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TOPIC 7-1: Isosceles Triangles and Special Segments in Triangles Term Definition Sketch Isosceles Triangle A triangle with 2 congruent sides. A EXAMPLE 1: legs _____________________ base ___________________ base angles ________________ vertex angle _______________ C B If two sides of a triangle are congruent, then the angles opposite them are also congruent. If two angles of a triangle are congruent, then the sides opposite them A are also congruent. If AB ≅ AC, then ________________. If ∠C ≅ ∠B, then ________________. B EXAMPLES: Find the value of ‘x’ in each. 2. 3. 30°° x°° x°° 40°° C 4. 5. 2x + 2 41 42 2x - 4 x+5 62°° 56°° x Term Definition Sketch angle bisector A ray that cuts an angle into 2 congruent angles. altitude Perpendicular segment from any vertex to its opposite side. Median Line segment drawn from any vertex to the midpoint of opposite side. Perpendicular Segment that is perpendicular bisector to a side at its midpoint. Every triangle has 3 of each of these. EXAMPLES: 6. BG is a median. Find the value of ‘x’ if BG = 4x + 10, A AG = 6x + 4, CG = 7x – 5. G B C 7. Given AF is an altitude, find the value of x if m∠ ∠1 = (8x+18)°. B A 1 F 2 C An isosceles triangle is a special case. In the picture below, the segment AD is an angle bisector, altitude, median, and perpendicular bisector. A m∠ ∠__________ = m∠ ∠__________ ________ ⊥ ________ ________ ≅ ________ B D C As long as it is drawn from the vertex angle, a segment that is one of the special segments is all four of the special segments. EXAMPLE 8: In isosceles ∆ABC below, BD is an angle bisector. Find the values of ‘x’, ‘y’, and ‘z’ if m∠ ∠1 = (6x + 7)°, m∠ ∠2 = (3x + 16)°, m ∠3 = (3y - 1)°, AD = 2z + 1, and DC = 5z – 8. B A D C