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Applying Congruent Triangles

Applying Congruent Triangles. OBJECTIVES. Special segments in triangles Congruence with right triangles Inequalities in triangles Relationship of sides and angles in a triangle. 3 Medians—from vertex to midpoint of opposite side

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Applying Congruent Triangles

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

  2. 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 Special segments in triangles

  3. 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’) Properties of Bisectors

  4. Right triangles LEG HYPOTENUSE LEG 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 2nd rt. Δ…  …THEN THE Δ’S ARE CONGRUENT

  5. 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 iffthere is some positive number c such that a = b + c Indirect Proof

  6. Inequalities -The Exterior Angle is > sum of 2 remote interior angles -If one side of aΔ 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.

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

  8. 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 3rd side has to be < 20 (? < 13 + 7 ) Answer: the 3rd side is between 6 and 20.

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