IV. (k) 19° (1) 35° (m) 80° 6° (0) 90° Interior and Exterior Angles of Polygons 1. We call angles within a triangle or parallelogram, interior angles to distinguish them from exterior angles which are outside the figure. A 2. Exterior angles of triangles. C (a) To form an exterior angle of a triangle or other polygon extend any side in one direction only. The angle formed by one side of the triangle and an adjacent side extended is an exterior angle. X ла (b) 4 X, Y, and Z are all exterior 4. Can you make others in these triangles? (c) Draw a figure as ▲ ABC. Cut out the at A and C and carefully fit them into the exterior ZX. Do this with several triangles, or until you feel sure that: An exterior angle at one vertex of a triangle is exactly equal to sum of the two interior angles at the other vertices. (d) Test this by measuring the 4 with your protractor. B. TRIANGLES FROM RECTANGLES 1. You have found that the diagonal of a square divides it into two equal triangles. 2. Draw and cut out a rectangle. Fold on one diagonal. Do the two parts fit? 3. Cut along the diagonal. Now can you make the two parts fit? D 4. What part of the rectangle is now in each ▲? C Ꭰ A B 5. (a) In our first formula for the area of a rectangle we used the names length and width because they were the names for the dimensions of the oblong block. We have other names. The side on which the rectangle stands is the base, as the line AB. The side perpendicular to the base is called the height or altitude, as the lines BC or AD. (b) We found that the area of the surface of a rectangle = length x width, or, the surface of a rectangle base x height. = Therefore we may write the formula thus: S = bh C. CLASSIFICATION OF TRIANGLES I. According to Size of Angles Triangles have different names according to the size of their angles. 1. A triangle having one angle a right angle is a right triangle. 2. A triangle having one angle an obtuse angle is an obtuse triangle. 3. A triangle having all angles acute is an acute triangle. An acute triangle having all of its angles equal is called an equiangular triangle. Why? Right Obtuse Acute Equiangular II. According to Equality of Sides Triangles have different names also according to the number of the sides that are equal. 1. A triangle having its 3 sides equal is an equilateral triangle. Equilateral means equal sided. Equilateral Isosceles Scalene 2. A triangle having 2 of its sides equal is an isosceles triangle. Isosceles means equal legs. 3. A triangle having no sides equal is a scalene triangle. Scalene means uneven. 4. We seldom use the word scalene; but when we mean 5. (a) Draw one or more of each kind of triangles as accurately as possible. (b) Label each kind. (c) Measure the angles in each. (d) Find the sum of the angles in each triangle. D. AREAS OF TRIANGLES I. Area of Right Triangle 1. We have seen that the triangles formed by the diagonal of a rectangle are two equal right triangles. 2. Those formed by the diagonal of a square are two equal isosceles right triangles. 4. Which sides of the right may be the base and height? 5. The two sides are called the legs of the rt. ▲. 6. The longest side, the one opposite the right angle, is called the hypotenuse. 7. Hypotenuse means stretched under. Which angle is it stretched under? 8. The hypotenuse may be considered the base of a right triangle. In that case, the altitude or height is a line drawn from the vertex of the rt. angle the hypotenuse. H B 9. How to find the altitude of any triangle (except certain altitudes of obtuse A). (a) By folding. (1) Draw and cut out a triangle. Make a fold from one vertex to the opposite side, so that the two parts of that side fall exactly together. Unfold and measure the on either side of the fold with your protractor or Z-paper. This fold is one of the altitudes of the ▲. (2) Make a similar fold from each of the other vertices. (3) Any side may be the base of a ▲, but with each base goes the to it from the opposite vertex. (4) Do the three altitudes you have folded meet in the same point? If not, try again, for they should meet. (b) By using the protractor. Slide the end of your protractor carefully along the base of your triangle until one side just reaches the opposite vertex. Then draw the altitude. (c) More exact constructions with a compass will be shown in Chapter Four. II. Area of Scalene Triangle. 1. (a) On squared paper, draw a rectangle whose base is 5 inches and whose height is 4 inches. Plain paper may be used, but you must be very careful to make exact right angles. |