MELANIE DOUGHERTY GEOMETRY JOURNAL 5

Describe what a perpendicular bisector is. Explain the perpendicular bisector theorem and its converse. A perpendicular bisector is a line perpendicular to the base of a triangle that bisects it. Perpendicular Bisector theorem:  If a point is on the perpendicular bisector of a segment, then it is equidistant form the endpoints of the segment. Converse:  if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

Perpendicular Bisector Examples AB = AC

AC = BC

PB Converse Examples LN = EN

AD = DC

CD = DB

Describe what an angle bisector is. Explain the angle bisector theorem and its converse. An angle bisector is a line that divides the angle. The angle Bisector theorem:  If a point is on the bisector of an angle, then it is equidistant from the sides of the angle Converse:  If a point is equidistant from the sides of an angle the it is on the bisector.

Angle Bisector theorem examples BF = FC

<UFK is congruent to <KFC

<EWR is congruent to <RWT

CONCURRENT When 3 or more lines intersect at one point

Concurrency of perpendicular bisector theorem of triangles The circumcenter of a triangle is equidistant from the vertices of the triangle. Circumcenter: where the 3 perpendicular bisectors of a triangle meet circumcenter

DA = DB = DC acute right obtuse

concurrency of angle bisectors of a triangle theorem Incenter of a triangle : where the 3 angle bisectors of a triangle meet Concurrency of a angle bisectors of a triangle theorem: the incenter of a triangle is equidistant from the sides of the triangle. incenter

DF = DG = DE ACUTE RIGHT OBTUSE

MEDIANS AND ALTITUDES OF TRIANGLES The median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side The centroid of a triangle is the point of concurrency of the medians of a triangle. Concurrency of medians of a triangle theorem: the centroid of a triangle is located 2/3 of the distance from each vertex to the midpoint of the opposite side.

EXAMPLES MEDIAN CENTROID CMTT

Concurrency of altitudes of triangles theorem Altitude: a perpendicular segment from a vertex to the line containing the opposite side Orthocenter: point where the 3 altitudes of a triangle meet. Concurrency of altitudes of triangles theorem: the lines containing the altitude are concurrent

Triangle Midsegment theorem A midsegment is a segment that joins the midpoints of two sides of a triangle Midsegment theorem: a midsegment of a triangle is parallel to a side of the triangle, and its length is half of that side. midsegment

AB ll EF, EF = ½ ABDE ll BC, DE = ½ BC DE ll AC, DE = ½ AC

Angle-Side Relationship in Triangles If none of the sides of the triangle are congruent then the largest side is opposite the largest angle. If none of the sides of the triangle are congruent then the shortest side is opposite the smallest angle.

EXAMPLES

Triangle Inequality The sum of the lengths of two sides of a triangle is greater than the length of the third side.

Writing an indirect proof 1. Identify what is being proven 2. Assume that the opposite of your conclusion is true 3. Use direct reasoning to prove that the assumption has a contradiction 4. Assume that if the 1 st assumption is false then what is being proved is true.

EXAMPLES Step 1 Given: triangle JKL is a right triangle Prove: triangle JKL doesn't have and obtuse angle Step 2 Assume <K is an obtuse angle Step 3 m<K + m<L = 90 m<K = 90 – m<L m – m 90 m<L <0 (this is impossible) Step 4 The original conjecture is true.

Hinge theorem If 2 sides of a triangle are congruent to 2 sides of an other triangle and included angles are not congruent, then the longer third side is across from the larger included angle. Converse: if 2 sides of 2 triangle are congruent to 2 sides of an other triangle and the third sides of an other triangle are not congruent, then the larger included angle is across from the longer third side.

Examples