GEOMETRY JOUORNAL FOR CHAPTER 5

FRANCIS MIFSUD

Perpendicular bisector The Perpendicular bisector is a line that cuts

through a triangle and makes a 90 degree angle.

Ex 1,2 and 3

Angle bisectorThe angle bisector is a line that cuts through the

middle of the angle and divides it into 2 equal parts.

The angle bisector theorem states that If a point is on the bisector of an angle it is equidistant from both sides of the angle.

Example 1,2 and 3

Concurrent means at an equal distance. A concurrency of perpendicular bisectors or the point of concurrency is knows as the Circumcenter of the triangle.

Ex 1

Ex2

ex3

MedianA median of a triangle is a segment whose

endpoints are a vertex of the triangle and the midpoint of the opposite side. The concurrent point is known as the centroid. It is always inside the triangle. The centroid is also known as the center of gravity because it the point where a triangle reason will balance.

Ex 2

Ex 3

Altitudethe altitude of a triangle is the

perpendicular segment from a vertex of to the line containing the opposite side. All triangles have 3 altitudes and they can be inside, outside or on the triangle. Where the 3 altitudes meet is known as the orthocenter. The orthocenter can also be anywhere including outside inside or on the triangle.

Ex 1

Ex 2

Ex 3

MIDSEGMENT AND IT’S THEOREMS

A midsegment is the segment that joins to midpoints of a line.

The midsegment theorem sais that “midsegment that connects two sides of a triangle is parallel to the third side and is half as long.”

EX 1:

MIDSEGMENT

EX2:MIDSEGMENT

MIDSEGMENT

Triangle relationships In triangles if two sides of a triangle are not

congruent, then the larger angle is opposite to the longer side.

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

3

5

7

Ex 1: this is not a triangle because two of the sides don’t add up to the third side of the triangle

Ex2: This is a triangle because the measures of two of the sides do add up to the measure of the third side

Ex 3 : This is a triangle because again

the measures of

two of the sides

add up to the measure of the third side

4 4

63

5

3

Angle inequalitiesThe exterior angle is supplementary to the

adjacent interior angle and it is greater than either of the non adjacent interior angles.

Triangles inequalitiesThe sum of the lengths of any two sides of a

triangle must be greater than the third side if not then it is not a triangle

This is a triangle because the measures of two of the sides do add up to the measure of the third side

Ex 2

Ex 3

This is a triangle because the measures of two of the sides do add up to the measure of the third side

This is a triangle because the measures of two of the sides do add up to the measure of the third side

Indirect proofsFirst you assume that the conclusion is false

Then you show that this assumption leads to a contradiction.

Proove: tri. JKL does not have an obtuse anglegiven : triangle jkl is a right triangle

Ang. K+ ang. L =90 degrees

Ang. K =90 degrees – angl. L

90 degrees – ang. M = 90

Ang. L is smaller than 0 degrees

The acute angles of a right triangle are complementary

Subtraction

Sbstitution

Subtraction

Statement Reason

k

j

l

Hindge theorem If two sides of tone triangle are congruent to

two sides of another triangle and the included angles are not congruent then the long third side is across from the larger included angle

Thrid sides are congruent