Construction of Rectangle 1
1.Segment AB
2.Perpendicular line to segment AB through point B.
3.New point C on the perpendiculr line.
4.Parallel line to segment AB through point C.
5.Perpendicular line to segment AB through point A.
6.Intersection point D
7.Polygon ABCD
8.Show interior angles the rectangle.
9.Showlength of the sides of the rectangle
10.Hide all the lines.
11.Save the construction.
Construction of an Equilateral Triangle 2
1.Segment AB
2.Circle with center A through B.
3.Circle with center B through A.
4.Intersect both the circles to get the point C.
5.Polygon ABC
6.Hide the circles.
7.Show interior angles the triangle.
8.Showlength of te sides of the triangle.
9.Save the construction.
Construction of Square 3
1.Segment AB
2.Perpendicular line to segment AB through point B.
3.Circle with center B through A.
4.Intersection point C of the circle and the perpendicular line.
5.Construct another perpendicular line to segment AB throuhg point A.
6.Circle with center A through B.
7.Intersection point D of the circle and the perpendicular line.
8.Polygon ABCD.
9.Hide the circles and the perpendicular lines.
10.Show interior angles the square.
11.Showlength of the sides of the square.
12.Save the construction.
Construction of Circumcircle of a Triangle 4
Draw a triangle and construct its circumcircle.
1.Using the Polygon tool construct triangle ABC.
2.Using Perpendicular bisector tool draw perpendicular bsectors to any two sides of the triangle.
3.Mark the intersecting point of the perpendicular bisectors which is the centre of the circumcrcle. ( Intersect Two Objects tool)
4.Choose the tool Circle with centre through point and click first at the centre, then at any vertex of the triangle.
5.Using the Move tool change the position of vertices.
6.Save the construction.
Construction of Circumcircle of a Triangle 4
Draw a triangle and construct its circumcircle.
1.Using the Polygon tool construct triangle ABC.
2.Using Perpendicular bisector tool draw perpendicular bsectors to any two sides of the triangle.
3.Mark the intersecting point of the perpendicular bisectors which is the centre of the circumcrcle. ( Intersect Two Objects tool)
4.Choose the tool Circle with centre through point and click first at the centre, then at any vertex of the triangle.
5.Using the Move tool change the position of vertices.
6.Save the construction.
Sum of the interior angles of a triangle
An angle is the amount of rotation of a ray. In Figure given below, the ray was rotated from A to C, and the amount of rotation is 60 degrees. We can say that the measure of angle ABC is 60 degrees.
An angle is the amount of rotation of a ray. In Figure given below, the ray was rotated from A to C, and the amount of rotation is 60 degrees. We can say that the measure of angle ABC is 60 degrees.
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The rotation from A to D forms a straight line and measures 180 degrees. Therefore, straight angle ABD measures 180 degrees. It is clear that a 180-degree rotation is a half-circle. Therefore, a complete rotation forming a circle is 360 degrees.
We can verify if our problem about the sum of the interior angles of a triangle by drawing a triangle on a paper, cutting the corners, meeting the corners (vertices) in one point such that the sides coincide with no gas and overlaps (see Figure 3). Notice that no matter what the size or shape of a triangle, as long as the previous condition is met, the two of its sides will be collinear as shown in the Figure below.
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However, this is not the proof. To discuss the proof, we are going to use Euclid's Fifth Postulate. Euclid’s Fifth Postulate tells us that if a parallel line is cut by a transversal, their corresponding angles are congruent. In the diagram below, lines p and q are parallel, and angles shown with the same names are corresponding angles, and hence congruent.
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Using the fifth postulate, we use on base of a triangle and extend it to form a line, and then draw a parallel line to the vertex which is not common to the base as shown in Figure given below. Now, angles a, b and c combined together form a straight angle and is therefore equal to 180 degrees. But a, b and c are also the interior angles of a polygon.
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Therefore, the sum of the interior angles of a triangle is 180 degrees.
Geogebra applet
Sum of the interior angles of a polygon
A polygon is a closed figure with finite number of sides. In the figures below, is a ABCDE polygon with 5 sides and (5 vertices) and ABE is a polygon with three sides (and three vertices). It is clear that the number of sides of a polygon is always equal to the number of its vertices.
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Any polygon maybe divided into triangles by drawing diagonals from one vertex to all of the non-adjacent vertices. In the second figure above, the pentagon was divided into three triangles by drawing diagonals from vertex E to the non-adjacent vertices B and C forming BE and CE. Now let ak , bk and ck, where k=1,2,3 be measures of the interior angles of the three triangles as shown on the second figure.
Calculating the angle sum of pentagon ABCDE we have
Notice that the angle measures in the first line of our equation is just a rearrangement of the measures of the interior angles of the three triangles. Hence, the angle sum of the pentagon is equal to the angle sum of the three triangles. Therefore, we can conclude that the sum of the interior angles of a polygon is equal to the angle sum of the number of triangles that can be formed by dividing it using the method described above. Using this conclusion, we will now relate the number of sides of a polygon, the number of triangles that can be formed by drawing diagonals and the polygon’s angle sum.
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From the table above, we observe that the number of triangles formed is 2 less than the number of sides of the polygon. This is true, because n-2 triangles can be formed by drawing diagonals from one of the vertices to n-3 non-adjacent vertices. Therefore, there the angle sum m of a polygon with n sides is given by the formula
m = 180(n-2)
A More Formal Proof
Theorem: The sum of the interior angles of a polygon with n sides is 180(n-2) degrees.
Proof:
Assume a polygon has n sides. Choose an arbitrary vertex, say vertex v. Then there are n-3 non-adjacent vertices to vertex v. If diagonals are drawn from vertex v to all non-adjacent vertices, then n-2 triangles will be formed. The sum the interior angles of n-2 triangles is 180(n-2). Since the angle sum of the polygon with n sides is equal to the sum the interior angles of n-2 triangles, the angle sum of a polygon with n sides is . 180(n-2).
