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Chapter 5: Relationships in Triangles Lesson 5.1 Bisectors, Medians, and Altitudes Perpendicular Bisector Definition Facts to Know Point of Concurrency A line, segment or ray that passes through the midpoint of the opposite side and is perpendicular to that side Any point on a perpendicular bisector is equidistant from the endpoints Circumcenter: The point where 3 perpendicular bisectors intersect - the circumcenter is equidistant from all vertices of the triangle Example A E B D BD = CD AD BC E is the circumcenterAE = BE = CE C Median Definition Facts to Know Point of Concurrency A segment that goes from a vertex of the triangle to the midpoint of the opposite side The median splits the opposite side into two congruent segments Centroid: The point where 3 medians intersect Example A E B Small = 1/3 median Big = 2/3 median 2 x small = big D BD = CD E is the centroidED = 1/3 AD AE = 2/3 AD 2 ED = AE C Angle Bisector Definition Facts to Know Point of Concurrency A line, segment, or ray that passes through the middle of an angle and extends to the opposite side Any point on an angle bisector is equidistant from the sides of the triangle Incenter: The point where 3 angle bisectors intersect Example A E F G -the incenter is equidistant from all sides of the triangle B D BAD = CAD G is the incenterEG = FG C Altitude Definition A segment that goes from a vertex of the triangle to the opposite side and is perpendicular to that side Facts to Know Point of Concurrency Example Orthocenter: The point where 3 altitudes intersect A B D AD BC C C. Find the measure of EH. A. B. Find QS. Find WYZ. In the figure, A is the circumcenter of ΔLMN. Find x if mAPM = 7x + 13. In the figure, point D is the incenter of ΔABC. What segment is congruent to DG? In ΔXYZ, P is the centroid and YV = 12. Find YP and PV. In ΔLNP, R is the centroid and LO = 30. Find LR and RO. Lesson 5.2 Inequalities and Triangles Foldable Fold the paper into three sections (burrito fold) Then fold the top edge down about ½ an inch Unfold the paper and in the top small rectangles label each column… Exterior Angle Inequality •Exterior Angle = Remote Int. + Remote Int. -The exterior angle is greater than either of the remote interior angles by themselves rem. Int. < ext. Ex: Inequality with Sides Inequality with Angles -The biggest side is across from the biggest angle -The smallest side is across from the smallest angle -The biggest angle is across from the biggest side/ the smallest angle is across from the smallest side Ex: Ex: List the angles of ΔABC in order from smallest to largest. List the sides of ΔABC in order from shortest to longest. ___ ___the relationship between the What is lengths of RS and ST? What is the relationship between the measures of A and B? Lesson 5.4 The Triangle Inequality Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side Triangle Inequality Theorem Problems Determine if the measures given could be the sides of a triangle. 16, 17, 19 16 + 17 = 33 yes, the sum of the two smallest sides is larger than the third side 6, 9, 15 6 + 9 = 15 no, the sum of the two smallest sides is equal to the other side so it cannot be a triangle Find the range for the measure of the third side given the measures of two sides. 7.5 and 12.1 12.1- 7.5 < x < 12.1 + 7.5 4.6 < x < 19.6 9 and 41 41-9 < x < 41 + 9 32 < x < 50 Determine whether it is possible to form a triangle with side lengths 5, 7, and 8. Is it possible to form a triangle with the given side lengths of 6.8, 7.2, 5.1? If not, explain why not. Find the range for the measure of the third side of a triangle if two sides measure 4 and 13. In ΔPQR, PQ = 7.2 and QR = 5.2. Which measure cannot be PR? A 7 B 9 C 11 D 13 Lesson 5.3 Indirect Proof Steps to Completing an Indirect Proof: Assume that ______________ (the conclusion is false) Then _______________ (show that the assumption leads to a contradiction) This contradicts the given information that ________________. Therefore, __________________ (rewrite the conclusion) must be true. B. State the assumption you would make to start an indirect proof for the statement 3x = 4y + 1. Example Indirect Proof Given: 5x < 25 Prove: x < 5 1. Assume that x 2. Then x= 9 And 5(9)= 45 5. 45> 25 This contradicts the given info that 5x < 25 3. Therefore, x < 5 must be true. Example Indirect Proof Given: m is not parallel to n Prove: m 3 m 2 1. Assume that m 3 n 2 m 3 = m 2 2. Then, angles 2 and 3 are alternate interior angles When alternate interior angles are congruent then the lines that make them are parallel. This contradicts the given info that m is not parallel to n 3. Therefore, m 3 m 2 must be true. an indirect proof to show that if – 2x + 11 < 7, then x > 2. Given: –2x + 11 < 7 Prove: x > 2 Write Write an indirect proof. Given: ΔJKL with side lengths 5, 7, and 8 as shown. Prove: mK < mL Lesson 5.5 Inequalities Involving Two Triangles On the other side of the foldable from Lesson 2 (3 column chart) SAS Inequality Theorem (Hinge Theorem) -When 2 sides of a triangle are congruent to 2 sides of another triangle, and the included angle of one triangle is greater than the included angle of the other triangle… Then, the side opposite the larger angle is larger than the side opposite the smaller angle SSS Inequality Theorem -When 2 sides of a triangle are congruent to 2 sides of another triangle, and the 3rd side of a triangle is greater than the 3rd side of the other triangle… Then, the angle opposite the larger side is larger than the angle opposite the smaller side Examples: Ex: Ex: Ex: A. Compare the measures AD and BD. B. Compare the measures ABD and BDC. ALGEBRA Find the range of possible values for a. Find the range of possible values of n.