Applying Congruent Triangles Special segments in triangles Congruence with right triangles Inequalities in triangles Relationship of sides and angles in a triangle OBJECTIVES

Special segments in triangles 3 Medians—from vertex to midpoint of opposite side 3 Altitudes—from vertex to opposite side, hitting it at a 90°angle (can be outside if it’s an obtuse Δ) 3 Angle bisectors—bisects the angle & hits side opposite 3 Perpendicular bisectors—bisects side at 90 °angle

Properties of Bisectors Perpendicular bisectors: Any point on the perpendicular bisector is equidistant from endpoints of the segments ( & ‘vice versa’) Angle bisectors: Any point on the angle bisector is equidistant from the sides of the angle (& ‘vice versa’)

Right triangles LL : IF the legs of one right Δ are to the corresponding legs of another rt. Δ … HA : IF the hypotenuse & an acute angle of one right Δ are to the hyp. & acute angle of a 2nd rt.Δ … LA : IF one leg and an acute angle of 1 rt. Δ are to the corr. leg and acute angle of a 2nd rt. Δ … HL : IF the hyp.& a leg of 1 rt. Δ are to the hyp. & corr. leg of a 2 nd rt. Δ…  …THEN THE Δ’S ARE CONGRUENT LEG HYPOTENUSE

Indirect Proof 1. Assume the conclusion is false. 2. Show that it leads to a contradiction or an impossible statement (type of working backwards to solve). 3. Point out the assumption (#1) must be incorrect  the conclusion is really true. Inequalities Definition: a > b iff there is some positive number c such that a = b + c

Inequalities - The Exterior Angle is > sum of 2 remote interior angles - If one side of a Δ is longer than another, then the opp the longer side > the opp the shorter side. - If an of a Δ is > another, the side opp the greater will be longer than the side opp the smaller - The | segment from a point to a line is the shortest distance(segment) from the pt to the line - The | segment from a point to a plane is the shortest distance (segment) from the pt to the plane.

Triangle Inequalities The sum of the lengths of any two sides of a Δ is greater than the 3 rd side. The smallest side of the Δ is opposite the smallest and the largest side is opposite the largest

Example: Triangle Inequality If two sides of a triangle are 7 & 13, between what two numbers must the third side be? - if 13 is the largest side then the smallest side had to be > 6 (13 < 7 + ? ) -If 7 & 13 are the 2 smaller sides, the 3 rd side has to be < 20 (? < ) Answer : the 3 rd side is between 6 and 20.