## Lesson: Triangles and Quadrilaterals Developing the Concept

Students have learned about triangles and quadrilaterals and their properties. Now they will apply these properties to solve some problems.

Materials: pencil, protractor, ruler, and several sheets of paper for each student

Preparation: Display the five sheets showing the five types of quadrilaterals and their properties created in the "Introducing the Concept" section.

• Say: You have learned about different types of triangles and rectangles. Today you will use what you know to solve some problems.
• Say: This is an isosceles right triangle. Some of the angle measures and side lengths have been labeled, but others are missing. I want you to find the missing measures.

Give students a few minutes to work on the problem. Then ask for a volunteer to come to the board to fill in the missing measures and explain the reasoning involved. (Since the sum of the angle measures of a triangle is 180 degrees, the sum of the missing angle measures must be 90 degrees. Since the angles are congruent, each must measure 45 degrees. The fact that the triangle is isosceles can be used to determine that the missing side length is 3 in.)

Draw the parallelogram below on the board.

• Say: This is a parallelogram. Take a few minutes to find the missing measures.

Give students a few minutes to work on the problem and then ask for a volunteer to come to the board. Have the student fill in the missing measures and explain the reasoning involved. (Opposite angles are congruent, so y measures 50 degrees. Angles with a common side are supplementary, so x and z each measure 130 degrees. Opposite sides are congruent, so v = 1 in. and w = 2 in.

Repeat the process above for the rectangle and rhombus below.

• Say: Now I am going to give you descriptions of some polygons, and I want you to try to draw them. If you think it is impossible to draw one of the polygons I describe, see if you can figure out why it is impossible. First try to draw an acute scalene triangle.

Give students a few minutes to work on their drawings. Then ask for a volunteer to show his or her drawing and explain why it is both acute and scalene.

• Say: Try to draw a parallelogram with four acute angles.

Students should realize that this is impossible. Ask for a volunteer to explain why.

Continue this process, asking students to draw an obtuse isosceles triangle, a trapezoid with two right angles, and a right triangle with an obtuse angle. (The right triangle with an obtuse angle is impossible.)

Wrap-Up and Assessment Hints
Asking students to construct polygons with various properties is a good way to see if they understand the concepts. For example, students should be able to tell you that you can't have an equilateral triangle with a right angle because all the angles in an equilateral triangle measure 60 degrees. Similarly, if you ask students to draw a rhombus that is not a square, they should be able to explain that the figure would have all four sides congruent but it would have no right angles.