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1.2 Congruent Figures The Idea: two things are called congruent if they are essentially the same, but are just sitting in a different position in space. • Two angles are congruent if they have the same measure. • Two line segments are congruent if they have the same length. • Two circles are congruent if they have the same radius. • Two triangles are congruent if all corresponding sides and angles are the same size. • All rays are congruent. • All lines are congruent. 1 Theorem 1.2.1: Vertically opposite angles are congruent. Proof: We use the symbol ≡ to denote congruence. Thus 4ABC ≡ 4DEF if and only if all six of the following hold: ∠A ≡ ∠D ∠B ≡ ∠E ∠C ≡ ∠F AB ≡ DE BC ≡ EF AC ≡ DF. Note that 4ABC ≡ 4DEF and 4ABC ≡ 4EF D are not the same. To check to see if two triangles are equivalent is it necessary to check all 6 of the above equivalences? 2 Three frequently used triangle congruence checks: Axiom 1.2.2, SAS Congruency: Two triangles are congruent if two sides and the included angle of one are congruent to two sides and the included angle of the the other. Theorem 1.2.3, SSS Congruency: Two triangles are congruent if the three sides of one are congruent to the three sides of the other. Theorem 1.2.4 ASA Congruency: Two triangles are congruent if two angles and the included side are congruent to two angles and the included side of the other. Are their others? Why SAS is an axiom and the others are theorems: 3 Axiom 1.2.5, The Triangle Inequality: The sum of the lengths of two sides of a triangle is always greater than the length of the remaining side. 4 Definition. An isosceles triangle is a triangle with two sides of equal length. Theorem 1.2.6, The Isosceles Triangle Theorem: In 4ABC, if AB = AC then ∠ABC = ∠ACB. Proof: 5 Theorem 1.2.7, (Converse of The Isosceles Triangle Theorem): Proof: 6 Theorem 1.2.9, The Angle-Side Inequality: In 4ABC if ∠ABC is larger than ∠ACB, then AC is longer than AB. Proof: 7 Theorem 1.2.10, (Converse of The Angle-Side Inequality:) Proof: Summary: 8