This chapter covers the rules of geometry most often used to solve engineering problems. The relationships of similar triangles, and the angles and sides of right triangles are often used in engineering calculations.
After completing this chapter, the student will be able to do the following.
An angle is formed by two lines converging to a single point. The lines are called sides of the angle and the point is called the vertex. The symbol, Ð , is used for the word, "angle". Angles are commonly named in two ways, by the letters on the two sides and the vertex (with the vertex in the middle) or by a letter or number written inside the angle. The angle in Fig. 1 could be called:
Fig. 1
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| An angle less than 90° is called an acute angle | |
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EXAMPLE 1
EXAMPLE 2
Intersecting lines form two pairs of opposite angles. These opposite angles of intersecting straight lines are equal.
Ð a =Ð c and Ð b = Ð d. Therefore, if any one of these angles is known, the remaining three can be calculated.
EXAMPLE 3
| GIVEN:
FIND: Ð a, Ð b, and Ð c. |
Ða = 70° (opposite angle) Ð a + Ð c = 180° Ð c = 180° - Ð a = 180° - 70° Ð c = 110° Ð b = 110° (opposite angle of Ð c) ANSWER: Ð a = 70° Ð b = 110° Ð c = 110° |
Two lines are parallel if they are in the same plane and do not intersect.
Two lines perpendicular to the same line are parallel to each other.
When two parallel lines are intersected by a third line, they form two pairs of alternate interior angles. In Fig. 8 one pair of alternate interior angles is labeled "a". The other pair is labeled "b".
Fig. 8
Alternate interior angles are always equal. Using the fact that opposite angles of intersecting lines are equal , we can label the remaining four angles in Fig. 9. Since all the lines in Fig. 9 are straight, we know that Ð a + Ð b = 180°
With that information, we can FIND all the angles if we know one.
Fig. 9
EXAMPLE 4
| GIVEN:
Ð a = 40° FIND: angles b thru h |
Ð d = Ð a ( opposite angles of intersecting lines ) Ð e = Ð d ( alternate interior angles ) Ð f = Ð e ( opposite angles of intersecting lines ) Ð a + Ð b = 180° ( supplementary angles ) Ð b = 180° - Ð a = 180° - 40° Ð b = 140° Ð c = Ð b ( opposite angles of intersecting lines ) Ð h =Ð c ( alternate interior angles ) Ð g = Ð h ( opposite angles of intersecting lines ) ANSWER: Ð a = Ð d = Ð e = Ð f = 40°
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| A triangle
is the union of three lines connecting three points that
are not in a straight line. This triangle is made of the
lines connecting points A, B, and C. The three angles of
a triangle total 180°. Since the three angles of a
triangle total 180° and one angle of a right triangle is
90°, the other two angles are complementary, they total
90°. Triangles are GIVEN special names with respect to the length of their sides and angles. |
| A scalene triangle has no equal sides. |  |
| An obtuse triangle has one obtuse angle. |  |
| An isosceles triangle has at least two equal sides. In this triangle sides, DE and EF are equal. |  |
| An equilateral triangle has three equal sides. Since the three angles are also equal, it is also called an equiangular triangle. |  |
| An acute triangle has three acute angles. |  |
| A right triangle has a right angle. |  |
| In a right triangle the side opposite the right angle is the longest side and is called the hypotenuse. |  |
| Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional. This means that if two triangles have equal angles, then the sides are proportional. It also means that if two triangles have proportional sides, then the angles are equal. |
The two triangles are similar. This means that Ð d' = Ð d and that Ð b' = Ð b. It also means that the ratios of the lengths of corresponding sides are the same. THE CORRESPONDING SIDES ARE THE SIDES OPPOSITE EQUAL ANGLES.
Sides X, Y, and Z correspond to sides X', Y', and Z'.
EXAMPLE 5
EXAMPLE 6
Solve these problems using the examples as a guide. When done contact your instructor for solutions and a self-grading guide.
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