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# Mathematics Alternative Assessment

What is the sum of the interior angles in a triangle?

by

Tweet## Debdeep Mukherjee

on 25 April 2010#### Transcript of Mathematics Alternative Assessment

Mathematics Alternative Assessment

Introduction Done By : Debdeep Mukherjee (09) (Group Leader)

Hsu Zhong Jun (14) History Conclusion Method 1. The right angle triangle method 90° 1. The sum of the angles on a straigh line is 180°, so a perpendicular bisector would cut this line into two equal angles- 90° which explains the right angle of the triangle. Two perpendicular lines b c a c b + c = 90° Perpendicular bisector 2. The other two angles of the right angle triangle are b and c. 3. When two parallel lines are drawn at the base and tip of the triangle, the alternate interior angle of c is seen, beside b. Method 2. The exterior angle method a b c d e f Method 3. The quadrilateral method a b c b a c Method 4. The rotation method a b c a b Method 5. Opposite angle method Method 6. Paper folding A triangle is a polygon, meaning that it is one of the many shapes in geometry. It has three sides which are connected together by three points, which form three vertices. These three vertices have three angles. The are many types of triangles and they all have the same properties, which are-they have three sides and three vertices, which form three angles and they add up to the same total. The different types of triangles are, right angle triangle, equilateral triangle, isosceles triangle, scalene triangle and obtuse triangle. The question is, what is the sum of the interior angles of a triangle? Let us find out using various methods.

a b c a a a a b c a b a a b c Right angle triangle Equilateral triangle Isosceles triangle Scalene triangle Obtuse triangle One of its angle is 90° Three equal angles and sides Two of its angles and sides are equal All three angles and sides are different One of its angles are bigger than 90° What is the sum of the interior angles of a triangle? The history of triangles date back to 3000 BC, where people from the Indus Valley and the Ancient Babylonia had discovered the different types of triangles and discovered the basic properties of these triangles which are-they all have three sides and three vetices which make three angles. But questions soon started to arise that what is the sum of these three interior angles of any triangle. It was Euclid, that had first proved what the sum of the interior angles of a triangle really is in 300 BC. These are the methods- So, we can conclude that the sum of the interior angles of a triangle is 180° as this is proven by all these methods. A B C a b c a + b + c = 180° 4. As the perpendicular bisector divides the angle on the straight line into two equal halves, one half of it is b + c = 90°. 5. Angle a, is 90° and the other two angles of the triangle, b and c, also add up to 90°. So, the three angles add up to 90° + 90° = 180°.

1. This method works for other triangles as well. 2. First divide the triangle into two right angle trangles. 3. Use the same method to find the total of the two right angle triangle. 4. Add up the two values and the subtract off the extra 180° from the sum as it is the angle on the straight line inside the triangle which is not one of the angles of the triangle. 1. The iterior and exterior angles of a triangle forms an angle on a straight line, which is 180°. 2. This will mean that the total sum of the three interior and exterior angles of the triangle would be 180° x 3 = 540°. 3. As the sum of all the exterior angles of any shape is always 360°, the sum of the exterior angles of this triangle is also 360°.

4. So, subtract this value from 540° and it would give the sum of the interior angles of the triangle which is 540° - 360° = 180°. Angle on a straight line-180°, inside the triangle, which is not one of the angles of the triangle. The three interior and exterior angles which are on a straight of the triangle adds up to 180° x 3 = 540° 1. Flip the triangle to form a quadrilateral. 2. This shows the positions of the angles equal to a, b and c in the other side of the quadrilateral. 3. As the sum of the angles in any quadrilateral is 360° and the quadrilateral has a pair of angles a, b and c of the triangle, the sum of the angles in the triangle is 360°/2 = 180°. 1. Draw a straight line aty the top of the triangle. 2. Cut the angles a and b out and rotate it to fit into the line beside the both sides of c. 3. This would form an angle on a straighrt line, which is 180°, the sum of the three angles of the triangle-a + b + c. Angle on a straight line-180° 1. This method is similar to mthods 4 and 5 as it requires the three angles to meet at a point as well. First, fold the top angle of the paper triangle, then the other two angles of the paper triangle to meet at one point. 2. This would form a straight line and as angles on a straight line is 180°, the three angles of the triangle, a, b and c adds up to 180°. a b c a b c a b c 1. Draw a line parallel to the base of the triangle. 2. Extend the two sides of the triangle. This woud show the opposite angles of b and c, which is on a straight line with angle a. 3. The three angles of the triangle a, b and c are on a straight line, so they add up to 180°. Reflection Through all my research, I have hound that there are many methods of finding what the sum of the interior angles of a triangle is. I have also found that most of the methods are based on one mathematical threorem that, the sum of the angles on a straight line is 180°. Finally I have found through these methods that the sum of the interior angles of a triangle is 180°. Credits Euclid Indus Valley People Babylonians Intoduction-Debdeep Mukherjee (Drawings searched by Debdeep Mukherjee from-http://home.avvanta.com/~math/triangles.html)

