Warm up Draw two congruent angles: Draw two adjacent angles:

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Presentation transcript:

Warm up Draw two congruent angles: Draw two adjacent angles: Draw two angles that are complementary: Draw an angle which also has a ray that bisects the angle:

Proving Statements about Segments and Angles Chapter 2.5

We learned these in Chapter 1! To recap (again..): 2 column proof: has numbered STATEMENTS on one side and corresponding REASONS on the other So far, we have proved statements using algebra operations and properties. Today we prove statements using geometric properties We learned these in Chapter 1!

Adding to our proof reason list (purple sheet) Segment Addition Postulate Segment Congruence Property (reflexive, symmetric, transitive) Angle Congruence Property (reflexive, symmetric, transitive) m <PQR = m <PQS + m<SQR Angle Addition Postulate

Congruent / Equal “Congruent” is different from “equal” M L K J Things that are EQUAL have a numeric value, like the measure of an angle in degrees, or the length of a line segment. m <ABC = 30⁰ LM = 45 cm L M 45 cm 30⁰ Shapes are CONGRUENT, things like angles, line segments, and polygons. Two shapes are congruent when they have the same measure. m<ABC = 30⁰ and m<XYZ = 30⁰, so <ABC and <XYZ are congruent LM = 45 cm and JK = 45 cm So LM and JK are congruent 30⁰ x y z J K 45 cm

Properties of Congruence

Properties of Congruence

Example.. 1. m < 2 = 145⁰ 2. < 1 and < 2 are supplementary 3. m < 1 + m < 2 = 180⁰ 4. m < 1 + 145⁰ = 180⁰ 5. m < 1 = 35⁰ 1. Given 2. Given 3. Definition of Supplementary Angles 4. Substitution Property of Equality 5. Subtraction Property of Equality

Example 1. < 1 is a complement of < 2 2. <2 is congruent to < 3 3. m < 1 + m < 2 = 90⁰ 4. m < 2 = m < 3 5. m < 1 + m < 3 = 90⁰ 6. < 1 is a complement of < 3 1. Given 2. Given 3. Definition of Complementary Angles 4. Definition of congruent angles 5. Substitution Property of Equality 6. Definition of Complementary Angles

Example Solve for x. Justify each step. 1. QR ≅ PQ 2. RS ≅ PQ 3. QR ≅ RS 4. QR = RS 5. (2x + 5) = (10 – 3x) 6. 5x + 5 = 10 7. 5x = 5 8. x = 1 1. Given 2. Given 3. Transitive Prop Equality 4. Definition of congruent segments 5. Substitution 6. Addition Prop Equality 7. Subtraction Prop Equality 8. Division Prop Equality