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# definition of same side interior angles

From Wikipedia

Internal and external angle

In geometry, an interior angle (or internal angle) is an angle formed by two sides of a polygon that share an endpoint. For a simple, convex or concave polygon, this angle will be an angle on the 'inner side' of the polygon. A polygon has exactly one internal angle per vertex.

If every internal angle of a simple, closed polygon is less than 180Â°, the polygon is called convex.

In contrast, an exterior angle (or external angle) is an angle formed by one side of a simple, closed polygon and a line extended from an adjacent side.

The sum of the internal angle and the external angle on the same vertex is 180Â°.

For example: x+35+75=180
x+110=180
x+110-110=180-110
x=70

The sum of all the internal angles of a simple, closed polygon can be determined by 180(n-2) where n is the number of sides. The formula can by proved using mathematical induction and starting with a triangle for which the angle sum is 180, and then adding a vertex and two sides, etc. A pentagon's internal angles add up to of 540 degrees (shown below)
180(n-2)= 180(5-2)= 180(3)= 540
Knowing this you can easily find the measure of each angle if it is a equiangular polygon with
\frac{180(n-2)}{n}.
So continuing from the above example with the pentagon:
\frac{540}{n}=\frac{540}{5}=108

(The exterior angle can be worked out by doing the following sum 360/number of sides in the equiangular polygon The interior angle can then be found by taking away the value of the exterior angle from 180. So in a pentagon there are 5 sides so to work out the exterior angle you do 360/5 =72 So the interior angle is 180-72=108 so the interior angle = 108 degrees)

The sum of the external angles of any simple closed (convex or concave) polygon is 360Â°.

The concept of 'interior angle' can be extended in a consistent way to crossed polygons such as star polygons by using the concept of 'directed angles'. In general, the interior angle sum in degrees of any closed polygon, including crossed (self-intersecting) ones, is then given by 180(n-2k) where n is the number of vertices and k = 0, 1, 2, 3. ... represents the number of total revolutions of 360o one undergoes walking around the perimeter of the polygon, and turning at each vertex, until facing in the same direction one started off from. In other words (or put differently), 360k represents the sum of all the exterior angles. For example, for ordinary convex and concave polygons k = 1, since the exterior angle sum = 360o and one undergoes only one full revolution walking around the perimeter.

Question:Has to be in a 2 column proof form and cant have any thereoms that aren't proven in basic geometry

Question:

Answers:Yes. they will be supplementary because even though the line is intersected, it will still equal 180 degrees.

Question:

Answers:No. The AA similarity postulate applies to triangles and basically says that if two angles of one triangle are the same as two angles of another triangle, then the traingles are similar (basically because all the angles are the same, the sides have to be proportionate or similar to each other). The Alternate Interior Angles theorem applies to two parallel lines cut by a transversal (completely different shape than a triangle) and that these two angles are congruent. BTW, two parallel lkines cut by a transversal basically looks like a H (or a sideways H) with a crooked line going through the sides of the H, instead of a straight line.

Question:I need a good definition. I'd also like a good definition for base angles (of an isosceles triangle)

Answers:To start, let's make sure you understand the definitions of the terms. As isosceles triangle has two congruent sides with a third side that is the base. A base angle of an isosceles triangle is one of the angles formed by the base and another side. Base angles are equal because of the definition of an isosceles triangle. A picture would probably help here: A . / \ ABC = ACB = 39 degrees / \ BAC = ?? / \ / \ / \ / \ / \ ._______________. B C base ABC is the isosceles triangle. AB is congruent to AC. Angle ABC is congruent to angle ACB. These are the base angles. Triangle is a convex polygon with three segments joining three non-collinear points. Each of the three segments is called a side, and each of the three non-collinear points is called a vertex. Triangles can be categorized by the number of congruent sides they have. For instance, a triangle with no congruent sides is a scalene triangle; a triangle with two congruent sides is an isosceles triangle; a triangle with three congruent sides is an equilateral triangle. Triangles can also be categorized by their angles. For instance, a triangle with three acute interior angles is an acute triangle; a triangle with one obtuse interior angle is an obtuse triangle; a triangle with one right interior angle is a right triangle; a triangle with three congruent interior angles is an equiangular triangle. One property of a triangle is that the sum of the measures of the three interior angles is always 180 degrees (or pi radians). In addition, the exterior angle of a triangle is the supplement of the adjacent interior angle. The measure of the exterior angle is also the sum of the measures of the two remote interior angles.