Journal 5 Nina Dorion 9-5

Perpendicular bisector: line that cuts a segment into two equal parts which measure 90. Converse: if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. Examples: AB is the perpendicular bisector of CD FE is perpendicular bisector of IH MN is perpendicular bisector of lk a f c d h i M b e L K N

d b Angle bisector: divides an angle into two equal parts The angle bisector in this example is AB which bisects <DAC . a c Angle bisector theorem: the angle bisected in a triangle will have the same measure as the other half of the triangle. Measure of triangle ABD is = to measure of triangle CBD because of BD which is the angle bisector of ABC A B C D

Three or more lines are concurrent if they intersect a single point. Concurrency of Perpendicular Bisectors of a Triangle Theorem:   The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle Circumcenter: the center of a circle that circumscribes a triangle.

Concurrency of Angle Bisectors of a Triangle Theorem:  The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. The incenter of a triangle is equidistant from all sides of the triangle.

A median is a line joining the vertex to the midpoint of the opposite side. The centroid is the point of congruency of the three medians of the triangle. Acute triangle right triangle obtuse triangle concurrency of Medians of a Triangle Theorem:   The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side

Altitude of a triangle is where the lines of a triangle intersect The orthocenter is the intersection of three altitudes of a triangle. Concurrency of Altitudes of a Triangle Theorem:   The lines containing the altitudes of a triangle are concurrent

a midsegment is parallel to the third part of a triangle and is half the length of the third side Midsegment Theorem :  The segment joining the midpoints of two sides of a triangle is parallel to the third side, and is half the length of the third side 5 10

The relationship between the sides and the angles are that, the opposite side of the shortest angle will be the shortest side, and the opposite side of the largest angle will be the largest side. In this example we can see that segment BC is the smallest because the measure of angle A is the smallest 90 40 50 A B C shortest 80 40 60 75 longest shortest 55 50

Exterior angle inequality: The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle. <ABC will always be greater than BAD. A D B c m h l k i p n o

Triangle inequality theorem: Any side of a triangle is always shorter than the sum of the other two sides . A B Could a triangle have sides of 4, 8, 2? And can you make a triangle with 5,6,7? C a + b > c 
b + c > a 
a + c > b

Indirect proof: to write an indirect proof you have to assume that the given is true, then stop when the given contradicts itself. Examples: Statement reason <BDA is a gven Straight angle M<BDA=180 def of straight angle AD is perpendi cular Bisector of BC given <BDA is right angle def of perpendicular M<BDA=90 def. of right angle M<BDA is not a straight angle! A B C D

Prove that triangles cant have 2 right angles Statement Assume angle A ans <B are right triangles Measure of <A=90 and m<B=90 Measure of <A+<B+<C= 180 90+90+<C=180 Measure of <C=0 <A and <B cannot be right angles Reason Given Definition of right angles Sum of angles Substitution Subtraction contradiction

Hinge theorem: 2 triangles have 2 sides that are congruent, but the 3 sides is not congruent, then the triangle with the larger included angle has the longer side. F EF < GF By the hinge theorem 82 E G H