Practice problem-how are right, acute and obtuse triangles related?Ī good exercise for beginning to think at the analysis level (Van Hiele level 2) is to consider when a triangle can have both a side length and an angle property. Isosceles and Scalene triangles are disjoint sets-they don't overlap at all because a triangle can never be both isosceles and scalene. If you have no sides the same length, however, you can never have two sides the same length, so we can show the relationships between equilateral, isosceles and scalene triangles this way:Įquilateral triangles are a subset of isosceles triangles, because every equilateral triangle is also isosceles (though some isosecles triangles are equilateral and some are not). Notice that if you have all 3 sides the same length, you automatically have at least 2 sides the same length. Obtuse triangles have one angle that measures more than 90° Right triangles have one angle that is equal to 90°Īcute triangles have all 3 angles less than 90° The second 3 are defined by angle properties: Isosceles triangles have at least 2 sides the sameĮquilateral triangles have all 3 sides the same Scalene triangles have all 3 sides different lengths The first 3 are usually defined by side length properties: There are 6 special named types of triangles: scalene triangles, isosceles triangles, equilateral triangles, right triangles, acute triangles, and obtuse triangles. Most of this work with triangles is appropriate for students grades 3 and above. This set of problems shows the sorts of thinking we want students to develop to be ready for High school math. The apothem of a regular polygon is also the height of an isosceles triangle formed by the center and a side of the polygon, as shown in the figure below.įor the regular pentagon ABCDE above, the height of isosceles triangle BCG is an apothem of the polygon.This discussion is aimed at a Van Hiele level 1-2 understanding, and thus is fairly sophisticated for elementary students. The length of the base, called the hypotenuse of the triangle, is times the length of its leg. When the base angles of an isosceles triangle are 45°, the triangle is a special triangle called a 45°-45°-90° triangle. Base BC reflects onto itself when reflecting across the altitude. Leg AB reflects across altitude AD to leg AC. The altitude of an isosceles triangle is also a line of symmetry. So, ∠B≅∠C, since corresponding parts of congruent triangles are also congruent. Based on this, △ADB≅△ADC by the Side-Side-Side theorem for congruent triangles since BD ≅CD, AB ≅ AC, and AD ≅AD. Using the Pythagorean Theorem where l is the length of the legs. ABC can be divided into two congruent triangles by drawing line segment AD, which is also the height of triangle ABC. Refer to triangle ABC below.ĪB ≅AC so triangle ABC is isosceles. The base angles of an isosceles triangle are the same in measure. Using the Pythagorean Theorem, we can find that the base, legs, and height of an isosceles triangle have the following relationships: The height of an isosceles triangle is the perpendicular line segment drawn from base of the triangle to the opposing vertex. The angle opposite the base is called the vertex angle, and the angles opposite the legs are called base angles. Parts of an isosceles triangleįor an isosceles triangle with only two congruent sides, the congruent sides are called legs. DE≅DF≅EF, so △DEF is both an isosceles and an equilateral triangle.
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