(2) Page 221 – 223 #21 - 24, 28, 31, 35 - 40

# Geometry/Trigonometry Name: Unit 4: Properties of Triangles Notes Date: Period:

(1) Page 221#1 – 8, 15 - 20

(2) Page 221 – 223 #21 - 24, 28, 31, 35 - 40

(3) Page 227 #1 – 10

(4) Page 227 #11 – 19, Page 229 #30 - 32

(5) Page 236 #1 – 16 All

(6) Page 236 – 237 #17 – 20, 23, 27, 28; FF #30

(7) Page 239 #1 – 20 all

(8) Page 243 #1 – 14 All

(9) Page 243 – 245 #15 – 23, #30, 32

(10) Page 249 #1 –10 All

(11) Page 249 – 250 #11 – 18, 24, 30, 32

(12) Page 259 – 261 #9–18, 20, 22, 23–26, 28, 30, 38, 40, 45-48, 50

(13) Page 263 #1 – 20 All

Geometry Notes 5.1 Exploring Parts of Triangles C In the figure, the line is the ______of because it is

______to , and it______.

Theorem 5.1 Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

Theorem 5.2 Perpendicular Bisector Converse: If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment.

The ______between a point and a line is the length of the ______from the point to the line.

Theorem 5.3 Angle Bisector Theorem: If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.

Theorem 5.4 Angle Bisector Converse: If a point is in the interior of an angle and equidistant from the sides of an angle, then it lies on the bisector of the angle.

Geometry Notes 5.2 Special Segments in a Triangle

Special Segments in a Triangle

(1) ______:

(a) Perpendicular to a______of a triangle

(b) Cuts that side into ______

i. Their______point (where all three meet) is called the ______

ii. The circumcenter is the ______of the ______of the triangle

iii. The circumcenter is______from the ______of a triangle

(d) Does ______to come from a vertex, but can in certain triangles

(2) ______:

(a) ______an angle of a triangle (i.e. cuts it into ______)

(b) Every triangle has _____

i. Their concurrent point is called the ______

ii. The incenter is the ______of the ______circle

iii. The incenter is ______from the_____ of the triangle

(3) ______:

(a) Segment whose endpoints are a ______and the ______of the opposite side.

(b) Bisects the ______(i.e. cuts it into ______)

i. Their concurrent point is the called the______

ii. The centroid is ______of the distance from each ______to the ______of the opposite side.

(d) ______come from a______

(4) ______(aka height of a triangle)

(a) Segment from a ______that is ______to the opposite side.

(b) Every triangle has ______

i. Their concurrent point is called the ______

ii. They may lie inside ______of the triangle

(d) ______be ______to the opposite side

Geometry Notes 5.3 Midsegment Theorem

Midsegment –

(a) A special segment inside a triangle

(b) A segment that ______the ______of two sides of the triangle.

(d) Midsegment Theorem

(i) It is ______to the third side

(ii) It is ______of the third side

Geometry Notes 5.4 Inequalities in One Triangle

We are able to compare the ______of the ______of a triangle with the ______of its ______.

The ______of a triangle is ______the ______, and the ______is ______of the ______.

Theorem 5.7: If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side.

Theorem 5.8: If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.

Theorem 5.9 Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Do these triangles exist? **Check to see if the sum of the smallest two sides is bigger than the third. If so, then, yes the triangle exists.

(E1) 1 cm, 2 cm, 5cm

(E2) 2 cm, 2 cm, 5 cm

(E3) 3 cm, 2 cm, 5 cm

(E4) 4 cm, 2 cm, 5 cm

Geometry Notes 5.5 Inequalities in Two Triangles

Theorem 5.10 Hinge Theorem (SAS Ineq): If two sides of a one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second.

Theorem 5.11 Converse of the Hinge Theorem (SSS Ineq): If two sides of a one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second.

This is also known as a proof by contradiction

For instance, suppose everyone in a class is here on the day I give a test, but one person does not put their name on their paper. Through an indirect proof, I can figure out to whom the paper belongs.

Strategies for Writing an Indirect Proof