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G-09 Congruent Triangles and their parts “I can name corresponding sides and angles of two triangles.” Reflexive Property AB = AB Symmetric Property • If A = B, then B = A Transitive Property • If A = B and B = C, then A = C Addition, Subtraction, Multiplication, Division Property (=) Distributive Property • If A(B + C), then AB + AC Or • If (B + C)A, then BA + CA Substitution • If A = B, then A can be substituted for any B in the expression Angle/Segment Addition Postulate Definition of Congruence • If AB = CD, then AB CD • Congruent segments are segments that have the same length. • Congruent angles are angles that have the same measure. Definition of Vertical Angles • Vertical angles are two nonadjacent angles formed by two intersecting lines. • Vertical Angles are congruent • 1 and 2 are vertical angles Definition of Perpendicular Lines • Perpendicular lines intersect to form 90 angles. • Perpendicular lines are form congruent angles If ABC CBD then ABC 90 and CBD 90 so ABC CBD Definition of Complementary/Supplementary Angles • Complementary Angles: 2 angles that add up to be 90° mA mB 90 • Supplementary Angles: 2 angles that add up to be 90° mA mB 180 Definition of Midpoint/Bisector • The midpoint M of AB is the pt that bisects, or divides, the segment into 2 congruent segments. • (segments) If M is the midpt of AB, then AM = MB • An angle bisector is a ray that divides an angle into two congruent angles. JK bisects LJM; thus LJK KJM. Definition of Right Angles • All right angles are congruent • If A and B are right angles, then A B Third Angle Theorem Definition of Congruent Triangles • If two or more triangles have corresponding angles and sides that are congruent, then those triangles are congruent. • In a congruence statement, the order of the vertices indicates the corresponding parts. Helpful Hint When you write a statement such as ABC DEF, you are also stating which parts are congruent. Example 1 A. Given: ∆PQR ∆STU Identify all pairs of corresponding congruent parts. • Angles: • Sides: Example 1 B. Given: ∆ABC ∆DEF Identify all pairs of corresponding congruent parts. • Angles: • Sides: Example 1 C. Given: ∆JKM ∆LKM Identify all pairs of corresponding congruent parts. • Angles: • Sides: Example 2 A. Given: polygon ABCD EFGH • BCD ________ Example 2 B. Given: polygon ABCD EFGH • BAD ________ Example 2 C. Given: polygon DEFGH IJKLM • MLK ________ Example 3a: • Given: K is the midpt. of JL, JM LM , JKM LKM , J L Prove: MKJ MKL Statement Reason Statement K is the midpt. of JL JK KL JM LM KM KM JKM LKM JKM and LKM are right angles JKM LKM J L KMJ KML MKJ MKL Reason Given Definition of Midpoint Given Reflexive Property Given Definition of Perpendicular lines Right angles are congruent Given Third Angle Thm. Definition of Congruent Triangles Example 3b Given: YWX and YWZ are right angles. YW bisects XYZ. W is the midpoint of XZ. XY YZ. Prove: ∆XYW ∆ZYW Example 3b: Statement Reason YWX and YWZ are right angles. Given YWX YWZ YW bisects XYZ XYW ZYW W is mdpt. of XZ Given Given XW ZW YW YW X Z XY YZ ∆XYW ∆ZYW Given Example 3c Given: AD bisects BE. BE bisects AD. AB DE, A D Prove: ∆ABC ∆DEC Example 3c: Statement A D BCA DCE ABC DEC AB DE AD bisects BE, BE bisects AD BC EC, AC DC ∆ABC ∆DEC Reason Given Given Given Example 3d • Given: PR and QT bisect each other. PQS RTS, QP RT • Prove: ∆QPS ∆TRS Example 3d: Statement QP RT PQS RTS PR and QT bisect each other QS TS, PS RS QSP TSR QSP TRS ∆QPS ∆TRS Reason Given Given Given Example 3e Use the diagram to prove the following. Given: MK bisects JL. JL bisects MK. JK ML. JK || ML. Prove: ∆JKN ∆LMN Example 3e: Statement JK ML JK || ML JKN NML JL and MK bisect each other. JN LN, MN KN ∆JKN ∆LMN Reason Given Given Given Vert. s Thm. Third s Thm. Def. of ∆s Example 4a Given: ∆ABC ∆DBC. 1. Find the value of x. 2. Find mDBC. Example 4b • Given: ∆ABC ∆DEF 1. Find the value of x. 2. Find mF. Example 4c • Given: ∆ABD ∆CBD 1. Find the value of x. 2. Find AD. Example 4d • Given: ∆RSU ∆TSU 1. Find the value of x. 2. Find UT.