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 Any point on the perpendicular bisector of a segment is equidistant from the endpoints So…AP is congruent to BP! So…AP is congruent to BP!

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. Stated another way, any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment. If AP is congruent If AP is congruent with BP, then P is with BP, then P is on the perpendicular on the perpendicular bisector bisector

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

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

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

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

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

5-3 Indirect Proof & Inequalities Steps for writing an Indirect Proof: 1. A ssume that the conclusion is false 2. S how that the assumption leads to a contradiction of the hypothesis 3. P oint 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 >)

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

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. 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?

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 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 If BC > AB, then angle A > angle C

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 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 If angle A > angle C, then BC > AB

5-4 Inequalities For Triangles The perpendicular segment from a point to a line is the shortest segment from the point to the line. 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?

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. 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! ( is not > 32)

5-4 Triangle Inequality Example: 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: Example: If sides of a figure are 34, 22, and 17 could the figure be a triangle? Answer: Yes! ( > 34)

Chapter 5 Proofs

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