Triangle Inequalities

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Presentation Description

Students learn inequality properties and how these properties apply to lengths of the sides of triangles.

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Presentation Transcript

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Triangle Inequalities § 7.1 Segments, Angles, and Inequalities § 7.4 Triangle Inequality Theorem § 7.3 Inequalities Within a Triangle § 7.2 Exterior Angle Theorem

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Segments, Angles, and Inequalities You will learn to apply inequalities to segment and angle measures. 1) Inequality

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Segments, Angles, and Inequalities The Comparison Property of Numbers is used to compare two line segments of unequal measures. The property states that given two unequal numbers a and b, either: a < b or a > b The same property is also used to compare angles of unequal measures.

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Segments, Angles, and Inequalities The measure of J is greater than the measure of K. The statements TU > VW and J > K are called __________ because they contain the symbol < or >. inequalities a < b a = b a > b

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Segments, Angles, and Inequalities SN DN 6 – (- 1) 6 – 2 7 4 > >

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Segments, Angles, and Inequalities AB > AC AB > CB A similar theorem for comparing angle measures is stated below. This theorem is based on the Angle Addition Postulate.

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Segments, Angles, and Inequalities A similar theorem for comparing angle measures is stated below. This theorem is based on the Angle Addition Postulate.

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Segments, Angles, and Inequalities mBDA mCDA 45° 40° + 45° < < Use theorem 7 – 2 to solve the following problem. 45° 85°

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Segments, Angles, and Inequalities For any numbers a, b, and c, 1) if a < b and b < c, then a < c. 2) if a > b and b > c, then a > c. if 5 < 8 and 8 < 9, then 5 < 9. if 7 > 6 and 6 > 3, then 7 > 3.

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Segments, Angles, and Inequalities For any numbers a, b, and c, For any numbers a, b, and c, 1) if a < b, then a + c < b + c and a – c < b – c. 2) if a > b, then a + c > b + c and a – c > b – c. 1 < 3 1 + 5 < 3 + 5 6 < 8

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Exterior Angle Theorem You will learn to identify exterior angles and remote interior angles of a triangle and use the Exterior Angle Theorem. 1) Interior angle 2) Exterior angle 3) Remote interior angle

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Exterior Angle Theorem In the triangle below, recall that 1, 2, and 3 are _______ angles of ΔPQR. interior Angle 4 is called an _______ angle of ΔPQR. exterior An exterior angle of a triangle is an angle that forms a _________ with one of the angles of the triangle. linear pair In ΔPQR, 4 is an exterior angle at R because it forms a linear pair with 3. ____________________ of a triangle are the two angles that do not form a linear pair with the exterior angle. Remote interior angles In ΔPQR, 1, and 2 are the remote interior angles with respect to 4.

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Exterior Angle Theorem In the figure below, 2 and 3 are remote interior angles with respect to what angle? 5

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Exterior Angle Theorem remote interior angles m4 = m1 + m2

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Exterior Angle Theorem

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Exterior Angle Theorem remote interior angles m4 > m1 m4 > m2

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Exterior Angle Theorem 1 and 3 Name two angles in the triangle below that have measures less than 74°. acute

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Exterior Angle Theorem

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Exterior Angle Theorem The feather–shaped leaf is called a pinnatifid. In the figure, does x = y? Explain. __ + 81 = 32 + 78 28 28° 109 = 110 No! x does not equal y

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Inequalities Within a Triangle You will learn to identify the relationships between the _____ and _____ of a triangle. sides angles Nothing New!

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Inequalities Within a Triangle in the same order LP < PM < ML mM < mP mL <

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Inequalities Within a Triangle in the same order JK < KW < WJ mW < mK mJ <

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Inequalities Within a Triangle greatest measure WY > XW 3 5 4 WY > XY

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Inequalities Within a Triangle The longest side is So, the largest angle is The largest angle is So, the longest side is

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Triangle Inequality Theorem You will learn to identify and use the Triangle Inequality Theorem. Nothing New!

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Triangle Inequality Theorem greater a + b > c a + c > b b + c > a

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Triangle Inequality Theorem Can 16, 10, and 5 be the measures of the sides of a triangle? No! 16 + 10 > 5 16 + 5 > 10