A line perpendicular to a segment at the segment’s midpoint.

Perpendicular Bisector Theorem – If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

Converse of the Perpendicular bisector Theorem – If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

Converse P.B. Thm

H A

Ex 1. f = g

Ex 2. d = e

Ex 3. b = c

TT

f = g; g = c; i = h

ce ab; bg cb

Ex 1. a = h

Ex 2. i = j

Ex 3. cg = bg

T

Angle Bisector - A ray that divides an Angle Bisector - A ray that divides an angle into two congruent angles.angle into two congruent angles.

Angle Bisector Theorem – If a point is on the Angle Bisector Theorem – If a point is on the bisector of an angle, then it is equidistant from bisector of an angle, then it is equidistant from the sides of the angle.the sides of the angle.

Converse of Angle Bisector Theorem – If a Converse of Angle Bisector Theorem – If a point in the interior of an angle is point in the interior of an angle is equidistant from the sides of the angle, equidistant from the sides of the angle, then it is on the bisector of the anglethen it is on the bisector of the angle

Ex 1. c bisects <abc; point L is equidistant from both sides.

Ex 2. i bisects <ghi; point J is equidistant from both sides.

Ex 3. e bisects <def; point K is equidistant from both sides.

HH

A.B. ThmConverse

Ex 1. L is equidistant from both sides; c must be an a.b.

Ex 2. J is equidistant from both sides; i must be an a.b.

Ex 3. K is equidistant from both sides; f must be an a.b.

To be To be concurreconcurrent nt means means for three for three or more or more lines to lines to intersect intersect at one at one point.point.

The point of concurreThe point of concurrency of perpendicular bncy of perpendicular bisectors form a circumisectors form a circumcenter.center.

Circumcenter - where three perpendiCircumcenter - where three perpendicular bisectors of a triangle are concucular bisectors of a triangle are concurrent. It is equidistant from the verticrrent. It is equidistant from the vertices of the trianglees of the triangle

Circumcenter D is the circumcenterD is the circumcenterof triangle abc, it is equidof triangle abc, it is equidistant from points a, b, aistant from points a, b, and c.nd c.

Right = circumcenter midpoint of hypotenuseRight = circumcenter midpoint of hypotenuse

Circumcenter L is the circumcenterL is the circumcenterof triangle jkl, it is equidiof triangle jkl, it is equidistant from points j, k, anstant from points j, k, and l.d l.

Acute = circumcenter insideAcute = circumcenter inside

CircumcenterH is the circumcenterH is the circumcenterof triangle efg, it is equidiof triangle efg, it is equidistant from points e, f, anstant from points e, f, and g.d g.

Obtuse = circumcenter outsideObtuse = circumcenter outside

The point of coThe point of concurrency of anncurrency of angle bisectors fogle bisectors forms an incenter.rms an incenter.

Incenter – The point of concurreIncenter – The point of concurrency of three angle bisectors of a ncy of three angle bisectors of a triangle. It is equidistant from ttriangle. It is equidistant from t

he sides of the triangle.he sides of the triangle.

Incenter

Incenter always inside triangleIncenter always inside triangle

D is the incenter of tD is the incenter of triangle abc, it is equiriangle abc, it is equidistant to all sides.distant to all sides.

Incenter

Incenter always inside triangleIncenter always inside triangle

A is the incenter of trA is the incenter of triangle xyz, it is equiiangle xyz, it is equidistant to all sides.distant to all sides.

Incenter

Incenter always inside triangleIncenter always inside triangle

D is the incenter of tD is the incenter of triangle abc, it is equiriangle abc, it is equidistant to all sides.distant to all sides.

Median – segments whose Median – segments whose endpoints are a vertex of the endpoints are a vertex of the triangle and the midpoint of triangle and the midpoint of the opposite side.the opposite side.

The point The point of concurrof concurrency of meency of medians is caldians is called a centrled a centr

oid.oid.

