Triangles: types, sides and angles, construction and… — 11–13
Classify triangles by sides and angles, link sides to angles, and know when a triangle can be built and when two are congruent.
Classifying triangles by sides and by angles
By sides: equilateral (three equal sides), isosceles (at least two equal sides) and scalene (three different sides). By angles: acute-angled (three acute angles), right-angled (one right angle) and obtuse-angled (one obtuse angle). Every equilateral triangle is also isosceles, but not every isosceles triangle is equilateral. And sides and angles are linked: equal sides face equal angles, and the longest side faces the largest angle.
Why sides and angles go together
Picture two sticks joined by a pin at one corner. The wider you open the angle between them, the further apart their ends get, that is, the longer the side that closes the triangle. So the longest side faces the largest angle. If the two sticks are equal, the triangle is symmetric, and the angles opposite the equal sides are equal. Problem: in a triangle with sides 6 cm, 6 cm and 9 cm, the angles opposite the 6 cm sides are equal, and the largest angle is opposite the 9 cm side.
Building a triangle with ruler and compasses
Build a triangle with sides 4 cm, 5 cm and 7 cm. Draw the 7 cm side, [AB]. With the compass point on A, open it to 5 cm and draw an arc. With the point on B, open it to 4 cm and draw another arc. C is where the arcs cross. Join A to C and B to C. Anyone who follows these steps gets a triangle that fits exactly on yours: they are congruent (criterion SSS, three sides). Two sides and the angle between them (SAS), or one side and the two angles at its ends (ASA), are also enough.
The trap: not every three lengths make a triangle
With 2 cm, 3 cm and 6 cm there is no triangle: the arcs of 2 cm and 3 cm drawn from the ends of the 6 cm side never reach each other (2 + 3 = 5, less than 6). With 3, 4 and 7 the arcs only touch on the side itself: you get a flat line, not a triangle. Rule: the longest side must be shorter than the sum of the other two. And three angles are not enough either: there are small and large triangles with the same three angles.
Where you see this outside school
Triangles do not deform. A shelf frame made of four bars tilts sideways, but one diagonal bar, which makes two triangles, holds it firm. That is why you see triangles in bridges, roof frames and cranes. Three given sides allow only one shape of triangle, which is the reason behind the SSS criterion.
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