Inequalities In Two Triangles Indirect Proofs

Hinge Theorem If two sides of a triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.

Converse of the Hinge Theorem If two sides of a triangle are congruent to two sides of another triangle, and the third side of the first triangle is longer than the third side of the second triangle, then the included angle of the first triangle is larger than the included angle of the second triangle.

Example Find the range of possible values for x. JH ≅ GH EH ≅ EH JE > GE m∠JHE > m∠GHE 6x + 15 > 65; x > 8.33 6x + 15 < 180; x < 27.5 8.33 < x < 27.5

Example Compare the given measures. WX and XY

Example Compare the given measures. WX and XY WZ ≅ YZ XZ ≅ XZ WZX < YZX WX < YX

Example Compare the given measures. angle FCD and angle BFC

Example Compare the given measures. angle FCD and angle BFC BF ≅ DC CF ≅ CF BC > DF BFC > FCD

Examples Find the range of possible values for x.

Examples Find the range of possible values for x. 6x +15 > 65

Examples Find the range of possible values for x.

Examples Find the range of possible values for x. 5x + 2 < 47

Indirect Proofs In an indirect proof or proof by contradiction, you temporarily assume that what you are trying to prove is false. By showing this assumption to be logically impossible, you prove your assumption false and the original conclusion true.

Indirect Proofs

Examples State the assumption necessary to start an indirect proof of each statement. A) If 6 is a factor of n, then 2 is a factor of n. B) Angle 3 is an obtuse angle.

Examples State the assumption necessary to start an indirect proof of each statement. A) If 6 is a factor of n, then 2 is a factor of n. 2 is not a factor of n B) Angle 3 is an obtuse angle. Angle 3 is not an obtuse angle

Examples State the assumption necessary to start an indirect proof of each statement. x > 5 Triangle XYZ is an equilateral triangle.

Examples State the assumption necessary to start an indirect proof of each statement. x > 5 x ≤ 5 Triangle XYZ is an equilateral triangle. Triangle XYZ is not an equilateral triangle

Examples Write an indirect proof to show that if -3x + 4 > 16, then x < -4. Given: -3x + 4 > 16 Prove: x < -4

Examples Write an indirect proof to show that if -3x + 4 > 16, then x < -4.

Examples Write an indirect proof to show that if 7x > 56, then x > 8.

Examples Write an indirect proof to show that if 7x > 56, then x > 8. x ≤ 8 7(8) = 56; 7(7) = 49; 7(6) = 42 Since none of the values are > 56, x ≤ 8 is false; therefore x > 8 is true