Centroid – Point of concCentroid – Point of concurrency of the medians urrency of the medians of a triangle. It is locateof a triangle. It is located 2/3 of the distance frod 2/3 of the distance from each vertex to the mim each vertex to the midpoint of the opposite sidpoint of the opposite si

dede

CentroidEx 1. g is the centroid of triangle abc

Ex 2. o is the centroid of triangle egc

Ex 3. k is the centroid of triangle adg

Centroid = Point of balance

Altitude – a perpendicular Altitude – a perpendicular segment from a vertex to segment from a vertex to the line containing the the line containing the opposite side.opposite side.

The point The point of of concurrencconcurrency of three y of three altitudes altitudes of a of a triangle triangle creates an creates an orthocenteorthocenter.r.

Orthocenter – Orthocenter – point where point where

three altitudes of three altitudes of a triangle are a triangle are concurrent. concurrent.

Nothing special.Nothing special.

Orthocenter

D is the orthocenter ofD is the orthocenter oftriangle abc with linestriangle abc with linese, d, and f as altitudes.e, d, and f as altitudes.

Orthocenter

D is the orthocenter ofD is the orthocenter oftriangle abc with linestriangle abc with linese, d, and f as altitudes.e, d, and f as altitudes.

Orthocenter

D is the orthocenter ofD is the orthocenter oftriangle abc with linestriangle abc with linese, d, and f as altitudes.e, d, and f as altitudes.

Midsegment of a triangle – a seMidsegment of a triangle – a segment that joins the midpoints gment that joins the midpoints of two sides of the triangle. Eveof two sides of the triangle. Every triangle has three midsegmery triangle has three midsegments, forming a midsegment triants, forming a midsegment tria

ngle.ngle.Triangle MidsegmeTriangle Midsegment Theorem – A midnt Theorem – A midsegment of a triangsegment of a triangle is parallel to a sile is parallel to a si

de of the triangle, ade of the triangle, and its length is half nd its length is half the length of that sithe length of that si

de.de.

Midsegment Ex 1. f Ex 1. f BC BCEx 2. d Ex 2. d AC ACEx 3. e Ex 3. e BA BAThey are all midsegments and arThey are all midsegments and are e parallel to the lines written besidparallel to the lines written besideethem.them.

In any triangle, In any triangle, the longest side the longest side

is always is always opposite to the opposite to the largest angle. largest angle. Small side = Small side = small angle.small angle.

Ex 3. •<a is the smalles angle, so its opposite angle is also the smallest.•Side b is the largest side so its opposite angle is the largest.

Ex 2.•<c is the largest angle so its opposite side is also the largest.•Side a is the smallest side so its opposite angle is also the smallest.

Ex. 1

•<a is the largest angle so therefore side a, its opposite side, must be the largest side.

•<c has the smallest angle so this means that side c is the smallest side in the triangle.

Exterior Angle Exterior Angle Inequality – The Inequality – The exterior angle of exterior angle of

a triangle is a triangle is larger than larger than

either of the either of the non-adjacent non-adjacent

angles.angles.

Ex 1. m<e is larger than all non adjacent angles such as b and y.Ex 2. m<c is greater than all non adjacent angles such as a and y.Ex 3. m<o is greater than all non adjacent angles such as a and b.

Triangle Triangle Inequality Inequality

Theorem – The Theorem – The two smaller sides two smaller sides of a triangle must of a triangle must add up to MORE add up to MORE

than the length of than the length of the 3the 3rdrd side. side.

Ex 1.Ex 1.

2.08 + 1.71 is less 2.08 + 1.71 is less than 5.86 so it than 5.86 so it doesn’t form a doesn’t form a triangle.triangle.

Ex 2.Ex 2.3.02 + 2.98 = 63.02 + 2.98 = 6But it doesn’t formBut it doesn’t forma triangle, it formsa triangle, it formsa line/segment.a line/segment.

Ex 2.Ex 2.

1.41 + 1.41 is 1.41 + 1.41 is greater than 2 so greater than 2 so therefore, it does therefore, it does form a triangle. form a triangle.

