About angles in a triangle

Arthur Lee, Dec 2, 2016

What can we learn from investigation of the angles in a triangle? This set of activities focuses on the angle relation within a triangle. You may find different ways to explore the angle sum property. By comparing these different ways of exploring and/or proving this property about the sum of interior angles, we may also appreciate different connections with other geometric properties and methods of reasoning. This collection is intended to suggest a variety of tasks beyond the common choice between a simple experiment for discovery/verification and a formal deductive proof. The learning objectives may include building concepts about angles in a geometric figure, noticing properties and relations, connecting intuitive understanding and reasoning.

Table of Contents

  1. get a sense of changing angles
    1. angles in a triangle (up and down)
    2. angles in a triangle (matching 1)
    3. angles in a triangle (matching 2)
    4. smallest angle
  2. reasoning based on manipulation and transformation
    1. Sum of Interior Angles of a Triangle
    2. angles in a triangle (tiling)
    3. angles in a triangle (zooming)
    4. triangle folding
    5. Angle Sum of a Triangle by Paper Folding
  3. reasoning based on parallel lines
    1. Interior & Exterior Angles of Triangles 三角形內角和及外角
    2. Interior and Exterior Angles of Triangles
    3. interior and exterior angle
    4. Clairaut explained triangle angle sum
    5. Angle Sum of Triangle
  4. further exploration
    1. make a triangle
    2. angle pairs
    3. angle pairs (more)

1.

get a sense of changing angles

These activities suggest learners to [list] [*]observe changes in the angle size; [*]make sense of the changes; [*]predict changes [*]match the angles and measurements [/list] Although learners may have some knowledge about different types of angles (acute, obtuse, right), they may not notice how an angle can vary inside a triangle, or how those angles can vary together when the sides or vertices change. For example, you may check whether they think the size of an angle depends on the length of its arms or the space between its arms. It requires careful attention to possible changes in these angles and how they are connected (freedom and constraint) in order to appreciate the constant in their sum despite those changes. These tasks may provide some opportunity to draw their attention to these aspects of angle variation before or during the discussion about the constant sum of angles.

  • 1. angles in a triangle (up and down)
  • 2. angles in a triangle (matching 1)
  • 3. angles in a triangle (matching 2)
  • 4. smallest angle

1.1.

angles in a triangle (up and down)

By moving point A, how can you make angle A smaller? What will happen to the other two angles then?[br][br]When using the arrow keys in the keyboard (click A and press the keys), you may restrict the movement of A in 4 directions. In particular, you can keep pressing a key to move A upwards, even when it goes outside the window. How will the size of angle A change? Can you estimate its size even when the vertex A is hidden above the viewing window? (Use the check box the hide or show its measurement when trying this estimation.) [br][br]You may also hide all angles during this exploration and estimation.
This is a task in a collection about [url=https://ggbm.at/nh3y2vQf]angles of triangle[/url].
Here is a quick demo.
– Arthur Lee

1.2.

–

1.3.

–

1.4.

–

2.

reasoning based on manipulation and transformation

The tasks here may simulate some physical manipulation related to verification of the angle sum property. They are therefore also related to possible proofs based on geometric transformation. The first activity in this section comes from the official GeoGebra resources.

  • 1. Sum of Interior Angles of a Triangle
  • 2. angles in a triangle (tiling)
  • 3. angles in a triangle (zooming)
  • 4. triangle folding
  • 5. Angle Sum of a Triangle by Paper Folding

2.1.

Sum of Interior Angles of a Triangle

Discover the sum of the angle measures in any triangle in this activity.
Putting It All Together
[i]Answer these open ended questions on your own or with others to form deeper math connections. [/i]
If you move the point to change the original triangle, what happens to angle [math]A[/math]?
What is the sum of all the interior angles of a triangle?
Do you think it is possible to have a triangle with angles 50°, 60°, and 80°? Explain.
– GeoGebra Content Team

2.2.

–

2.3.

–

2.4.

–

2.5.

–

3.

reasoning based on parallel lines

More formal discussion or reasoning about the angle sum property would involve knowledge about parallel lines. The tasks and activities may facilitate this kind of discussion. However, it does not mean that the emphasis is put on the formal writing of a proof. Some of them can be related to other tasks in the precious section where transformation is considered. The first activity in this section comes from the gMath project: (https://www.gmath.hk/s2/polygons) .

  • 1. Interior & Exterior Angles of Triangles 三角形內角和及外角
  • 2. Interior and Exterior Angles of Triangles
  • 3. interior and exterior angle
  • 4. Clairaut explained triangle angle sum
  • 5. Angle Sum of Triangle

3.1.

Interior & Exterior Angles of Triangles 三角形內角和及外角

點擊各角可顯示它們的大小。
GeoGebra 互動數學教室 [url]http://www.geogebra.hk[/url]
– Alex CHIK

3.2.

–

3.3.

–

3.4.

–

3.5.

–

4.

further exploration

Some examples are given here to test students' understanding of the angle relation inside a triangle.

  • 1. make a triangle
  • 2. angle pairs
  • 3. angle pairs (more)

4.1.

make a triangle

[size=150]Can you make a triangle with 3 of the following angles? [/size][br][br][size=150][size=100][math]40^\circ[/math] , [math]50^\circ[/math] , [math]60^\circ[/math] , [math]70^\circ[/math] , [math]80^\circ[/math] [/size][/size]
– Arthur Lee

4.2.

–

4.3.

–

Information