Geometry
Chapter 4 Answers
Triangles
A tessellation is a repeating pattern that completely fills a plane region with congruent figures that do not overlap. Use triangle tessellations to learn about the properties of triangles.
Part A, Angles Inside the Triangle
1. Read through the explanations for creating two different types of tessellations with triangles at the Polygon tessellations Web site.
a. Make a small triangle template from cardboard. You can choose the dimensions of the triangle (but don't make it equilateral). Using this template, create two tessellations, one using each of the methods described.
b. Label the angles on your triangle template A, B, andC. Pick a point in each of your tessellations where six triangle vertices meet. Label angles that meet at the point you've chosen with A, B, or C to indicate the angle of the template with which it corresponds.
[There should be a section on each tessellation which looks similar to those shown below:
]
c. Use the labeled sections of each of your tessellations to find the sum of the measures of the interior angles of your triangle template. In other words, find
and explain how you know that your answer is correct.
Part B, Angles Outside the Triangle
2. Refer to your first tessellation. Using a different colored pen or pencil, highlight the sides of one angle of a triangle and its exterior angle. Use this section of your tessellation to find the relationship between the measure of an exterior angle of a triangle and its two remote interior angles.
Answer:
Triangles are used in many architectural constructions, including bridges. Below, you will examine the use of congruent triangles in bridges.
Part A, Correspondence and Congruence
1. A closer look at congruent triangles.
a. Each of the three rectangular sections (made up of the thick tubing) of the Walnut Street Bridge in Hellertown, PA contains two pairs of congruent triangles. Draw any one pair of these congruent triangles and label the vertices.
b. Write a statement indicating a triangle congruence for the triangles you drew and labeled in a.
c. Write six congruence statements for the sides and angles of the two triangles from a.
Part B, Congruent Triangles
2. Look at a picture of the Tunkhannock Creek Bridge (Nicholson Viaduct). A basic component of this bridge is a design that
looks like:
Suppose you knew the cables were the same length and the metal towers
were the same height. Which congruence postulate could you use to show 
Explain how you know.
Hint: Drawing the triangles separately may help.
Part D, Corresponding Parts
4. Look at the picture in 2 again. Suppose that the only information
you have about the Tunkhannock Creek Bridge is that its cables bisect each
other at point X. How could you prove that the medal towers are the
same height?
Hint: Prove that
first.
Answer:
If you take a close look at the frame of a bicycle, you will probably be able to identify triangular shapes within its structure. Many other structures also have triangles incorporated into their design because of the structural stability of this shape.
Part A, Isoceles Triangles
1. Examine the triangular shapes in the bicycle pictures.
a. Look at the bicycles shown on each of the two sites listed below. Identify the isoceles triangle(s) in each frame. Sketch the portion of each frame which contains the isoceles triangles. Indicate the congruent sides of each triangle.
Hint: Take a close look at the back wheel.
- Site #1: Exploratorium
Look at the first bicycle drawing.
Answer:
- Site #2: Brodie Bikes
Answer:
b. In the two isoceles triangles that you have sketched, measure the angles opposite the congruent sides. What can you conclude about the angles opposite the congruent sides of an isoceles triangle?
[Approximate measurements are shown above for angles. Students should be able to see that the angles opposite the congruent sides of an isoceles triangle are also congruent.]
Part D, Lines Associated With Triangles
2. Print out a copy of the mountain bike and racing bike shown at the sites below.
a. Estimate the location of the center of gravity, or centroid, for the triangle which makes up the main part of the frame in each of the two bicycles.
b. How is the center of gravity different for the mountain bike and the racing bike?
[The center of gravity on the mountain bicycle is lower than the center of gravity on the racing bike.]c. How would this difference be important for the function of each of the bicycles?
[The lower center of gravity aids in stability for mountain biking over the rough terrain.]