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
The first enclosed shape to examine is the triangle.
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 hasone curved side. is not a closed shape, 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 four of the smaller triangles.
There are a total of seven triangles of this size, including the inverted one in the center of the diagram.Next, count the triangles that are formed by nine of the smaller triangles. There are three of this size. And finally, there is one triangle formed by the 16 smaller triangles. Adding these numbers together, we get .
Classifying by Angles
Angles can be classified 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: When a triangle has one right angle.
Obtuse Triangle: When a triangle has one obtuse angle.
Acute Triangle: When all three angles in the triangle 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
These classifications have to do with the sides of the triangle and their relationships to each other.
Scalene Triangle: When a triangles sides are all different lengths.
Isosceles Triangle: When at least two sides of a triangle are congruent.
Equilateral Triangle: When all the sides of a triangle are congruent.
Note that by the definitions, an equilateral triangle is also an isosceles triangle.
Example 4: Classify the triangle by its sides and angles.
Solution: We are told there are two congruent sides, so it is an isosceles triangle. By its angles, they all look acute, so it is an acute triangle. Typically, 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 planar 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. The easiest way to identify a polygon is to look for a closed figure with no curved sides.
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 cannot be polygons. has all straight sides, but one of the vertices is not at the endpoint of the adjacent side, so it is not a polygon either. 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. Think of the term concave as referring to a cave, or “caving in”. A concave polygon has a section that “points inward” toward the middle of the shape. All stars are concave polygons.
A convex polygon does not share this property.
Diagonals: Line segments that connects 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. You could add here that concave polygons have at least one diagonal outside the figure.
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 can always be named by the number of sides. See the chart below.
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 | ? | |
gon | (where ) | ? |
Example 10: Name the three polygons below by their number of sides and if it is convex or concave.
Solution:
A. This shape has six sides and concave, so it is a concave hexagon.
B. This shape has five sides and is convex, so it is a convex pentagon.
C. This shape has ten sides and is convex, so it is a convex decagon.
Know What? Revisited The pentagon is divided up into 10 sections, all quadrilaterals. More specifically, there are 5 rectangles and 5 kites. None of these dividing lines are diagonals because they are not drawn from vertices.
Review Questions
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?
- Construction Construct an equilateral triangle with sides of 3 cm. Start by drawing a horizontal segment of 3 cm and measure this side with your compass from both endpoints.
- What must be true about the angles of your equilateral triangle from #9?
In problems 11-16, 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?
- Use your answers from 17-19 to figure out how many diagonals you can draw from one vertex of an gon?
- Determine the number of total diagonals for an octagon, nonagon, decagon, undecagon, and dodecagon. Do you see a pattern? BONUS: Find the equation of the total number of equations for an gon.
For 23-30, determine if the statement is ALWAYS true, SOMETIMES true, or NEVER true.
- Obtuse triangles are isosceles.
- A polygon must be enclosed.
- A star is a concave polygon.
- A right triangle is acute.
- An equilateral triangle is equiangular.
- A quadrilateral is a square.
- You can draw triangles from one vertex of a polygon.
- A decagon is a 5-point star.
In geometry it is important to know the difference between a sketch, a drawing and a construction. A sketch is usually drawn free-hand and marked with the appropriate congruence markings or labeled with measurement. It may or may not be drawn to scale. A drawing is made using a ruler, protractor or compass and should be made to scale. A construction is made using only a compass and ruler and should be made to scale.
For 31-36, draw, sketch or construct the indicated figures.
- Sketch a convex heptagon with two sides congruent and three angles congruent.
- Sketch a non-polygon figure.
- Draw a concave pentagon with exactly two right angles and at least two congruent sides.
- Draw an equilateral quadrilateral that is NOT a square.
- Construct a right triangle with side lengths 3 cm, 4 cm and 5 cm.
- Construction Challenge Construct a angle. (Hint: Think about an equilateral triangle.)
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