And there are ten angles... So, the measure of the angle of a regular decagon is 144 degrees. The measure of the central angles of a regular decagon: To find the measure of the central angle of a regular decagon, make a circle in the middle (I'll let you do the picture)...  A circle is 360 degrees around...  Divide that by ten angles... So, the measure of the central angle of a regular decagon is 36 degrees. Polygons:  Properties of 11-gons Sum of the Interior Angles of an 11-gon: Using the same methods as above (I'll let you do the pictures)... To find the sum of the interior angles of an 11-gon, divide it up into triangles... There are nine triangles...  Because the sum of the angles of each triangle is 180 degrees...  We get So, the sum of the interior angles of an 11-gon is 1620 degrees. Regular 11-gons: The properties of regular 11-gons: All sides are the same length (congruent) and all interior angles are the same size (congruent). To find the measure of the angles, we know that the sum of all the angles is 1620 degrees (from above)...  And there are eleven angles... So, the measure of the interior angle of a regular 11-gon is about 147.27 degrees. The measure of the central angles of a regular 11-gon: To find the measure of the central angle of a regular 11-gon, make a circle in the middle (I'll let you do the picture)...  A circle is 360 degrees around...  Divide that by eleven angles... So, the measure of the central angle of a regular 11-gon is about 32.73 degrees. Polygons:  Properties of Dodecagons Sum of the Interior Angles of a Dodecagon: Using the same methods as above (I'll let you do the pictures)... To find the sum of the interior angles of a dodecagon, divide it up into triangles... There are ten triangles...  Because the sum of the angles of each triangle is 180 degrees...  We get So, the sum of the interior angles of a dodecagon is 1800 degrees. Regular Dodecagons: The properties of regular dodecagons: All sides are the same length (congruent) and all interior angles are the same size (congruent). To find the measure of the angles, we know that the sum of all the angles is 1800 degrees (from above)...  And there are twelve angles... So, the measure of the interior angle of a regular dodecagon is 150 degrees. The measure of the central angles of a regular dodecagon: To find the measure of the central angle of a regular dodecagon, make a circle in the middle (I'll let you do the picture)...  A circle is 360 degrees around...  Divide that by twelve angles... So, the measure of the central angle of a regular dodecagon is 30 degrees.

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