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Transcript

Chapter 9 Congruence, Symmetry, and Similarity Section 9.2 Applications of Transformations Congruent Angles When 2 or 3 lines intersect in certain patterns congruent angles are formed. These particular types of angles come up so often we give them their own names. Vertical Angles When two lines in a plane intersect at a point they form two pairs of vertical angles. Vertical angles have a common vertex (the point of intersection of the lines) but no side in common. The pairs of vertical angles are congruent. A B AOB and COD are vertical angles O C AOC and BOD are vertical angles AOB COD and AOC BOD D The congruence of vertical angles can be established by formal deduction as in van Hiele level 4 as follows: mAOC + mAOB = 180 and mAOB + mBOD = 180 mAOC + mAOB = mAOB + mBOD mAOC = mBOD Corresponding and Alternate Interior Angles When a pair of parallel lines is intersected by another line, the line that intersects the two parallel lines is called a transversal (or transversal line). When this happens several pairs of congruent angles are formed. The pairs of congruent angles are referred to so often we give them special names. In the picture below to the right line l is parallel to line m (i.e. l || m). The line t is the transversal line. t The orange angles are all congruent (i.e. 1 2 1,3,5,7). The green angles are all l 4 3 congruent (i.e. 2,4,6,8). m 5 8 6 7 Pairs of Alternating Interior Angles: Pairs of Corresponding Angles: 3 and 5 4 and 6 1 and 5 2 and 6 Pairs of Alternating Exterior Angles: 3 and 7 4 and 8 1 and 7 2 and 8 The intuitive reasons for why this is true come from translating (i.e. applying a translation transformation) the angle shaped formed by lines l and t along t until it exactly coincides (matches) with the angle formed by the lines m and t. The importance of these relationships is that it allows a student studying geometry to be able to reason a great deal of information about a shape even though they were given only a small amount of information about the shape to start with. This is illustrated by the example below. AB Given parallel to CD x A B w 42 v C 78 58 y z D v = 42 This is because the 42 angle and the v angle are vertical angles. w = 78 This is because the 78 angle and the w angle are corresponding angles. x = 60 The sum of interior angles of a triangle is 180. We get: 78 + 42 + x = 180 y = 42 This is because the v angle and the y angle are alternate interior angles. Z = 80 The sum of interior angles of a triangle is 180. We get: 58 + 42 + x = 180 Congruent Shapes and Parts Shapes that are congruent have the parts that correspond and are congruent. Below consider the example of the two congruent triangles. A B D C E F We say ABC DEF and we have the following parts of the triangles that correspond () and are congruent (): Vertices Sides Angles AD AB DE and AB DE CAB FDE and CAB FDE BE AC DF and AC DF ABC DEF and ABC DEF CF BC EF and BC EF BCA EFD and BCA EFD In shapes that are congruent corresponding parts are congruent (usually we think of this as sides and angles in polygons) and if all corresponding parts are congruent then so are the shapes.