Properties of shapes | Lesson (article) | Khan Academy (2024)

Properties of shapes deal with the definitions of shapes and how shapes are classified. Additionally, we will use our knowledge of shapes to identify counterexamples, determine whether three lines of specified length can form a triangle, and solving problems by drawing diagrams.

• Recognizing properties of common shapes
• Identifying a counterexample to a mathematical statement about shapes
• Using the triangle inequality rule
• Drawing diagrams to solve problems

Triangles are polygons with three sides and three interior angles.

Isosceles triangles have two sides with the same length. The two angles opposite these two sides have the same measure.

Equilateral triangles have three sides with the same length. Each interior angle of an equilateral triangle measures ${60}^{\circ }$‍.

Quadrilaterals are polygons with four sides and four interior angles.

Squares are parallelograms with four sides of equal length and four ${90}^{\circ }$‍ angles. While all squares are both rectangles and parallelograms, not all parallelograms are squares and not all rectangles are squares.

A mathematical statement is composed of two parts: a condition and a conclusion.

Showing that a mathematical statement is true requires a formal proof. However, showing that a mathematical statement is false only requires us to find one example where the statement isn't true. Such an example is called a counterexample because it is an example that counters, or goes against, the statement's conclusion.

Consider the statement "All rectangles are squares":

• Condition: is a rectangle
• Conclusion: is a square

The shape below is one of many counterexamples of the statement. It is a rectangle, but it is not a square.

To find a counterexample:

1. Identify the condition and conclusion of the statement.
2. Eliminate any choices that do not satisfy the statement's condition.
3. For the remaining choices, counterexamples are those that do not satisfy the statement's conclusion.

The triangle inequality rule states that the longest side of a triangle must be shorter than the combined lengths of the two other sides. In other words, for a triangle with side lengths $a$‍, $b$‍, and $c$‍:

For a triangle with only two known side lengths, $a$‍ and $b$‍, the unknown side length, $x$‍, must meet one of the following conditions:

• Shorter than the sum of the two known side lengths
• Longer than the positive difference of the two known side lengths

For some geometry questions, drawing a rough sketch is the best way to visualize the problem. These sketches do not need to be precise but should help us to see which choices can be eliminated.

Try sketching a few shapes to answer the question below!

Which of the following shapes can be formed using two congruent right triangles?

• A triangle
• A rectangle
• A square
• A pentagon (five-sided polygon)

Triangles are polygons with three sides and three interior angles.

• Isosceles triangles have two sides with the same length. The two angles opposite these two sides have the same measure.
• Equilateral triangles have three sides with the same length. Each interior angle of an equilateral triangle measures ${60}^{\circ }$‍.

Quadrilaterals are polygons with four sides and four interior angles.

• Parallelograms are quadrilaterals with two pairs of parallel sides and two pairs of angles with the same measure. The opposite sides have the same length, and adjacent angles are supplementary.
• Rectangles are quadrilaterals with four ${90}^{\circ }$‍ angles. The adjacent sides are perpendicular. While all rectangles are parallelograms, not all parallelograms are rectangles.
• Squares are quadrilaterals with four sides of equal length and four ${90}^{\circ }$‍ angles. While all squares are both rectangles and parallelograms, not all parallelograms are squares and not all rectangles are squares.

To find a counterexample:

1. Identify the condition and conclusion of the statement.
2. Eliminate any choices that do not satisfy the statement's condition.
3. For the remaining choices, counterexamples are those where the statement's conclusion isn't true.

The longest side of a triangle must be shorter than the combined lengths of the two other sides. For a triangle with side lengths $a$‍, $b$‍, and $c$‍:

For a triangle with only two known side lengths, $a$‍ and $b$‍, the unknown side length, $x$‍, must meet one of the following conditions:

• Shorter than the sum of the two known side lengths
• Longer than the positive difference of the two known side length

Draw diagrams helps us solve geometry questions.