## Learn about the classifications of triangles, their different properties, and relationships between them. Examine concepts such as triangle inequality, triangle rigidity, and side–side–side congruence, and look at the conditions that cause them. Compare how these concepts apply to quadrilaterals. Explore properties of triangles and quadrilaterals through practical applications such as building structures.

### In This Session

Part A: Different Triangles
Part C: Building Towers
Homework

In this session, you will build triangles and quadrilaterals to explore their properties. Classification is an important part of geometry and other areas of mathematics. By creating and manipulating triangles and quadrilaterals, you will develop a sense of logical classifications and how different figures are related.

For information on required and/or optional materials for this session see Note 1 below.

### Learning Objectives

In this session, you will learn the following:

• How to classify triangles according to some of their features
• A rule that describes when three given lengths will make a triangle and when they will not
• A rule that describes when four given lengths will make a quadrilateral and when they will not
• How to use what you’ve learned about geometry and the properties of figures to help with a construction task

### Previously Introduced

Altitude: An altitude of a triangle is a line segment connecting a vertex to the line containing the opposite side and perpendicular to that side.

Perpendicular Bisector: The perpendicular bisector of a line segment is perpendicular to that segment and bisects it; that is, it goes through the midpoint of the segment, creating two equal segments.

### New in This Session

Acute Triangle: An acute triangle is a triangle with all three angles less than 90°.

Congruent: Two figures are congruent if all corresponding lengths are the same, and if all corresponding angles have the same measure. Colloquially, we say they “are the same size and shape,” though they may have different orientation. (One might be rotated or flipped compared to the other.)

Equilateral Triangle: An equilateral triangle is a triangle with three equal sides.

Isosceles Triangle: An isosceles triangle is a triangle with two equal sides.

Obtuse Triangle: An obtuse triangle is a triangle with one angle more than 90°.

Right Triangle: A right triangle is a triangle with one right (90°) angle.

Scalene Triangle: A scalene triangle is a triangle with all three sides unequal.

Side-Angle-Side (SAS) Congruence: Side-angle-side (SAS) congruence states that if any two sides of a triangle are equal in length to two sides of another triangle and the angles bewteen each pair of sides have the same measure, then the two triangles are congruent; that is, they have exactly the same shape and size.

Side-Side-Side (SSS) similarity: he side-side-side (SSS) similarity test says that if two triangles have all three pairs of sides in proportion, the triangles must be similar.

Similar Triangles: Similar triangles are triangles that have the same shape but possibly different size. In particular, corresponding angles are congruent, and corresponding sides are in proportion.

Triangle Inequality: The triangle inequality says that for three lengths to make a triangle, the sum of the lengths of any two sides must be greater than the third length.

### Notes

Part A Notes: Different Triangles
Part C Notes: Building Towers

Materials Needed

• scissors
• box of toothpicks
• bag of mini-marshmallows
• meter stick or ruler You can make your own linkage strips from stiff paper with evenly spaced holes and paper fasteners, or you can purchase them from ETA/Cuisenaire under the name polystrips.

For homework, you’ll also need a ruler marked in inches and a protractor. Collect this set of materials for each group or individual working alone.

Polystrips

ETA/Cuisenaire
500 Greenview Court
Vernon Hills, IL 60061
800-445-5985 847-816-5050
800-875-9643 847-816-5066 (fax)
(Product not available online)