Chapter 5 Relationships in Triangles

Warm - Up Textbook – Page – 11 (all) This will prepare you for today’s lesson

Perpendicular Bisector A line, segment, or ray that passes through the midpoint of the side and is perpendicular to that side Special Properties – Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment – Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment.

Terms When three or more lines intersect at a common point, the lines are called concurrent lines Their point of intersection is called the point of concurrency The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter

Circumcenter Theorem The circumcenter of a triangle is equidistant from the vertices of the triangle

Theorems Any point on the angle bisector is equidistant from the sides of the angle Any point equidistant from the sides of the angle lies on the angle bisector

Angle Bisectors There are three angle bisectors in any triangle The angle bisectors of a triangle are concurrent, ant their point of concurrency is called the incenter of a triangle.

Incenter Theorem The incenter of a triangle is equidistant from each side of the triangle.

Terms A median is a segment whose endpoints are a vertex of a triangle and the midpoint of the side opposite the vertex. – Each triangle has three medians The medians of a triangle also intersect at a common point, called the centroid – The centroid is the point of balance for any triangle

Centroid Theorem The centroid of a triangle is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median.

Terms An altitude of a triangle is a segment form a vertex to the line containing the opposite side and perpendicular to the line containing that side Every triangle has three altitudes The intersection point of the altitudes of a triangle is called the orthocenter

Concept Summary NameTypePoint of Concurrency Perpendicular Bisectors Line, segment, or ray Circumcenter Angle BisectorLine, segment, or ray Incenter MedianSegmentCentroid AltitudeSegmentOrthocenter

Inequality For any real numbers a and b, a > b if and only if there is a positive number c such that a = b + c Example: – If 6 = 4 + 2, then 6 > 4 and 6 > 2

Example Determine which angle has the greatest measure

You Do It Determine which angle has the greatest measure

Exterior Angle Inequality Theorem If an angle is an exterior angle of a triangle, then its measure is greater than the measure of either of its corresponding remote interior angles.

Example Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition – All angles whose measures are less than angle 8 – All angles whose measures are greater than angle 2

You Do It Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition – All angles whose measures are less than angle 14 – All angles whose measures are greater than angle 5

Theorem If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle

Example Determine the relationship between the measures of the given angles – Angle ABD, Angle DBA – Angle CDA, Angle CBA

You Do It Determine the relationship between the measures of the given angles – Angle RSU, Angle SUR – Angle TSV, Angle STV – Angle RSV, Angle RUV

Classwork Worksheet