Project AMP Dr. Antonio Quesada – Director, Project AMP

Isosceles triangles

Lesson Summary: Students will investigate the properties of isosceles triangles. Angle bisectors, perpendicular bisectors, midpoints, and medians are also examined in this lesson. A nice introduction is included.

Key Words: Isosceles triangle, midpoint, median, angle bisectors, perpendicular bisectors

Existing Knowledge: The students should be familiar with angle bisectors, perpendicular bisectors, midpoints of a segment, and medians of a triangle. Students should be able to use Cabri to draw lines, plot points, find a midpoint, and measure angles or segments.

Existing Knowledge for Extension: The student should be comfortable with small proofs, particularly proofs dealing with congruent triangles.

Materials: Computer(s) with Cabri Geometry II or equivalent software

Objectives: 1. The student will be able to define an isosceles triangle 2. The student will identify the following: Legs of an isosceles triangle Base of an isosceles triangle Base angles of an isosceles triangle Vertex angle of an isosceles triangle 3. The students will deduce the relationship between the base angles of an isosceles triangle. 4. The student will deduce the relationship between the median of the base, perpendicular bisector of the base and the angle bisector of the vertex.

Extension Objectives: 1. The student will word with the properties of the line of symmetry. 2. The student will deduce the properties of a kite. 3. The student will derive the formula for the area of a kite.

Revisit Objective: This lesson also includes a tie in between isosceles triangles and constructions. If your curriculum covers constructions before this lesson, cover this tie in now. If not, save the tie in until constructions are covered in your curriculum.

Project AMP Dr. Antonio Quesada – Director, Project AMP

Procedures: Before the lab: Draw an isosceles triangle on the board or overhead:

Try to avoid using an equilateral triangle. Be sure that it is obvious that the base is not the same length as the 2 legs. Then begin to label and define the sides and angles of the triangle.

Define an isosceles triangle as any triangle with 2 congruent sides Define a leg as one of the congruent sides (AB and AC) Define the vertex angle as the angle containing the congruent sides ( A) Define the base as the side opposite the vertex (BC) *A review of sides opposite an angle may be necessary Define the base angles as the angles opposite the legs ( B and C) * Do NOT mention that these angles are congruent this will be discovered in the lab

Transition to the lab by stating the students will now construct an isosceles triangle, and explore some of its properties using Cabri.

Use a method of your choice to group students. Project AMP Dr. Antonio Quesada – Director, Project AMP

Isosceles Triangles

Team members’ names: ______

File name: ______

Goal: to explore an isosceles triangle, in particular the base angles and line of symmetry.

Part 1: Constructing an isosceles triangle 1. Draw a circle of any size. Label the center A. [Use circle tool] 2. Draw any 2 radii, and label their endpoints on the circle B and C. [Use segment tool] 3. Construct the segment BC. [Use segment tool] Verify that this will be an isosceles triangle. [Use the measure tool]

Now change the size of the circle, and move the two points on the circle. Is the triangle always isosceles?

Now hide the circle and the measurements. Make your triangle large enough to take up the screen.

Part 2: Exploring properties of an isosceles triangle 1. Measure the base angles of your isosceles triangle. What property do you notice? ______[Use the measure tool]

2. Find the median between the vertex and the base. Label the midpoint of the base as D. (The median is the segment connecting the midpoint and the vertex.) [Use the midpoint and segment tools]

3. Find the measure of segments BD and DC. Then find the measure of ADB and ADC. What do these measurements tell you about the relation of AD and BC? [Use the measure tool]

4. Find the measure of BAD and DAC. What does the measure of these angles tell you about the relationship between the BAC and the line segment AD? [Use the measure tool]

5. Use the properties of line ¬¾AD ® to find another method to construct an isosceles triangle.

Project AMP Dr. Antonio Quesada – Director, Project AMP

Extension: In the exploration of the properties of an isosceles triangle you may have realized that the median of the base and vertex, perpendicular bisector of the base and angle bisector of the vertex is the same line. This is called the line of symmetry.

Exploring the line of symmetry:

Create 2 intersecting circles of different sizes. Plot the intersecting points of the two circles. Then draw 2 triangles with the intersection points and each of the circle’s centers. Then connect the 2 centers. [Use the circle tool and the segment tool]

Now Hide the circles and explore the line segment AB.

1. If two isosceles triangles share the same base, do they share the same line of symmetry? Support your answer with measurements from your construction.

2. What relation exists between ADB and ACB?

3. The quadrilateral ADBC is a kite. Try to determine what characteristics of this figure make it a kite. Move the circles and points. What properties remain the same?

4. Try to determine a formula for the area of a kite based on this picture. Project AMP Dr. Antonio Quesada – Director, Project AMP

Extension: In the exploration of the properties of an isosceles triangle you may have realized that the median of the base and vertex, perpendicular bisector of the base and angle bisector of the vertex is the same line. This is called the line of symmetry.

Theorem: In an isosceles triangle, if any 2 of the following facts are true about a line, then all 4 are true, and the line is the line of symmetry.

1. The line passes through the vertex of the triangle. 2. The line bisects the vertex angle of the triangle. 3. The line passes through the midpoint of the base of the triangle. 4. The line is perpendicular to the base of the triangle.

To prove this theorem, you must prove all 6 possibilities:

a. Assume 1 and 2 are true, and prove the line is the line of symmetry. b. Assume 1 and 3 are true, and prove the line is the line of symmetry. c. Assume 1 and 4 are true, and prove the line is the line of symmetry. d. Assume 2 and 3 are true, and prove the line is the line of symmetry. e. Assume 2 and 4 are true, and prove the line is the line of symmetry. f. Assume 3 and 4 are true, and prove the line is the line of symmetry.

See how many of these parts you can prove. Remember you can assume that the triangle is isosceles, so the base angles and legs are congruent

Project AMP Dr. Antonio Quesada – Director, Project AMP

Revisiting isosceles triangles

This is designed to relate isosceles triangles to common constructions.

As shown in the enrichment, isosceles triangles with the same base have the same line of symmetry, and it is the line defined by the vertices of the 2 isosceles triangles. This theorem is the basis of most constructions involving perpendicular lines and angle bisection. For example the construction for an angle bisectors may look like the figure on the right:

If you construct DE, DF and FE, as shown on the left, you can see that the bisector of BAC is the line of symmetry of the two isosceles triangles ADE FDE.

You can also show this for constructing perpendicular lines, like construct a perpendicular line through a line and a point not on the line.

This activity shows the students how different parts of geometry are related.