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Geometry Lesson 1
Introduction to Geometry

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Instruction 1-3

Definitions | Measuring Angles | Line and Angle Relationships | Triangles | Definitions of Figures

Lines and Angles: Intersecting, Parallel and Transversal Lines, Exterior, Interior, and Corresponding Angles

Intersecting Lines

If two lines or segments lie in a plane, they can have only two types of relationships. Either they are parallel to or they intersect each other. If they intersect without completely overlapping, they form four angles around the point of intersection (Figure 1.10).

parallel lines
Figure 1.10

Vertical Angles
Look at Figure 1.11. Here, we named the angles formed by the intersecting segments IQ and CD as 1, 2, 3, and 4. These angles can be grouped in two sets (1, 3) and (2, 4). Pairs of angles in each group only share a common vertex. They do not have any common sides. Such angles are called vertical angles. Vertical angles are always are congruent. That is,
1 3 and 2 4. For example, if m1 = 125˚, then m3 = 125˚.

Figure 1.11

Straight Angle

If two sides of an angle lie on a straight line, it is called a straight angle (Figure 1.12). The measure of any straight angle is always 180˚. So by knowing m
1 in Figure 1.11, we can determine m3 and m4. We see that 1 and 2 together form a straight angle we can calculate m3 = 180˚ 125˚ = 65˚. Similarly, m4 = 180˚ 125˚ = 65˚.

Figure 1.12

Adjacent Angles

Two angles are called adjacent if:

  1. they have a common vertex and side
  2. their non-common sides lie opposite areas of the common side.

For example, JKM and MKN are adjacent angles (Figure 1.13).

Figure 1.13

Supplementary Angles

If the measures of two angles sum up 180˚, they are called supplementary angles. Each of a pair of supplementary angles is called a supplement of the other.

Complementary Angles

If the measures of two angles sum up 90˚, they are called complementary angles. Each of the two complementary angles is called a complement of the other.

Parallel Lines and Transversals

If two or more lines or segments lie on a plane and never intersect each other, they are called parallel lines or parallel segments. Two parallel lines are always in the same distance from each other. We denote parallel lines using ||. In Figure 1.14,   and are parallel. We denote these segments as  .

If a line or a segment intersects two or more parallel lines, it is called a transversal. For example, is a transversal segment with respect to parallel segments and (Figure 1.14).

Figure 1.14

Exterior and Interior Angles
When a transversal intersect two parallel lines, eight angles are formed around the points of intersection (Figure 1.15). The angles that lie outside the area bounded by the parallel lines are called exterior angles. For example,
1, 2, 6, and 7 in Figure 1.15 are exterior angles. The angles that lie inside the area bounded by the parallel lines are called interior angles. In Figure 1.15, 3, 4, 5, and 8 are interior angles.

Figure 1.15

Alternate Interior Angles

If two interior angles located on the opposite sides of the transversal, they are called alternate interior angles. In Figure 1.15, pairs of (4, 5) and (3, 8) are alternate interior angles. Each pair of alternate interior angles are always congruent. That is, 4 5 and 3 8.

Alternate Exterior Angles
If two exterior angles lie on the opposite sides of the transversal, they are called alternate exterior angles. In Figure 1.15, (
1, 6) and (2, 7) are pairs of alternate exterior angles. Each pair of alternate exterior angles are always congruent. That is, 1 6 and 27.

Corresponding Angles

If an alternate exterior angle and an alternate interior angle lie on the same side of the transversal, they are called corresponding angles. In Figure 1.15, pairs (1, 8), (2, 5), (3, 6), and (4, 7) are corresponding angles. Each pair of corresponding angles are supplementary.

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for Students, Parents and Teachers

Now let's do Practice Exercise 1-3 (top).


Next Page:  Triangles (top)