Chapter 5Delia Coloma 9-5

Journal

What is a perpendicular

bisector?Perpendicular bisector theorem: This theorem says that if there is a point on the perpendicular bisector, then the distance will be the same from the endpoints of the segment.

Converse of the p.b.t: The converse says that if the distance is the same from the endpoints of the segment then it is on the perpendicular bisector.

Examples

CONVERSE O

F THE

PERPEN

DICULAR

BISECTO

R

THEOREM

Real life examples (:

What is an angle bisector?Angle bisector theorem: This theorem says that if a point is on the bisector of an angle, then it will have the same measure from the sides of the angle.

Converse of A.B.T: if a point in the interior of the angle has the same distance then it is on the bisector of an angle.

Examples

CONVERSE OF THE ANGLE BISECTOR THEOREM

Real life examples:

Concurrency Concurrency is when three or more lines intersect at a point. There are many types of them.

Intersect at one point

examples

Real life example:

Concurrency of Perpendicular bisectors of a triangle theorem

In the circumcenter theorem you will see that the three perpendicular bisectors of the triangle are concurrent.

What is the circumcenter?! The circumcenter theorem says that the circumcenter of a triangle has the same distance from the vertices of the triangle.

Where perpendicular bisectors meet!

Examples:Acute triangles:

Right triangles:

Obtuse angles:

What is a median?A median of a triangle is a segment on which one endpoint is a vertex and the other is the midpoint of the other/opposite side.

Incenter theorem: The incenter of a triangle has the same distance from the sides to the triangle.

THE INCENTER WILL ALWAYS BE INSIDE OF THE TRIANGLE.

Were the angle bisectors meet! It can be useful when putting something in the middle of highways.

Examples:Acute tiangle:

Obtuse triangle:

Right triangle:

Real life example:

Centroid:

Centroid theorem: states that the centroid of a triangle is located 2/3 of the distance from a vertex to the midpoint of the opposite side.

Centroid: the point of concurrency of the three medians of a triangle.

It balances it so its distance from the vertex to the other is doubled.

Can be helpful when building something so that it is balanced.

Examples:

concurrency of medians of a triangle theorem

This means that it is the point on which the medians intersect.

Altitude of a triangle:A perpendicular segment from the vertex to the opposite side line. TRIANGLES HAVE 3 ALTITUDESIt can be inside, outside or on it.

Orthocenter:The two lines containing the altitudes are concurrent to the line intersecting.

concurrency of altitudes of a triangle theorem

Concurrency of altitudes in a triangle means/is the point where the altitudes intersect.

Midsegment:A segment that joins the midpoints of two sides of a triangle.TRIANGLES HAVE 3 AND THEY FORM A MIDSEGMENT TRIANGLE.

Triangle Midsegment theorem:The midsegment of a triangle is parallel to the opposite side of the triangle, and it is half its size.

2 in

4 in

5 cm

10 cm

6 yd

3 yd

Real life example:

20 cm

10 cm

relationship =sides of triangle and

their opposite anglesif two sides of a triangle are not congruent, then the larger angle is opposite the longer side.

If two angles of a triangle are not congruent, then the longer side is opposite the larger angle.

Examples:

Exterior angle inequality:This means that the non-adjacent interior angles are smaller than the exterior angle. When you add them up you get the measure of the exterior angle.

A B C

60

60

12070

70

140

Real life example:

40 4080

Triangle inequality: This theorem says that the sum of any two sides of a triangle is greater that the other side.

A B

C

AC+CB > AB

X

W Y

WY<WY+YX

T

U

V

5

5

6

TU+TV>UV

Real life example:

A

B

CCA+AB>CB

INDIRECT PROOFS :OIndirect proofs are used when something is not possible to be proved directly.

STEPS WHEN DOING AN IDIRECT PROOF:1. Assume that what you are proving is false.2. Use that as your given, and start proving it. 3. When you come to a contradiction you have

proved its true.

Example 1: statements reasonsAssume a triangle has 2 right triangles <1+<2

Given

m<1= m<2 = 90 Def. of Right trianglem<1+ m<2 = 180 Substitutionm<1+ m<2 + m<3= 180

Triangle sum theorem

M<3=0 Contradiction

Prove: a triangle cannot have 2 right triangles.

So it cannot have 2 right angles.

Example 2:Write an indirect proof that the supplementary of an acute angle cannot be an acute angle. statements reasonsAssuming that the supp. of an acute < is an acute <.

Given

2 <´s added= 180 Def of supplementaryAcute < less than 90 Def of acuteAngle addition property contradictionSo the supplement of an acute angle cannot be an

acute <.

Example 3:Prove: A triangle has two obtuse angles.

Statements reasonsAssume a triangle has 2 obtuse angles <1+<2

given

<1=<2= 91+ Def of obtuse angle<1+<2=182+ substitutionA triangle´s angles only measure up to 180

contradiction

A triangle cannot have 2 obtuse angles.

Hinge theorem: This theorem says that if two sides of a triangle are congruent to two sides of another and the included angles are not, then the long third side is across from the larger included angle.

BC>DF

BC>EF

ED>AB

Converse of the hinge theorem:If two sides if a triangle are congruent to another two sides

of other triangle and the third sides are not congruent, then the larger included angle is across from the third side.

M<BAC > M<EDF

m<BAC > m<DEF

M<EFD > M<ACB

45-45-90 TRIANGLESIt is called the Theorem 5-8-1 it states that in a 45-45-90 triangle both legs are congruent, the length of the hypotenuse us the length of a leg times √2.

To do it just multiply a leg times √2. Your answer could be in radical form.

Radical=√

Example: find the value of x

5

5

x

5x √2= 5√2X= 5√2

7

7

x

7x√2= 7√2X= 7√2

9√2

x

x

9√2 /√2 = 9X= 9

30-60-90 TRIANGLES:It is called the theorem 5-8-2, it says that the length of a hypotenuse is 2 times the length of the shorter leg. The length of the longer leg is the length of the shorter leg times √3.

Examples: Not drawn in scale

√3

18

9

9√3

612

6√3

Real life example:

7

14

7√3

THE END