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6.6/6.7 Isosceles Triangles, Altitudes and Medians Learning Objective: To apply the Isosceles triangle theorem and its converse, write proofs using isosceles triangles, and to identify and explore the properties of medians and altitudes. Warm-up (IN) Complete with < or >. B 65º 1. AB___BC < 2. BC___AC < A 60º 55º Solve each equation. 3. 4x 6 3x 9 15 4. 5x 11 x 3 2 5. 9x 12x 4 4/3 C Learning Objective: To apply the Isosceles triangle theorem and its converse, write proofs using isosceles triangles, and to identify and explore the properties of medians and altitudes. Notes Isosceles Triangle Theorem - If 2 sides of a triangle are congruent, then the angles opposite those sides are congruent. A If AT AD, then T D. Vertex angle legs Converse - T Base angles base D If 2 angles of a triangle are congruent, then the sides opposite those angles are congruent. Learning Objective: To apply the Isosceles triangle theorem and its converse, write proofs using isosceles triangles, and to identify and explore the properties of medians and altitudes. EX 1 – Given: AB CB AE CD Prove: 1 2 B Paragraph proof!! A 1 D 2 E C If AB CB, then A B, by the Isosceles Thm. Because it's given that AB CB and AE CD, ABE CBD by the SAS postulate. Therefore, BE BD because of CPCTC. So 1 2, by the Isosceles Thm. Learning Objective: To apply the Isosceles triangle theorem and its converse, write proofs using isosceles triangles, and to identify and explore the properties of medians and altitudes. Median of a Triangle Segment from a vertex to the midpoint of the opposite side Altitude of a Triangle A perpendicular segment from a vertex to the line that contains the opposite side. Learning Objective: To apply the Isosceles triangle theorem and its converse, write proofs using isosceles triangles, and to identify and explore the properties of medians and altitudes. *In an isosceles triangle, the median, altitude and angle bisector (from the vertex angle) are all the same segment. Learning Objective: To apply the Isosceles triangle theorem and its converse, write proofs using isosceles triangles, and to identify and explore the properties of medians and altitudes. EX 2 – In ABC , mAEC 65 , AC AE , C B 5 X A AD is and altitude, and BE bisects AEC. D Find each measure or length. Explain your reasoning. E a. mDXE b. CE CKC p. 321!! On separate paper! Out – Compare and contrast the altitude and the median of a triangle. POW!! Summary – Today, I understand… Or I’m not too sure about… HW – p. 317#9-15 odd, 20, p. 322 #1,2,6,7,11,12