Geometry CP Lesson 5-1: Bisectors, Medians and Altitudes Page 1 of 3

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1 Geometry CP Lesson 5-1: Bisectors, Medians and Altitudes Page 1 of 3 Main ideas: Identify and use perpendicular bisectors and angle bisectors in triangles. Standard: 12.0 A perpendicular bisector of a side of a triangle is a line, segment, or ray that passes through the midpoint of the side and is perpendicular to that side. Construction A) Construct the perpendicular bisector of the segment below. a) Open the compass greater than half the length of the segment, and place its point on one of the endpoints. b) Draw an arc above and below the segment. c) Place the compass on the other endpoint, and draw an arc above and below the segment. d) Draw a line that passes through both sets of arcs. That is your perpendicular bisector. B) Choose a point on the perpendicular bisector and draw two segments that connect it to both endpoints of the original segment. C) Measure the length of each of the new segments. What do you notice? Points of Perpendicular Bisectors Theorem: Converse of the Points of the Perpendicular Bisector Theorem: Example 1: a) If YV is the perpendicular bisector of UVW, what is the relationship between UV and WV? c) b) What is the relationship between UYV and WYV? d) What is the relationship between UY and YW? e) True or false? Because YV is the perpendicular bisector of UW, UY YV.

2 Geometry CP Lesson 5-1: Bisectors, Medians and Altitudes Page 2 of 3 Example 2: What is an angle bisector? Points on Angle Bisectors Theorem: Converse of the Points on Angle Bisectors Theorem: Example 3: If CD is the bisector of ACB, what is the relationship between ACD and DCB? What is the relationship between AE and EB? What is the relationship between AD and BD? What is the relationship between AC and CB? If AC=2x+18 and CB=30, what is the value of x? Points of concurrency When three or more lines intersect at a common point, the lines are called concurrent lines, and their point of intersection is called the point of concurrency. The perpendicular bisectors of a triangle are concurrent, and their point of concurrency is called the circumcenter. The point of concurrency of the three angle bisectors is called the incenter of a triangle. A median is a segment whose endpoints are a vertex of a triangle and the midpoint of the side opposite of the vertex. The point of concurrency of the three medians in a triangle is called the centroid. The centroid is the point of balance for any triangle.

3 Geometry CP Lesson 5-1: Bisectors, Medians and Altitudes Page 3 of 3 An altitude of a triangle is a segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side. The point of intersection of the three altitudes of a triangle is called the orthocenter. Example 4: Draw and label a median and an altitude in the triangle given: A C U Theorems related to the points of concurrency:

4 Geometry CP Lesson 5-2: Inequalities and Triangles Page 1 of 3 Main ideas: Recognize and apply properties of inequalities to the measures of angles of a triangle. Recognize and apply properties of inequalities to the relationships between angles and sides of a triangle. Standards: 12.0 and 13.0 What do the following symbols mean? > Example: < Example: = Example: Complete the following sets of comparisons with the correct symbol: 1) ) Example 1: A) Find m3 if m1=43 o and m2=45 o. B) What is 3 called? C) Complete the following comparisons with the correct symbol: a) m3 m1 + m2 b) m3 m1 c) m3 m1

5 Geometry CP Lesson 5-2: Inequalities and Triangles Page 2 of 3 Exterior Angle Inequality Theorem Example 2: Example 3: Example 4: Measure the sides and angles of the triangle below. Arrange the names of the segments in order from least to greatest. Arrange the names of the angles in order from least to greatest. What do you notice about the measures of the angles and the measures of the sides in a triangle?

6 Geometry CP Lesson 5-2: Inequalities and Triangles Page 1 of 3 Theorem 5.9 ( Opposite Angle Measure Theorem) Theorem 5.10 (Converse of the Opposite Angle Measure Theorem ) Example 5: Example 6:

7 Geometry CP Lesson 5-3: Indirect Proof Page 1 of 2 Objective: Use indirect proof with algebra and geometry CA Standard: 2 The proofs you have written so far use direct reasoning, in which you start with a true hypothesis and prove that the conclusion is true. If AC MN, then AB + BC = MN. Indirect reasoning is another strategy used to prove things. Writing an Indirect Proof 1. Assume that the conclusion is false. 2. Show that this assumption leads to a contradiction of the hypothesis or some other accepted fact. 3. Point out that because the false conclusion leads to an incorrect statement, the original conclusion must be true. Example: Stating Assumptions State the assumption you would make to start an indirect proof of each statement. 1. If then AB MN Assumption: 2. If then CAT is obtuse Assumption: 3. If 9 is a factor of n, then 3 is a factor of n. Assumption: 4. If then x < 4 Assumption: Algebraic Indirect Proofs given Given: x + y = 20 and x = 5 Prove: y 10 prove Statements 1. x + y = 20 and x = y = = = Our assumption that y = 10 led to a false statement 15 = 20, therefore, y 10. Reasons Given: 3x + 5 > 8 Prove: x > 1 Statements 1. 3x + 5 > x x Our assumption that x 1 led to a contradiction of the given, therefore, x > 1. Reasons

8 Geometry CP Lesson 5-3: Indirect Proof Page 1 of 2 Geometric Indirect Proofs Given: ABC is a right triangle, where A is a right angle. Prove: ABC doesn t have more than 1 right angle. Statements Reasons 1. ABC is a right triangle, where A is a 1. Given right angle. 2. Assume ABC has more than 1 right angle 2. Assumption (A and B are both right angles) 3. ma = 90, mb = ma + mb + mc = Substitution Given: D F Prove: DE EF E D Statements F Reasons

9 Geometry CP Lesson 5-4: Triangle Inequality Page 1 of 1 Triangle Inequality Theorem The sum of the lengths any two sides of a triangle is greater than the length of the third side. Ex: Determine if the given measures can be the lengths of the sides of a triangle. 3, 4, 6 6, 9, 15 8, 8, 8 4, 8, 16 Ex: The lengths of a triangle are 1, 6 and x. What is the range for the unknown side? Ex: Determine the range for the measure of the third side given the measures of two sides of a triangle. 8 and and and and 8 SAS Inequality Theorem (Hinge Theorem) Lesson 5-5: Inequalities Involving Two Triangles Page 1 of 2 If 2 sides of one triangle are to 2 sides of another triangle, and the included angle of the first triangle is larger than the included angle of the 2 nd triangle, then the 3 rd side of the first triangle is longer than the 3 rd side of the second triangle. x x > y y Ex: Write an inequality for the given pair of segment measures.

10 SSS Inequality Theorem Lesson 5-5: Inequalities Involving Two Triangles Page 2 of 2 If 2 sides of one triangle are to 2 sides of another triangle, and the 3 rd side of the first triangle is longer than the 3 rd side of the 2 nd triangle, then the included angle of the 1 st triangle is larger than the included angle of the 2 nd triangle. 11 A 10 B ma > mb Ex: Write an inequality for the given pair of angle measures. Ex: Write an inequality to describe the possible values of x. Ex: Given: G is the midpoint of DF Prove: ED > EF m1 > m2 Statements 1. G is the midpoint of DF 1. m1 > m ED > EF 4. Reasons