INEQUALITIES Sides/Angles of Triangles

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Presentation transcript:

INEQUALITIES Sides/Angles of Triangles Geometry 5.2 Objectives: Learn the Exterior Angle INEQUALITY Theorem To Recognize the LONGEST Side from the Largest Angle To Recognize the LARGEST Angle from the Longest Side

A REMINDER about INEQUALITIES If a > b if there exists a positive number c such that b + c = a If a < b if there exists a positive number c such that a + c = b

EXTERIOR ANGLE INEQUALITY Theorem The measure of an EXTERIOR Angle of a Triangle is GREATER Than the measure of EITHER of its REMOTE INTERIOR Angles.

EXTERIOR ANGLE INEQUALITY Theorem The measure of an EXTERIOR Angle of a Triangle is GREATER Than the measure of EITHER of its REMOTE INTERIOR Angles. C 3 2 1 A D B

INEQUALITY Algebra Assume: a > b 1. If b > c, then a > c (Transitive)

INEQUALITY Algebra Assume: a > b 1. If b > c, then a > c (Transitive) 2. If c = d, then a + c > b + d (Algebra)

INEQUALITY Algebra Assume: a > b 1. If b > c, then a > c (Transitive) 2. If c = d, then a + c > b + d (Algebra) 3. If c > d, then a + c > b + d (Algebra)

INEQUALITY Algebra Assume: a > b 1. If b > c, then a > c (Transitive) 2. If c = d, then a + c > b + d (Algebra) 3. If c > d, then a + c > b + d (Algebra) 4. If b = c, then a > c (Substitution)

LONGEST SIDE - BIGGEST ANGLE Theorem IF ONE SIDE of a Triangle is LONGER than another side, THEN the Angle OPPOSITE the Longer Side is GREATER than the Angle OPPOSITE the Shorter Side.

LONGEST SIDE - BIGGEST ANGLE Th. R Q P Given: Prove: WITHOUT using the BIGGEST Angle LONGEST Side Theorem

TRICHOTOMY Property For any TWO Real Numbers a and b, Only ONE of THREE Possible Relationships Exists: a < b or a = b a > b

BIGGEST ANGLE - LONGEST SIDE Th. IF ONE Angle of a Triangle is GREATER THAN a SECOND Angle, THEN The Side OPPOSITE the GREATER Angle is LONGER than the Side OPPOSITE the Smaller Angle.

BIGGEST ANGLE - LONGEST SIDE Th. R Q P Given: Prove: RQ > RP

HUGE D 58 56 A 63 59 70 54 B C Name the LONGEST Segment.

S 65 60 R 55 70 P 60 50 Q The LONGEST Segment is?

R 1 Q P S Given: Prove:

D C 1 3 2 B A Given: Prove: BD > AD

Geometry 5.2 You should be able to: State the Exterior Angle INEQUALITY Theorem Identify the LONGEST Side from the Largest Angle Identify the LARGEST Angle from the LONGEST Side