2. Angles and Triangles

The angle formed by segments BA and BC (or rays BA and BC) is denoted by the point B is its vertex. If no other angle with vertex at B is under consideration, may be shortened to just Triangle ABC is denoted by

Figure 4a

Figure 4b

The sum of the lengths of two sides of a triangle is always greater than the length of the third side. For example, a + b > c in Figures 4a and 4b. Angles are measured either in degrees or radians. The degree measure of a right angle is 90^o and its radian measure is The sum of the degree measures of the angles of a triangle is 180^o and the sum of their radian measures is An angle of a triangle is acute, right, or obtuse depending on whether its degree measure is respectively, less than , equal to, or greater than 90^o. The notation means that the degree measure of is 30^o and means that the radian measure of is

A triangle with one 90^o angle is called a right triangle. The side opposite the right angle is called the hypotenuse. In Figure 4b, and side c is the hypotenuse. The famous Theorem of Pythagoras states that a triangle is a right triangle if and only if the square of the length of one side equals the sum of the squares of the lengths of the other two sides. (For example, c² = a² + b² in Figure 4b.) The main steps of a proof follow:

Let have a right angle at C. Place a square of side c externally on the hypothenuse AB. Then place copies of on the other three sides of the square. All together, we now have a big square whose sides have length a + b. See Figure 5.

Now we get

Figure 5

Calculator Example 1.2.1

When laying out the foundation of a new building, stakes are driven at the four corners. Measuring the proper distances between the stakes is relatively easy, but measuring the angles properly is much more difficult. After driving the stakes, the builder always measures the diagonal distance to make sure it satisfies the Theorem of Pythagoras thereby insuring that the angles are 90^o. Suppose that the points A, B, and C in Figure 4b are three of the four corner stakes of a rectangular foundation, a = 36 ft, and b = 40 ft. What must c be to insure Give your answer to the nearest 10th of an inch.

Solution: We will assume the calculator has already been set to Fix 1 display mode (see Preface). We will let the calculator keep track of the units for us, so the first sequence is to get to the appropriate units menu: RS UNITS F2-LENGTH. For each of the given lengths we will enter the value, attach the units, then square it: 36 F5-ft LS x² 40 F5-ft LS x². We now have the squares of a and b on the stack. To find c we must add these, take the square root, then convert the result to inches: + LS F6-in. We see the answer 645.8_in on the display. For complete instructions on the use of units see "Operations with Units" starting on page 3-17 of UG.