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