Lesson 5 Measured Reasoning Practice Understanding
For each of the following polygons, pick one vertex and draw all of the possible diagonals of the polygon that include that vertex.
What conjecture might you make about the relationship between the number of diagonals drawn from a vertex and the number of sides of the polygon. Can you explain why this is so?
Practice using geometric reasoning in computational work.
How do I look for structure in a diagram so I can use familiar features of the diagram to find the measures of unknown sides and angles?
What measurements do I need to calculate first, in order to calculate additional measurements?
Open Up the Math: Launch, Explore, Discuss
In the diagram, lines
Find the measures of all missing sides and angles by using geometric reasoning, not rulers and protractors. If you think a measurement is impossible to find, identify what information you are missing.
Identify all similar triangles in the diagram. How do you know they are similar, by dilation or by AA similarity?
Identify at least three different quadrilaterals in the diagram.
Find the sum of the interior angles for each quadrilateral.
Make a conjecture about the sum of the interior angles of a quadrilateral.
Identify at least three different pentagons in the diagram. (Hint: The pentagons do not need to be convex.)
Find the sum of the interior angles for each pentagon.
Make a conjecture about the sum of the interior angles of a pentagon.
Do you see a pattern in the sum of the angles of a polygon as the number of sides increases? If so, write a conjecture.
Ready for More?
How can you convince yourself that the pattern you have noticed for the sum of the interior angles of a triangle, a quadrilateral, a pentagon, and a hexagon holds for all
Using the theorem that the sum of the interior angles of a triangle is
This diagram can be used to illustrate why this is true:
This theorem is true for both convex and concave polygons.
Finding missing sides and angles in a complex diagram requires that I look for structure in the diagram. Some of the geometric structures I used today included:
- concave and convex
- Bold terms are new in this lesson.
In this lesson, we drew upon a variety of theorems to support the computational work of finding missing sides and angles. To identify which theorems to use, we had to examine the available features of the diagram. For many measurements, multiple strategies could be used. We also used the diagram, along with our computed measurements, to develop and justify a conjecture for the sum of the interior angles of any polygon, similar to the theorem we proved previously about the sum of the interior angles in a triangle.
Find the missing side lengths for the right triangles. Then determine if the triangles are similar or not.
Missing side in small triangle =
Missing side in large triangle =
Solve each equation.