Picture. Right Triangle. Acute Triangle. Obtuse Triangle

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1 Name Perpendicular Bisector of each side of a triangle. Construct the perpendicular bisector of each side of each triangle. Point of Concurrency Circumcenter Picture The circumcenter is equidistant from each vertex of the triangle. Right Triangle Acute Triangle Obtuse Triangle

2 Angle Bisectors of each vertex of a triangle. Construct the angle bisector of each angle in the triangles pictured. Incenter The incenter is equidistant from each side of the triangle. Right Triangle Acute Triangle Obtuse Triangle

3 Median of each side of a triangle. A median has a vertex as an endpoint and cuts the opposite side of the triangle in half (forms a midpoint). Centroid** The medians of a triangle intersect at a point called the centroid that is two thirds of the distance from each vertex to the midpoint of the opposite side. Construct the median to each side of the given triangle.

4 Geometry UNIT 5 Relationships in Triangles Day 1 Day 2 Day 3 Day 4 Incenter, Circumcenter, Centroid Constructions and Definitions Using Incenter, Circumcenter, and Centroid (2 days?) Finding Equations of Perpendicular Bisectors Inequalities in One Triangle Quiz Review QUIZ Day 6 Day 7 Day 8 The Triangle Inequality Inequalities in Two Triangles The Indirect Proof Review TEST

5 Day 2 Using Incenter (angle bisector), Circumcenter (perpendicular bisector), and Centroid (median) *If there is a ray equidistant (and creates a 90 o angle) from each side of an angle, then the ray is an angle bisector. *If there is an angle bisector, then the ray is equidistant (and creates a 90 o angle) from each side of the angle. *The perpendicular bisector in a triangle begins at the vertex and divides the base into two congruent parts connected by a right angle.

6 *If medians are drawn, or the centroid point is names, the medians are divided into 3 equal parts. The vertex to the centroid is 2 parts and from the centroid to the side is 1 part. Find the coordinates of the centroid of the triangle with the given vertices. X( 5, 7) Y(9, -3), Z(13, 2)

7 Practice: Using Incenter (angle bisector), Circumcenter (perpendicular bisector), and Centroid (median) Use COMPLETE SENTENCES!!!!!

8 Day 3 WRITING EQUATIONS OF PERPENDICULAR BISECTORS Write an equation in slope-intercept form for the perpendicular bisector of the segment with the given endpoints. Justify your answer. C(-4, 5) D(10, -2)

9 Write an equation in slope-intercept form for the perpendicular bisector of the segment with the given endpoints. Justify your answer. C(-3, 1) D(4, 3)

10 Practice: WRITING EQUATIONS OF PERPENDICULAR BISECTORS

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12 QUIZ REVIEW 1b. Name a perpendicular bisector. 1b. 1c. Name a median. 1c.

13

14 Day 6 THE TRIANGLE INEQUALITY Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Short cut: Add the two sides and subtract the two sides to determine a length in between the answers. Examples: 1. Is it possible to form a triangle with the given side lengths? If not, explain why not. a. 5 cm, 7 cm, 10 cm b. 3 in, 4 in, 8 in 2. If the measures of two sides of a triangle are 5 yards and 9 yards, what is the least possible measure of the third side if the measure is an integer? a. 5 yd b. 4 yd c. 14 yd d. 6 yd 3. Find the range for the measure of the third side of a triangle given the measures of two sides. a. 5 m, 11 m b. 3.8 in, 9.2 in 4. Find the range of possible outcomes of x if each set of expressions represents measures of the sides of a triangle. a. x, 4, 6 b. x + 2, x + 4, x + 6

15 5. Determine whether the given coordinates are the vertices of a triangle. Explain. F(-4, 3), G(3, -3), H(4, 6) Practice:

16 Day 7 INEQUALITIES IN TWO TRIANGLES Inequality in Two Triangles ( Hinge Theorem ): If two sides of a 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 triangle, then the third side of the first triangle is longer than the third side of the second triangle Converse: If the 3 rd side of one triangle is longer than the 3 rd side of another triangle, then the two sides of the 1 st triangle are congruent to the two sides of the 2 nd triangle and the included angle of the 1 st triangle is larger than the included angle of the 2 nd triangle.

17 Practice: INEQUALITIES IN TWO TRIANGLES

18 Day 8 PROOF BY CONTRADICTION INDIRECT PROOF How to write an indirect proof: a. Identify the conclusion you are asked to prove. b. Assume the opposite of the conclusion is true. c. Use logical reasoning to show that this assumptions leads to a contradiction of the hypothesis. d. State that since the assumption leads to a contradiction, the original conclusion must be true. *We will only be asked to determine what should be assumed for an indirect proof.* Examples:

19 Unit 5 TEST REVIEW

20

21 17.

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