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Applying Congruent Triangles

Applying Congruent Triangles. “Six Steps To Success”. 5-1 Special Segments in Triangles. Any point on the perpendicular bisector of a segment is equidistant from the endpoints So…AP is congruent to BP!. 5-1 Special Segments in Triangles.

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Applying Congruent Triangles

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  1. Applying Congruent Triangles “Six Steps To Success”

  2. 5-1 Special Segments in Triangles • Any point on the perpendicular bisector of a segment is equidistant from the endpoints • So…AP is congruent to BP!

  3. 5-1 Special Segments in Triangles • Stated another way, any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment. • If AP is congruent with BP, then P is on the perpendicular bisector

  4. 5-1 Special Segments in Triangles • Any point on the bisector of an angle is equidistant from the sides of the angle. • WE and AB are perpendicular • WE is congruent with WF

  5. 5-2 Right Triangles • LL Theorem – To prove two right triangles congruent when you know the two legs.

  6. 5-2 Right Triangles • HA Theorem - To prove two right triangles congruent when you know the hypotenuse and an acute angle of both triangles.

  7. 5-2 Right Triangles • LA Theorem - To prove two right triangles congruent when you know the leg and an acute angle of both triangles.

  8. 5-2 Right Triangles • HL Postulate - To prove two right triangles congruent when you know the hypotenuse and leg of both triangles.

  9. 5-3 Indirect Proof & Inequalities • Steps for writing an Indirect Proof: • Assume that the conclusion is false • Show that the assumption leads to a contradiction of the hypothesis • Point out that the assumption must be false, and therefore the conclusion must be true. • Definition of Inequality – a relationship between two numbers that are not equal to each other. (Example: < or >)

  10. 5-3 Indirect Proof & Inequalities • Exterior Angle Inequality Theorem – “If an angle is an exterior angle of a triangle, then its measure is greater than the measure of either remote interior angle. • Angle 4 > Angle 2 or Angle 4 > Angle 3

  11. 5-3 Indirect Proof & Inequalities • Working Backward – after assuming that the conclusion is false, you work backward from the assumption to show that for the given information, the assumption itself is false. Example: What is the original number if you multiply it by three and then add nine to get thirty?

  12. 5-4 Inequalities For Triangles • If one side of a triangle is longer than another side, then the angle opposite the first side will be greater than the angle opposite the second. • If BC > AB, then angle A > angle C

  13. 5-4 Inequalities For Triangles • If one angle of a triangle is longer than another angle, then the side opposite the first angle will be greater than the side opposite the second angle. • If angle A > angle C, then BC > AB

  14. 5-4 Inequalities For Triangles • The perpendicular segment from a point to a line is the shortest segment from the point to the line. Example: What is the shortest distance between ST and point V?

  15. 5-4 Triangle Inequality • Triangle Inequality Theorem – the sum of the lengths of any two sides of a triangle is greater than the length of the third side. Example: If sides of a figure are 15, 32, and 16 could the figure be a triangle? Answer: NO! (15 + 16 is not > 32)

  16. 5-4 Triangle Inequality • Example: If sides of a figure are 3, 12, and 7 could the figure be a triangle? Answer: NO! (3 + 7 is not > 12) • Example: If sides of a figure are 34, 22, and 17 could the figure be a triangle? Answer: Yes! (17 + 22 > 34)

  17. Chapter 5 Proofs

  18. Chapter 5 Proofs • More to come soon!!!

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