By: Ana Cristina Andrade Chapter 5 Journal By: Ana Cristina Andrade

Perpendicular Bisector: Definition of Perpendicular bisector: a line perpendicular to a segment at the line segment´s midpoint Perpendicular bisector theorem: If a line is perpendicular, then it is equidistant from the endpoints of a segment. Converse of perpendicular bisector theorem: If a point is equidistant from the endpoint of a segment, then it is perpendicular line.

Perpendicular bisector theorem Examples: Perpendicular bisector theorem =

Converse of perpendicular bisector theorem Examples: Converse of perpendicular bisector theorem

Angle Bisector: Angle bisector theorem: a ray or line that cuts an angle into 2 congruent angles. It always lies on the inside of an angle Converse of angle bisector theorem: If a point is equidistant from the sides of a angle, then it lies on the bisector angle.

Examples: Angle bisector theorem A B C A B C A B C AB = CB

Examples: <ADB = <CDB Converse of angle bisector theorem B A D C (Congruent ,not equal)

Concurrency: Definition of concurrency: Where three or more lines intersect at one point.

concurrency of Perpendicular bisectors: Concurrency of perpendicular bisectors: Point where the perpendicular bisectors intersect.

Circumcenter: Definition of Circumcenter: the point of congruency where the perpendicular bisectors of a triangle meet. The circumcenter theorem: The circumcenter of a triangle is equidistant from the vertices of the triangle.

Examples:

concurrency of angle bisectors: Concurrency of angle bisectors: Point where the angle bisectors intersect.

Incenter: Definition of incenter: The point where the angle bisector intersect of a triangle Always occur on the side of triangle Incenter theorem: The incenter of a triangle is equidistant from the side of a triangle

Examples:

Median: Definition of Median: segment that goes from the vertex of a triangle to the opposite midpoint.

Centroid: Centroid: The point where the medians of a triangle intersect. The distance from the vertex to the centroid is double the distance from the centroid to the opposite midpoint.

Examples:

concurrency of medians: Concurrency of medians: point where the medians intersect.

Altitude: Definition of altitude: a segment that goes from the vertex perpendicular to the line containing the opposite side.

Examples:

Orthocenter: Definition of Orthocenter: Where the altitudes intersect If the triangle is acute, the orthocenter is on the inside of the triangle If it is right orthocenter is on the vertex of the right angle.

Examples:

concurrency of altitudes: Concurrency of altitudes: point where the altitudes intersect.

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

Examples:

midsegment theorem: Triangle midsegment theorem: A midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side.

relationship between the longer and shorter sides of a triangle: Hinge theorem: If 2 triangles have 2 sides that are congruent, but the third side is not congruent, then the triangle with the larger included angle has the longer third side. Converse of Hinge theorem: If two sides of a triangle are congruent to the two sides of the other triangle but the other sides are not congruent then, the largest included angle is across from the largest side.

Examples: HI>BC AC > XZ Hinge Theorem J>A B H J <B > <Y HI>BC A I C J>A A B C H I J AC > XZ

Examples: <D> <A Converse of Hinge Theorem FE > CB, FD=CA, DE = AB (congruent) <D> <A

Relationship between opposite angles of a triangle: Triangle side-angle relationship theorem: In any triangle, the longest side is always opposite from the largest angle and vice versa.

Examples: Longest side Shortest side

exterior angle inequality: The non-adjacent interior angles are smaller than the exterior angle A+B = exterior angle (c) A B C A C B A B C

Triangle inequality: Triangle inequality theorem: the 2 smaller sides of a triangle must add up to more than the length of the 3rd side.

Examples: 3, 1.1, 1.7 1.1+1.7= 2.8 NO 4, 7, 10 4+7=11 YES 2, 9, 12 2+9=11 NO

indirect proof: Indirect proof: used when it is not possible to prove something directly. Steps: Assume that what you are proving is false Use that as your given, and start proving it When you come to a contradiction you have proved that it is true.

Examples: Prove: A triangle cannot have 2 right angles A triangle has 2 right angles (<1 & <2) Given M<1=m<2=90 Def. right angle M<1+m<2=180 Substitution M<1+m<2+m<3=180 Triangle sum theorem M<3=0 contradiction

Examples: Proove: a right triangle cannot have an obtuse angle A right triangle can have an obtuse angle (<A) Given M<A + m<B= 90 Substitution M<A =90 – m<b Subtraction prop. M<A> 90° Def. obtuse triangle 90° - m<b > 90 substitution m<b = 0 contradiction

A triangle cannot have 4 sides Examples: A triangle cannot have 4 sides A triangle can have 4 sides Given A square is a shape with 4 sides Def of square A triangle is a shape with only 3 sides Def of triangle A triangle cannot have 4 sides contradiction

special relationships in the special right triangles: 45° - 45° - 90° triangle theorem: In this kind of triangle, both legs are congruent and the hypotenuse is the length of a leg times √2 30° - 60° - 90° Triangle theorem: In this kind of triangle the longest leg is √3 the shorter leg and the hypotenuse is √2 the shortest side of the triangle.

Examples: 45° - 45° - 90° triangle theorem A B C X BC=AC=X AB=X√2

Examples: 45° - 45° - 90° triangle theorem 45° 14 X X=14√2

Examples: 30° - 60° - 90° triangle theorem 16=2a 8=a B=a√3 B=8√3 20=2x Y 20 20=2x 10=x Y=a√3 Y=10√3 B 16 16=2a 8=a B=a√3 B=8√3 d 100 100=2d 50=d H=d√3 H=50√3