Classify triangles by sides and angles

Learning goal: Describe similarities and differences among triangles using side lengths and angle measures. A triangle can receive one classification by sides and another by angles. Its orientation or apparent resemblance to a familiar picture does not determine the class.

A triangle is a closed polygon with three straight sides. A scalene triangle has three different side lengths. An isosceles triangle has at least two equal side lengths in the inclusive convention used here. An equilateral triangle has all three sides equal, so it is also isosceles under that convention. State the convention when a classroom uses the word differently, and focus on the actual equality relationships.

Side lengths can be marked with matching ticks. Two sides with the same tick mark are equal; a side with a different mark is not being identified as equal to those two. Do not infer equality only from how close the lengths look on a screen. Measurement tools and given properties provide stronger evidence than appearance.

Angle classification uses a different set of properties. An acute triangle has three angles smaller than 90 degrees. A right triangle has one angle of exactly 90 degrees. An obtuse triangle has one angle greater than 90 degrees. A small square marks a right angle. The angles of a triangle total 180 degrees, so a triangle cannot have two right angles or two obtuse angles.

An equiangular triangle has three equal angles. Because their total is 180 degrees, each is 60 degrees. In the flat (plane) geometry studied here, an equilateral triangle is equiangular, and an equiangular triangle is equilateral. Its three angles are acute, so it also belongs to the acute group. The labels describe connected properties rather than unrelated names to memorize.

A triangle can be both right and isosceles. For example, a triangle with equal perpendicular legs has one 90-degree angle and two equal 45-degree angles. A right triangle with side lengths 3, 4 and 5 is scalene because all side lengths differ. These two right triangles share an angle property but differ in their side classifications.

An isosceles triangle can also be obtuse, such as one with angles 40, 40 and 100 degrees. An acute scalene example can have angles 50, 60 and 70 degrees. These examples show why one label is usually insufficient for a complete description. Do not claim that all isosceles triangles are acute just because the first example you saw was acute.

When comparing two triangles, choose a property and apply the same test to both. Ask which angles are right, acute or obtuse; then compare side equalities. Rotating a triangle does not change side lengths or angle measures. A right angle remains right even if its legs do not appear horizontal and vertical on the page.

A sorting chart can have angle types as rows and side types as columns. Some combinations are possible and some are not. Equilateral-right is impossible because equilateral angles are all 60 degrees. Isosceles-right is possible, as shown by the equal-leg example. Justify an empty chart cell with a property rather than leaving it blank without explanation.

Worked comparison: Triangle A has angles 45, 45 and 90 degrees; Triangle B has 30, 60 and 90 degrees. Both are right triangles. A has two equal sides opposite equal angles, while B has three unequal sides. Thus A is isosceles and B is scalene.

Worked impossibility: A triangle cannot have two 100-degree angles because those already total 200 degrees, more than the entire 180-degree angle sum. This supports the claim that an obtuse triangle has only one obtuse angle.

Guided investigation: Draw, cut out and rotate several triangles. Use a protractor and ruler to test angle and side classifications, recording reasonable measurement uncertainty rather than pretending hand drawings are exact. Compare with exact provided angle measures.

Independent performance: Classify four triangles using both angle and side properties, provide a shared property and a difference for a pair, and explain one impossible combination. Create a rotated right-isosceles example and annotate its defining properties. Assess property reasoning separately from drawing neatness.