Chapter 5: Inequalities!

By Zach Teplansky

Properties of Inequalities The “Three Possibilities” Property

-Either a > b, a = b, or a < b • The Transitive Property

-If a > b, and b > c, then a > c The Addition property

-If a > b, then a + c > b + c The Subtraction Property

-If a > b, then a – c > b - c The Multiplication Property

-If a > b, and c > 0, then ac>bc The Division Property-

-If a>b, and c>0, then a / c > b / c The Addition Theorem of Inequality

-If a > b and c > d, then a + c > b + d The “Whole Greater than Part” Theorem

-If a > 0, b > 0, and a + b = c, then c > a and c > b

=

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The Exterior Angle Inequality Theorem

An exterior angle of a triangle is an angle that forms a linear pair with an angle of the triangle.

Theorem- An Exterior angle of a triangle is greater than either remote interior angle.

A

B CD

<ACD is an exterior angle of ABC.

• <ACD > <A and <ACD > <B

Triangle Side and Angle Inequalities

Theorem- If two sides of a triangle are unequal , the angles opposite them are unequal in the same order.

A

BC

line segment AB > line segment AC so <C > <B

Triangle Side and Angle Inequalities

(Converse) Theorem- If two angles of a triangle are unequal, the sides opposite them are unequal in the same order.

A

BC

<C > <B, so line segment AB > line segment AC

The Triangle Inequality Theorem

Theorem- The sum of any two sides of a triangle is greater than the third side

A

BC

AB + BC > AC BC + AC > AB AB + AC > BC

The Hinge Theorem The Hinge Theorem-If two sides of

one triangle are equal to two sides of a second triangle and if the included angle of the first triangle is larger than the included angle in the second triangle, then the third side of the first triangle is longer than the third side in the second triangle. (Also known as the SAS inequality Theorem)

The Converse to the Hinge Theorem

The Converse to the Hinge Theorem-If two sides of one triangle are equal to two sides of a second triangle and if the third side of the first triangle is larger than the third side of the second triangle, then the angle between the pair of congruent sides of the first triangle is larger than the corresponding included angle in the second triangle.

LAB: Discovering Triangle Inequalities

In the lab we proved the exterior angle inequality theorem- an exterior angle is greater than either remote interior angles.

We discovered that the sum of the two shorter sides of a triangle must be greater than the larger side – The Triangle inequality theorem.

We also constructed triangles to discover that if two sides of a triangle are unequal then their opposite angles are unequal in the same order – Triangle Side and Angle Inequalities ( Theorem + Converse)

Summary Theorems

Exterior Angle TheoremTriangle Side and Angle Inequality Theorem

and ConverseThe Triangle Inequality TheoremThe Hinge Theorem and Converse

• Concepts Exterior angles The Properties of the Inequalities