Date: _____________________________ Section 5 – 1: Bisectors, Medians, and Altitudes Notes – Part A Perpendicular Lines: Bisect: Perpendicular Bisector: a line, segment, or ray that passes through the __________________ of a side of a ________________ and is perpendicular to that side Points on Perpendicular Bisectors Theorem 5.1: Any point on the perpendicular bisector of a segment is _____________________ from the endpoints of the _________________. Example: Concurrent Lines: _____________ or more lines that intersect at a common _____________ Point of Concurrency: the point of ___________________ of concurrent lines Circumcenter: the point of concurrency of the _____________________ bisectors of a triangle 1 Circumcenter Theorem: the circumcenter of a triangle is equidistant from the ________________ of the triangle Example: Points on Angle Bisectors Theorem 5.4: Any point on the angle bisector is ____________________ from the sides of the angle. Theorem 5.5: Any point equidistant from the sides of an angle lies on the ____________ bisector. Incenter: the point of concurrency of the angle ________________ of a triangle Incenter Theorem: the incenter of a triangle is _____________________ from each side of the triangle Example: 2 Example #1: RI bisects ∠SRA. Find the value of x and m∠ IRA . Example #2: QE is the perpendicular bisector of MU . Find the value of m and the length of ME . Example #3: EA bisects ∠DEV . Find the value of x if m∠ DEV = 52 and m∠ AEV = 6x – 10. 3 Example #4: Find x and EF if BD is an angle bisector. Example #5: In ∆DEF, GI is a perpendicular bisector. a.) Find x if EH = 19 and FH = 6x – 5. b.) Find y if EG = 3y – 2 and FG = 5y – 17. c.) Find z if m∠EGH = 9z. 4 Date: _____________________________ Section 5 – 1: Bisectors, Medians, and Altitudes Notes – Part B Median: a segment whose endpoints are a ______________ of a triangle and the ___________________ of the side opposite the vertex Centroid: the point of concurrency for the ________________ of a triangle Centroid Theorem: The centroid of a triangle is located _________ of the distance from a ____________ to the __________________ of the side opposite the vertex on a median. Example: Example #1: Points S, T, and U are the midpoints of DE, EF , and DF , respectively. Find x. 1 Altitude: a segment from a _______________ to the line containing the opposite side and _______________________ to the line containing that side Orthocenter: the intersection point of the ____________________ Example #2: Find x and RT if SU is a median of ∆RST. Is SU also an altitude of ∆RST? Explain. Example #3: Find x and IJ if HK is an altitude of ∆HIJ. 2 Date: _____________________________ Section 5 – 2: Inequalities and Triangles Notes Definition of Inequality: For any real numbers a and b, ____________ if and only if there is a positive number c such that _________________. Example: Exterior Angle Inequality Theorem: If an angle is an ________________ angle of a triangle, then its measures is ________________ than the measure of either of its ________________________ remote interior angles. Example: Example #1: Determine which angle has the greatest measure. Example #2: Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition. a.) all angles whose measures are less than m∠8 b.) all angles whose measures are greater than m∠2 1 Theorem 5.9: If one side of a triangle is ________________ than another side, then the angle opposite the longer side has a _______________ measure than the angle opposite the shorter side. Example #3: Determine the relationship between the measures of the given angles. a.) ∠∠RSU, SUR b.) ∠∠TSV, STV c.) ∠∠RSV, RUV Theorem 5.10: If one angle of a triangle has a ________________ measure than another angle, then the side opposite the greater angle is ________________ than the side opposite the lesser angle. Example #4: Determine the relationship between the lengths of the given sides. a.) AE, EB b.) CE, CD c.) BC, EC 2 Date: _____________________________ Section 5 – 4: The Triangle Inequality Notes Triangle Inequality Theorem: The sum of the lengths of any two sides of a _________________ is _________________ than the length of the third side. Example: Example #1: Determine whether the given measures can be the lengths of the sides of a triangle. a.) 2, 4, 5 b.) 6, 8, 14 Example #2: Find the range for the measure of the third side of a triangle given the measures of two sides. a.) 7 and 9 b.) 32 and 61 1 Theorem 5.12: The perpendicular segment from a ____________ to a line is the _________________ segment from the point to the line. Example: Corollary 5.1: The perpendicular segment from a point to a plane is the ________________ segment from the point to the plane. Example: 2 .