Parallel Lines and Angles Definition Two different given lines L1 and L2 on a plane are said to be parallel if they will never intersect each other no matter how far they are extended Definition Two angles are called vertical angles if they are opposite to each other and are formed by a pair of intersecting lines A Theorem Any pair of vertical angles are always congruent B Parallel Lines and Angles Definition Given two line L1 and L2 not necessarily parallel on the plane a third line T is called a transversal of L1 and L2 if it intersects these two lines L1 L2 T Definitions Let L1 and L2 be two lines not necessarily parallel on the plane and T be a transversal a a and form a pair of corresponding angles b c and form a pair of corresponding angles etc L1 a c L2 T Definitions Let L1 and L2 be two lines not necessarily parallel on the plane and T be a transversal c c and form a pair of alternate interior angles d d and form a pair of alternate interior angles L1 c d L2 T Definitions Let L1 and L2 be two lines not necessarily parallel on the plane and T be a transversal e a and form a pair of alternate exterior angles f b and form a pair of alternate exterior angles L1 a L2 T Theorem Let L1 and L2 be two lines on the plane and T be a transversal If L1 and L2 are parallel then a any pair of corresponding angles are congruent b any pair of alternate interior angles are congruent c any pair of alternate exterior angles are congruent L1 L2 T Theorem Let L1 and L2 be two lines on the plane and T be a transversal a if there is a pair of congruent corresponding angles then L1 and L2 are parallel b if there is a pair of congruent alternate interior angles then L1 and L2 are parallel c if there is a pair of congruent alternate exterior angles then L1 and L2 are parallel L 1 L2 T Congruence of Triangles Definition Given two triangles ABC and XYZ If AB is congruent to XY A is congruent to X BC is congruent to YZ B is congruent to Y CA is congruent to ZX C is congruent to Z then we say that ABC is congruent to XYZ and we write ABC XYZ Y B C A X Z Congruence of Triangles Side Angle Side Principle Given two triangles ABC and XYZ If AB is congruent to XY B is congruent to Y BC is congruent to YZ then ABC is congruent to XYZ Z B Y C A X Congruence of Triangles Angle Side Angle Principle Z Given two triangles ABC and XYZ If A is congruent to X AC is congruent to XZ C is congruent to Z then ABC is congruent to XYZ B A Y C X Congruence of Triangles Side Side Side Principle Given two triangles ABC and XYZ If AB is congruent to XY BC is congruent to YZ CA is congruent to ZX then ABC is congruent to XYZ Z B Y C A X Theorem If ABC is congruent to XYZ then AB is congruent to XY BC is congruent to YZ CA is congruent to ZX and A is congruent to X B is congruent to Y C is congruent to Z In short corresponding parts of congruent triangles are congruent Example 14 5 Show that the diagonals in a kite is perpendicular to each other Recall that a kite is a quadrilateral with 2 pairs of congruent adjacent sides In particular for the following figure AB AD and CB CD A D E C B We first need to show that ADC and ABC are congruent This is true because AD AB b c it is a kite DC BC b c it is a kite AC AC b c they are the same side A and we have the SSS 1 2 congruence principle D C B Therefore click 1 is congruent to 2 Now we only consider ADE and ABE They should be congruent because AD AB 1 2 AE AE Hence SAS principle applies A 1 2 D E B C AED is congruent to AEB and they both add up to 180 hence each one is 90 Similarity of Triangles Definition Given ABC and XYZ If A is congruent to X B is congruent to Y C is congruent to Z and AB XY BC YZ CA ZX then we say that ABC is similar to XYZ and the notation is ABC XYZ Y B A C X Z Similarity of Triangles SSS similarity principle Given ABC and XYZ If AB XY BC YZ CA ZX then ABC is similar to XYZ Y Z B A C X Similarity of Triangles AAA similarity principle Given ABC and XYZ If A is congruent to X B is congruent to Y C is congruent to Z then ABC is similar to XYZ X B A C Z Y Similarity of Triangles AA similarity principle Given ABC and XYZ If A is congruent to X B is congruent to Y then ABC is similar to XYZ because the angle sum of a triangle is always 180 o X B A C Z Y Similarity of Triangles SAS similarity principle Given ABC and XYZ If AB XY BC YZ and B is congruent to Y then ABC is similar to XYZ Y B A C X Z Indirect Measurements If the shadow of a tree is 37 5 m long and the shadow of a 1 5m student is 2 5 m long How tall is the tree assuming that they are all on level ground Indirect Measurements What is the distance between the two points A and B on the rim of the pond 47 m A B o 40 C 58m Indirect Measurements How far is the boat from the point A on shore B How do we measure angles 38 36m C 72 A land Artillery Rangefinder A large 3 5m optical rangefinder mounted on the flying bridge of the USS Stewart a Destroyer Escort A Transit is a surveying instrument to measure horizontal angles Glass reticle on both models has stadia lines for measuring distance Stadia ratio 1 100 Measure the height of Devils Tower in Wyoming The first proper ascent was in 1937 when some of America s best climbers took on the project The Weissner Route was the result a 5 7 decent VS classic on which he placed a single per runner on the crux pitch A bizarre incident took place only a couple of years later in 1941 when local air ace Charles George Hopkins decided to parachute onto the top of the tower to advertise his aerial show He came prepared with a length of rope a block and tackle as well as a sharpened axle from a Model T Ford to act as an anchor for his planned escape His parachute descent went OK but on arrival he found that …