Equal sides, equal angles
The end wall of a hut is an isosceles triangle. Two of its sides are equal. These are the two sloping sides. What can you say about the corners of the triangle?
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- Triangle KLM, with KL = KM.
- From K, draw the altitude KN, square to the base. Square here means at 90°.
- ΔKLN and ΔKMN are congruent by RHS.
- So the angles at L and M are equal.
The rule
The two corners across from the equal sides are equal. An equilateral triangle has all three sides equal.
Draw the altitude from the top of an isosceles triangle. An altitude runs from a corner, square to the opposite side. It cuts the triangle into two right-angled triangles. They are congruent by RHS, as shown below. So the angles across from the equal sides are equal. Any two sides of an equilateral triangle are equal. So all three angles are equal. Each is 60°, because they add to 180°.
The two halves share KN. They both have a right angle. Their hypotenuses KL and KM are equal. That is RHS. RHS also gives LN = MN, so N is the middle of LM.
An equilateral triangle has angles of 60°. An obtuse triangle has an angle over 90°. So an equilateral triangle is never right-angled or obtuse.
An isosceles triangle can be right-angled or obtuse. Try a top angle of 90° or 120° in the lab.