Learning Objectives
- Define triangle and polygon.
- Classify triangles by their sides and angles.
- Understand the difference between convex and concave polygons.
- Classify polygons by number of sides.
Review Queue
- Draw a triangle.
- Where have you seen 4, 5, 6 or 8 - sided polygons in real life? List 3 examples.
- Fill in the blank.
- Vertical angles are always _____________.
- Linear pairs are _____________.
- The parts of an angle are called _____________ and a _____________.
Know What? The pentagon in Washington DC is a pentagon with congruent sides and angles. There is a smaller pentagon inside of the building that houses an outdoor courtyard. Looking at the picture, the building is divided up into 10 smaller sections. What are the shapes of these sections? Are any of these division lines diagonals? How do you know?
Triangles
Triangle: Any closed figure made by three line segments intersecting at their endpoints.
Every triangle has three vertices (the points where the segments meet), three sides (the segments), and three interior angles (formed at each vertex). All of the following shapes are triangles.
You might have also learned that the sum of the interior angles in a triangle is . Later we will prove this, but for now you can use this fact to find missing angles.
Example 1: Which of the figures below are not triangles?
Solution: is not a triangle because it has one curved side. is not closed, so it is not a triangle either.
Example 2: How many triangles are in the diagram below?
Solution: Start by counting the smallest triangles, 16.
Now count the triangles that are formed by 4 of the smaller triangles, 7.
Next, count the triangles that are formed by 9 of the smaller triangles, 3.
Finally, there is the one triangle formed by all 16 smaller triangles. Adding these numbers together, we get .
Classifying by Angles
Angles can be grouped by their angles; acute, obtuse or right. In any triangle, two of the angles will always be acute. The third angle can be acute, obtuse, or right. We classify each triangle by this angle.
Right Triangle: A triangle with one right angle.
Obtuse Triangle: A triangle with one obtuse angle.
Acute Triangle: A triangle where all three angles are acute.
Equiangular Triangle: When all the angles in a triangle are congruent.
Example 3: Which term best describes below?
Solution: This triangle has one labeled obtuse angle of . Triangles can only have one obtuse angle, so it is an obtuse triangle.
Classifying by Sides
You can also group triangles by their sides.
Scalene Triangle: A triangles where all three sides are different lengths.
Isosceles Triangle: A triangle with at least two congruent sides.
Equilateral Triangle: A triangle with three congruent sides.
From the definitions, an equilateral triangle is also an isosceles triangle.
Example 4: Classify the triangle by its sides and angles.
Solution: We see that there are two congruent sides, so it is isosceles. By the angles, they all look acute. We say this is an acute isosceles triangle.
Example 5: Classify the triangle by its sides and angles.
Solution: This triangle has a right angle and no sides are marked congruent. So, it is a right scalene triangle.
Polygons
Polygon: Any closed, 2-dimensional figure that is made entirely of line segments that intersect at their endpoints.
Polygons can have any number of sides and angles, but the sides can never be curved.
The segments are called the sides of the polygons, and the points where the segments intersect are called vertices.
Example 6: Which of the figures below is a polygon?
Solution: The easiest way to identify the polygon is to identify which shapes are not polygons. and each have at least one curved side, so they are not be polygons. has all straight sides, but one of the vertices is not at the endpoint, so it is not a polygon. is the only polygon.
Example 7: Which of the figures below is not a polygon?
Solution: is a three-dimensional shape, so it does not lie within one plane, so it is not a polygon.
Convex and Concave Polygons
Polygons can be either convex or concave. The term concave refers to a cave, or the polygon is “caving in”. All stars are concave polygons.
A convex polygon does not do this. Convex polygons look like:
Diagonals: Line segments that connect the vertices of a convex polygon that are not sides.
The red lines are all diagonals.
This pentagon has 5 diagonals.
Example 8: Determine if the shapes below are convex or concave.
Solution: To see if a polygon is concave, look at the polygons and see if any angle “caves in” to the interior of the polygon. The first polygon does not do this, so it is convex. The other two do, so they are concave.
Example 9: How many diagonals does a 7-sided polygon have?
Solution: Draw a 7-sided polygon, also called a heptagon.
Drawing in all the diagonals and counting them, we see there are 14.
Classifying Polygons
Whether a polygon is convex or concave, it is always named by the number of sides.
Polygon Name | Number of Sides | Number of Diagonals | Convex Example |
---|---|---|---|
Triangle | 3 | 0 | |
Quadrilateral | 4 | 2 | |
Pentagon | 5 | 5 | |
Hexagon | 6 | 9 | |
Heptagon | 7 | 14 | |
Octagon | 8 | ? | |
Nonagon | 9 | ? | |
Decagon | 10 | ? | |
Undecagon or hendecagon | 11 | ? | |
Dodecagon | 12 | ? | |
n-gon | (where ) | ? |
Example 10: Name the three polygons below by their number of sides and if it is convex or concave.
Solution: The pink polygon is a concave hexagon (6 sides).
The green polygon convex pentagon (5 sides).
The yellow polygon is a convex decagon (10 sides).
Know What? Revisited The pentagon is divided up into 10 sections, all quadrilaterals. None of these dividing lines are diagonals because they are not drawn from vertices.
Review Questions
- Questions 1-8 are similar to Examples 3, 4 and 5.
- Questions 9-14 are similar to Examples 8 and 10
- Question 15 is similar to Example 6.
- Questions 16-19 are similar to Example 9 and the table.
- Questions 20-25 use the definitions, postulates and theorems in this section.
For questions 1-6, classify each triangle by its sides and by its angles.
- Can you draw a triangle with a right angle and an obtuse angle? Why or why not?
- In an isosceles triangle, can the angles opposite the congruent sides be obtuse?
In problems 9-14, name each polygon in as much detail as possible.
- Explain why the following figures are NOT polygons:
- How many diagonals can you draw from one vertex of a pentagon? Draw a sketch of your answer.
- How many diagonals can you draw from one vertex of an octagon? Draw a sketch of your answer.
- How many diagonals can you draw from one vertex of a dodecagon?
- Determine the number of total diagonals for an octagon, nonagon, decagon, undecagon, and dodecagon.
For 20-25, determine if the statement is true or false.
- Obtuse triangles can be isosceles.
- A polygon must be enclosed.
- A star is a convex polygon.
- A right triangle is acute.
- An equilateral triangle is equiangular.
- A quadrilateral is always a square.
- A 5-point star is a decagon
Review Queue Answers
- Examples include: stop sign (8), table top (4), the Pentagon (5), snow crystals (6), bee hive combs (6), soccer ball pieces (5 and 6)
- congruent or equal
- supplementary
- sides, vertex