Mathematics Year 6: Triangles: the sum of interior and exterior angles
Mathematics, Year 6 (Portugal): Conjecture from cut-outs that the interior angles add up to 180° and the exterior ones to 360°, explain why, and use it in triangle problems.
Cutting off the corners of a triangle
Draw any triangle, colour its three corners and cut them off. Put them side by side with the vertices at the same point: they always form a straight angle. Conjecture: the interior angles of a triangle add up to 180°. Now the exterior angle at a vertex: the angle between one side and the extension of the other side that meets it there. Take one exterior angle at each vertex, cut them out and fit them together: they form a full turn, 360°. An exterior angle and the interior angle at the same vertex are supplementary.
Why the exterior ones give 360° and what that rules out
Each exterior angle is supplementary to the interior angle at the same vertex: exterior = 180° − interior. Adding the three: 3 × 180° − 180° = 540° − 180° = 360°. So if the interior angles add up to 180°, the exterior ones add up to 360°: the two conjectures explain each other. And 180° is a short budget: a triangle cannot have two right angles, nor a right angle and an obtuse one, because they would already use up 180° or more. So no triangle is both right-angled and obtuse.
A case: two known angles
A triangle has interior angles of 50° and 65°. The third: 180° − (50° + 65°) = 180° − 115° = 65°. The exterior ones: 180° − 50° = 130°, 180° − 65° = 115° and 115°; the sum is 130° + 115° + 115° = 360°. All three angles are acute, so it is an acute triangle. Another case: a right-angled isosceles triangle. The two equal angles add up to 180° − 90° = 90°, so each measures 90° ÷ 2 = 45°.
The trap: the sum does not depend on the size
Size does not change the sum: a tiny triangle and a huge one both add up to 180°. If you measured 178° or 183° with a protractor, you have not found an exception: it is measuring error, and the exact sum is 180°. Another slip: counting two exterior angles at each vertex. Count one per vertex: 360°; if you counted both (they are equal), you would get 720°.
Where you see this outside school
The two set squares in a geometry kit have angles 45°, 45°, 90° and 30°, 60°, 90°: in each one the sum is 180°. An equilateral triangle has three equal angles, so each measures 180° ÷ 3 = 60°. Anyone building a frame made of triangles can find the third angle from the other two, without a protractor, and check the work: if the three do not add up to 180°, something was measured badly.
Test what you learned
You cut off the three corners of a triangle and put them side by side, with the vertices at one point. What angle do they form?
- a) A right angle, 90°
- b) A straight angle, 180°
- c) A full turn, 360°
- d) It depends on the triangle
Correct answer: b) A straight angle, 180° — Yes: the three corners fit into a straight angle. That suggests the interior angles of any triangle add up to 180°.
A triangle has two interior angles of 72° and 38°. How much does the exterior angle at the vertex of the third angle measure?
- a) 110°
- b) 70°
- c) 250°
- d) 290°
Correct answer: a) 110° — Yes: the third interior angle is 180° − 110° = 70°, and its exterior angle is 180° − 70° = 110°.
Ana measures the angles of a big triangle with a protractor and gets 179° in total. What is the best conclusion?
- a) Big triangles have a smaller sum than small ones
- b) This triangle is an exception to the rule
- c) A small measuring error: the exact sum is 180°
- d) The 180° rule only works for equilateral triangles
Correct answer: c) A small measuring error: the exact sum is 180° — Right: measured angles are never perfect, and 179° is close to 180°. The exact sum for any triangle is 180°.
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