Steps for indirect proofs:Steps for indirect proofs:

1.1.Assume the opposite Assume the opposite of the conclusion is of the conclusion is truetrue

2.2.Get the contradictionGet the contradiction

3.3.Prove it is trueProve it is true

Ex 1.Ex 1.

Assuming that the supplement of an acute angle is another acute angle.

Given

Two angles add up to form 180 degrees

Definition of supplementary angles

Acute angles are less than 90 degrees

Definition of acute angles

89 + 89 does not equal 180

Angle addition postulate

Ex 2.Ex 2.Prove: a supplement of an acute angle cannot be another acute angle

Contradiction Contradiction

Assuming that a scalene triangle has two congruent midsegments.

Given

A midsegment is half the length of the line it is parallel to

Definition of a midsegment

A midsegment can be congruent if there are congruent sides.

definition of a midsegment.

a scalene triangle doesn’t have congruent sides

Definition of a scalene triangle.

Prove: a scalene triangle cannot have two congruent midsegments.

Ex 3.Ex 3.Assuming that triangle abc has a base angle that is a right angle. let angle a be the right angle. Triangle abc is an isosceles triangle.

Given

Angle c is congruent to angle a, so angle c is also a right angle.

Isosceles triangle theorem.

M<a and m<c are both 90

Definition of right angles

M<a + m<b + m<c have to equal 180

Triangle sum theorem

90 + m<b + 90 have to equal 180. m<b must equal 0

Substitution

Prove: an isosceles triangle cannot have a base angle that is a right angle

Prove: triangle abc cannot have a base angle that is a right triangle

Given: triangle abc is an isosceles triangle with base ac.Assuming that triangle abc has a base angle that is a right angle. let angle a be the right angle

By the isosceles triangle theorem, angle c is congruent to angle a, so angle c is also a right angle.

By the difinitio of right angle m<a = 90 and m<r = 90.

By the triangle sum theorem m<a + m<b + m<c =180.

By substitution, 90 + m<b + 90 = 180 so m<b = 0.

The Hinge Theorem – if two The Hinge Theorem – if two triangles have 2 sides triangles have 2 sides

congruent, but the third side congruent, but the third side is not congruent, then the is not congruent, then the

triangle with the larger triangle with the larger included angle has the longer included angle has the longer

33rdrd side. side.

Converse – if two triangles Converse – if two triangles have 2 congruent sides, but have 2 congruent sides, but

the third side is not the third side is not congruent, then the triangle congruent, then the triangle

with the larger 3with the larger 3rdrd side has the side has the longer 3longer 3rdrd angle. angle.

Ex 1. As <a is larger than <b,side d is larger than side c

Ex 2.As <a is smaller than <b,side e is smaller than side b

Ex 3.As <b is greater than <aside f is larger than side b

Hinge Converse

Ex 1. As side d is larger than side c,<a must be larger than <b

Ex 3.As side b is smaller than side f,<a must be smaller than <b.

Ex 2.As side b is larger than side e,<b must also be larger than <a.

45-45-90 triangle - in all 45-45-45-45-90 triangle - in all 45-45-90 triangles, both leg are 90 triangles, both leg are congruent, and the length of congruent, and the length of the hypotenuse is the length of the hypotenuse is the length of a leg radical 2.a leg radical 2.30-60-90 – In all 30-60-90 – In all

30-60-90 30-60-90 triangles, the triangles, the length of the length of the hypotenuse is hypotenuse is twice the length twice the length of the short leg; of the short leg; the longer side is the longer side is the length of the the length of the smaller leg radical smaller leg radical 3.3.

45-45-90Ex 1.Ex 1.b = 4b = 4√2√2

Ex 3.Ex 3.b = 6.92b = 6.92√2√2

Ex 2.Ex 2.b = 4b = 4√2√2

30-60-90

Ex 1.Ex 1.b = 4.62b = 4.62c = 4c = 4

Ex 2.Ex 2.h = 4.62h = 4.62i = 4i = 4

Ex 3.Ex 3.e = 8e = 8f = 6.93f = 6.93