GEOMETRY JOUORNAL FOR CHAPTER 5

1. GEOMETRY JOUORNAL FOR CHAPTER 5 FRANCIS MIFSUD

2. Perpendicular bisector • The Perpendicular bisector is a line that cuts through a triangle and makes a 90 degree angle. Ex 1,2 and 3

3. Angle bisector The 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.

4. Example 1,2 and 3

5. Concurrent means at an equal distance. A concurrency of perpendicular bisectors or the point of concurrency is knows as the Circumcenter of the triangle. • Concurrent means a equal distance. A concurrency of perpendicular bisectors or the point of concurrency is knows as the Circumcenter of the triangle. The circumcenter theorem Ex 1

6. Ex2 ex3

7. Median • A 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.

8. Ex 2 Ex 3

9. Altitude • the 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.

10. Ex 1 Ex 2

11. Ex 3

12. 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

13. EX2: MIDSEGMENT MIDSEGMENT

14. 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. 7 Ex 1: this is not a triangle because two of the sides don’t add up to the third side of the triangle 5 3

15. 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 3 3 6 5

16. Angle inequalities • The exterior angle is supplementary to the adjacent interior angle and it is greater than either of the non adjacent interior angles.

17. Triangles inequalities • The 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

18. 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

19. Indirect proofs • First you assume that the conclusion is false • Then you show that this assumption leads to a contradiction.

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

21. 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