History-http://www.google.com/search?q=interior+angles+of+a+triangle+history&hl=en&tbo=p&ei=ZwfTS-WVCcufrAe0wf2pDg&sa=X&oi=timeline_other_dates&ct=timeline-other-dates&ved=0CIYBEKUC&tbs=tl:1,tlul:4000BC,tluh:2010

(Link provided by Hsu Zhong Jun.)

Pictures-Google Images (Searched by Debdeep Mukherjee)

Method 1-http://www.youtube.com/watch?v=FBuMhhWTldc (Searched by Debdeep)

Method 2-http://math4allages.wordpress.com/2009/12/26/the-exterior-angle-theorem/ (Thought of it myself at first and I founded this in this websits as well.)

Method 3-http://www.coolmath.com/reference/polygons.html (Thought of it myself, then Hsu Zhong Jun provided this methodand I founded it in this website as well.)

Method 4-Material provided by Hsu Zhong Jun

Method 5-http://www.cut-the-knot.org/triangle/pythpar/AnglesInTriangle.shtml (Thought of it myself, then Hsu Zhong Jun provided the method and found it in th website as well.)

Method 6-http://demonstrations.wolfram.com/TheSumOfTheInteriorAnglesOfATriangleEquals180DegreesByPaperF/ (Searched by Debdeep Mukherjee)

Conclusion-Debdeep Mukherjee

Reflection-Debdeep Mukherjee

Compiled into the presentation from the raw materials by-Debdeep Mukherjee

Full transcriptIntroduction Done By : Debdeep Mukherjee (09) (Group Leader)

Hsu Zhong Jun (14) History Conclusion Method 1. The right angle triangle method 90° 1. The sum of the angles on a straigh line is 180°, so a perpendicular bisector would cut this line into two equal angles- 90° which explains the right angle of the triangle. Two perpendicular lines b c a c b + c = 90° Perpendicular bisector 2. The other two angles of the right angle triangle are b and c. 3. When two parallel lines are drawn at the base and tip of the triangle, the alternate interior angle of c is seen, beside b. Method 2. The exterior angle method a b c d e f Method 3. The quadrilateral method a b c b a c Method 4. The rotation method a b c a b Method 5. Opposite angle method Method 6. Paper folding A triangle is a polygon, meaning that it is one of the many shapes in geometry. It has three sides which are connected together by three points, which form three vertices. These three vertices have three angles. The are many types of triangles and they all have the same properties, which are-they have three sides and three vertices, which form three angles and they add up to the same total. The different types of triangles are, right angle triangle, equilateral triangle, isosceles triangle, scalene triangle and obtuse triangle. The question is, what is the sum of the interior angles of a triangle? Let us find out using various methods.

a b c a a a a b c a b a a b c Right angle triangle Equilateral triangle Isosceles triangle Scalene triangle Obtuse triangle One of its angle is 90° Three equal angles and sides Two of its angles and sides are equal All three angles and sides are different One of its angles are bigger than 90° What is the sum of the interior angles of a triangle? The history of triangles date back to 3000 BC, where people from the Indus Valley and the Ancient Babylonia had discovered the different types of triangles and discovered the basic properties of these triangles which are-they all have three sides and three vetices which make three angles. But questions soon started to arise that what is the sum of these three interior angles of any triangle. It was Euclid, that had first proved what the sum of the interior angles of a triangle really is in 300 BC. These are the methods- So, we can conclude that the sum of the interior angles of a triangle is 180° as this is proven by all these methods. A B C a b c a + b + c = 180° 4. As the perpendicular bisector divides the angle on the straight line into two equal halves, one half of it is b + c = 90°. 5. Angle a, is 90° and the other two angles of the triangle, b and c, also add up to 90°. So, the three angles add up to 90° + 90° = 180°.

