Journal Chapter 5

PERPENDICULAR BISECTOR

A line that bisects a segment and is perpendicular to that segment.

Any point that lies on the perpendicular bisector, is equidistant to both of the endpoints of the segment.

If it is equidistant from both of the endpoints of the segment, then it is on the perpendicular bisector.

Examples of Perpendicular Bisector

A

B

C

D

EBE = DEAE = CEm<E = 90

ANGLE BISECTOR A angle bisector is a ray that cuts an angle

into 2 congruent angles. It always lies on the interior of the angle.

Any point that lies on the angle bisector is equidistant to both of the sides of the angle.

If it is equidistant, then it lies on the angle bisector.

Examples of Angle bisector

A

B

C

D

AB = BCADB = CDBM AB = M BC

CONCURRENCY Concurrent is when three or more lines

intersect at one point. The concurrency of the perpendicular

bisector is the circumcenter because it is where 3 lines meet.

Circumcenter is the point of concurrency where the perpendicular bisectors of a triangle meet. It is equidistant to the 3 vertices.

Examples of concurrency

A voting post would be an example of circumcenter because it is equidistant to all vertices so a voting post needs to be equidistant to 3 towns

INCENTER The incenter is the point of concurrency of

the angle bisector because it is where 3 lines meet.

Incenter is the point where the angle bisector of a triangle intersect. It is equidistant to the sides of the triangle.

Examples of Incenter

A restaurant in the Middle of 3 highways Is an example of incenterBecause it is equidistant to The 3 sides or highways

MEDIAN Median is the segment that goes from the

vertex of a triangle to the opposite midpoint.

Centroid is the point where the medians of a triangle intersect.

When the median goes from the vertex to the opposite midpoint you can see that it makes to congruent parts so one side is concurrent to the other.

Examples of Median

The cockpit of a jetWould be an example ofCentroid because it is the Center of balance.

ALTITUDE The altitude of a triangle is a segment that

goes from the vertex perpendicular to the line containing the opposite side.

The orthocenter is where the altitudes intersect.

The concurrency of the altitudes is the orthocenter because it is where 3 lines meet.

Altitude

MIDSEGMENT Is a point that joins two midpoints of the

sides of the triangles. The midsegment is parallel to the opposite

side and the midsegment is half as long as the opposite side.

Midsegment

TRIANGLE SIDE ANGLE

RELATIONSHIP In any triangle the longest side is always

opposite the biggest angle, the shortest side is opposite the shortest angle.

Triangle side angle relationship

A

B

C

AB > BC, m of angle C > m of angle A

X Y

Zm< Z < m< Y, XY>XZ

D

E

F

DE > EF, m< D < m< F

EXTERIOR ANGLE INEQUALITY

The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles

Exterior angel inequality

1

2

3 4

M<4= m<1+m<2

60

6060M<4 = 120

90

45

45

M<4= 135

TRIANGLE INEQUALITY

The two smaller sides of a triangle must add up to more than the length at the length of the 3rd side.

Triangle inequalityA

B

C

AB+BC > ACBC+AC > ABAC+AB > BC

INDIRECT PROOFS Indirect proofs are proofs that you use to

proof something that is not right by contradicting yourself at one point.

1. First you assume what you are proving is false.

2. Second use that as your given and start proving

3. Last you find a contradiction and prove it.

Indirect proofsStatement Reason

1.FH is a medianof triangle DFGM<DHF > M< GHF

GIVEN

2. DH ≅ GH Definition of median3. FH ≅ HF Reflexive property4. DF > GF Hinge Theorem

Statement Reason

1. A triangle hasTwo right angles<1+<2

GIVEN

2. M<1=m<2=90 Def. right <

3. m<1+m<2 =180Substitution

4.M<1+m<2 +m<3=180Triangle sumTheorem

5. m<3 =o A triangleCan’t have2 right <‘sStatement Reason

1. a>0 so 1/a<0 Given2. 1/a<0 Given3. 1<0 Multiplicative prop.

Therefore if a>0 1/a>0

HINGE THEOREM If two triangles have two sides that are

congruent, but the third side is not congruent, then the triangle with the larger included angle has the longer third side.

If the triangle with the larger included angle has the longer third side, but the third side is not congruent, then two triangles have two sides that are congruent.

Hinge theorem

A

B

CD

E

F

m<A > m<DBC > EF

57

53

6

6

K

L

N

KL<MN 7 7

5.3 5.1P

Q

RS

m<PQS > m<RQS

TRIANGLES 45-45-90, 30-60-90

In a 45-45-90 triangle, both legs are congruent, and the length of the hypotenuse is the length of a leg times √2

In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg and the length of the longer leg is the length of the shorter leg times √3

Triangles 45-45-90, 30-60-90

45

45l√2

l

l7

x45

X= 7√2

30

60s

2ss√3

30

60

16

y

x

16=2x8=xy=x√3y=8√3