1. This method works for other triangles as well. 2. First divide the triangle into two right angle trangles. 3. Use the same method to find the total of the two right angle triangle. 4. Add up the two values and the subtract off the extra 180° from the sum as it is the angle on the straight line inside the triangle which is not one of the angles of the triangle. 1. The iterior and exterior angles of a triangle forms an angle on a straight line, which is 180°. 2. This will mean that the total sum of the three interior and exterior angles of the triangle would be 180° x 3 = 540°. 3. As the sum of all the exterior angles of any shape is always 360°, the sum of the exterior angles of this triangle is also 360°.

4. So, subtract this value from 540° and it would give the sum of the interior angles of the triangle which is 540° - 360° = 180°. Angle on a straight line-180°, inside the triangle, which is not one of the angles of the triangle. The three interior and exterior angles which are on a straight of the triangle adds up to 180° x 3 = 540° 1. Flip the triangle to form a quadrilateral. 2. This shows the positions of the angles equal to a, b and c in the other side of the quadrilateral. 3. As the sum of the angles in any quadrilateral is 360° and the quadrilateral has a pair of angles a, b and c of the triangle, the sum of the angles in the triangle is 360°/2 = 180°. 1. Draw a straight line aty the top of the triangle. 2. Cut the angles a and b out and rotate it to fit into the line beside the both sides of c. 3. This would form an angle on a straighrt line, which is 180°, the sum of the three angles of the triangle-a + b + c. Angle on a straight line-180° 1. This method is similar to mthods 4 and 5 as it requires the three angles to meet at a point as well. First, fold the top angle of the paper triangle, then the other two angles of the paper triangle to meet at one point. 2. This would form a straight line and as angles on a straight line is 180°, the three angles of the triangle, a, b and c adds up to 180°. a b c a b c a b c 1. Draw a line parallel to the base of the triangle. 2. Extend the two sides of the triangle. This woud show the opposite angles of b and c, which is on a straight line with angle a. 3. The three angles of the triangle a, b and c are on a straight line, so they add up to 180°. Reflection Through all my research, I have hound that there are many methods of finding what the sum of the interior angles of a triangle is. I have also found that most of the methods are based on one mathematical threorem that, the sum of the angles on a straight line is 180°. Finally I have found through these methods that the sum of the interior angles of a triangle is 180°. Credits Euclid Indus Valley People Babylonians Intoduction-Debdeep Mukherjee (Drawings searched by Debdeep Mukherjee from-http://home.avvanta.com/~math/triangles.html)

History-http://www.google.com/search?q=interior+angles+of+a+triangle+history&hl=en&tbo=p&ei=ZwfTS-WVCcufrAe0wf2pDg&sa=X&oi=timeline_other_dates&ct=timeline-other-dates&ved=0CIYBEKUC&tbs=tl:1,tlul:4000BC,tluh:2010

(Link provided by Hsu Zhong Jun.)

Pictures-Google Images (Searched by Debdeep Mukherjee)

Method 1-http://www.youtube.com/watch?v=FBuMhhWTldc (Searched by Debdeep)

Method 2-http://math4allages.wordpress.com/2009/12/26/the-exterior-angle-theorem/ (Thought of it myself at first and I founded this in this websits as well.)

Method 3-http://www.coolmath.com/reference/polygons.html (Thought of it myself, then Hsu Zhong Jun provided this methodand I founded it in this website as well.)

Method 4-Material provided by Hsu Zhong Jun

Method 5-http://www.cut-the-knot.org/triangle/pythpar/AnglesInTriangle.shtml (Thought of it myself, then Hsu Zhong Jun provided the method and found it in th website as well.)

Method 6-http://demonstrations.wolfram.com/TheSumOfTheInteriorAnglesOfATriangleEquals180DegreesByPaperF/ (Searched by Debdeep Mukherjee)

Conclusion-Debdeep Mukherjee

Reflection-Debdeep Mukherjee

Compiled into the presentation from the raw materials by-Debdeep Mukherjee