CHAPTER 5 RELATIONSHIPS WITHIN TRIANGLES

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1 HPTR 5 RLTIONSHIPS WITHIN TRINGLS In this chapter we address three ig IS: 1) Using properties of special segments in triangles ) Using triangle inequalities to determine what triangles are possible 3) xtending methods for justifying and proving relationships Section: ssential Question 5 1 Midsegment Theorem What is a midsegment of a triangle? Warm Up: Key Vocab: Midsegment of a Triangle segment that connects the midpoints of two sides of the triangle. xample: MO, MN, NO are midsegments M N O Theorem: Midsegment Theorem The segment connecting the midpoints of two sides of a triangle is a. parallel to the third side b. and is half as long as that side. 1 and Student Notes Geometry hapter 5 Relationships within Triangles KY Page #1

2 Show: x 1: In the diagram of an -frame house, G and F. G and H are midsegments of. Find G 6 ft F 0 ft 10 ft G H F 1 ft x : In the diagram, RS TS and RW VW. xplain why VT WS R S T RS TS and RW VW, so S and W are the midpoints of RT and RV, and SW is a midsegment of RTV. Therefore, VT WS by the Midsegment Theorem. W U V Try it: x 3: Use RST to answer each of the following a. If If UV = 13, find RT. RT = 13 6 b. If the perimeter of RST 68 inches, find the perimeter of UVW. 1 Perimeter UVW inches c. If VW x 4 and RS 3x 3, what is VW? (x 4) 3x 3 4x8 3x 3 x 5 VW (5) 4 6 Student Notes Geometry hapter 5 Relationships within Triangles KY Page #

3 Section: ssential Question 5 Use Perpendicular isectors How do you find the point of concurrency of the perpendicular bisectors of the sides of triangle? Warm Up: *Review efinitions for uler Line Project: points of concurrency circumcenter centroid orthocenter Key Vocab: n Perpendicular isector segment, ray, line, or plane, that is perpendicular to a segment at its midpoint. Line n is bisector of ircumcenter The point of concurrency of the three perpendicular bisectors of the triangle Point is the circumcenter Student Notes Geometry hapter 5 Relationships within Triangles KY Page #3

4 Theorems: In a plane, if a point is on the perpendicular bisector of a segment, Perpendicular isector Theorem then it is equidistant from the endpoints of the segment. Point is on the bisector of In a plane, if onverse of the Perpendicular isector Theorem then a point is equidistant from the endpoints of a segment, it is on the perpendicular bisector of the segment. Point is on the bisector of The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangles ircumcenter Theorem Student Notes Geometry hapter 5 Relationships within Triangles KY Page #4

5 Show: x 1: In the diagram, RS is the perpendicular bisector of PQ. Find PR. P 8x-9 8x9 6x S Q 6x R x 9 x 4.5 PR 8(4.5) 9 7 x : In the diagram, JM is the perpendicular bisector of HK. a. Which lengths in the diagram are equal? H HM KM ; HJ KJ; HL KL 1.5 b. Is L on JM? L 1.5 M J Yes K x 3: Prove the Perpendicular isector Theorem Given: is on the perp. bis. of Prove: Statements Reasons 1. is on the perp. bis. of 1. Given.. ef of segment bisector Reflexive Prop. 4. and are right 's 4. ef of perp Rt. ngles cong. Thm SS Post PT Student Notes Geometry hapter 5 Relationships within Triangles KY Page #5

6 Section: ssential Question 5 3 Use ngle isectors of Triangles When can you conclude that a point is on the bisector of an angle? Warm Up: Key Vocab: Incenter The point of concurrency of the three angle bisectors of the triangle. I Point I is the incenter. Theorem: If a point is on the bisector of an angle, ngle isector Theorem then it is equidistant from the two sides of the angle. bisects F F F Student Notes Geometry hapter 5 Relationships within Triangles KY Page #6

7 If a point is in the interior of an angle and is equidistant from the sides of the angle, onverse of the ngle isector Theorem then it lies on the bisector of the angle. F bisects F F The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. Incenter Theorem I I I I Show: x 1: Find the measure of. x : For what value of x does P lie on the bisector of? xx P - 3x+18 x 3x 18 x 3x 18 0 ( x 6)( x 3) 0 x 6, x 3 Student Notes Geometry hapter 5 Relationships within Triangles KY Page #7

8 x 3: In the diagram, G is the incenter of S RST. Find GW. 1 GU 13 R V U G 13 W GU GU GU GV GW GW 5 1 T F x 4: Prove the ngle isector Theorem Given: is on the angle bis. of Prove: F Statements 1. is on the angle bis. of 1. Given Reasons. raw and F. Perp. Post. 3. and F are right 's 3. ef of Perp. 4. F 4. Right ngles ong. Thm Reflexive Prop. 6. F 6. ef of ang. bisector 7. F 7. S Post 8. F 8. PT Student Notes Geometry hapter 5 Relationships within Triangles KY Page #8

9 Section: ssential Question 5 4 Use Medians and ltitude How do you find the centroid of a triangle? Warm Up: Key Vocab: Median of a Triangle segment from one vertex of the triangle to the midpoint of the opposite side G I entroid The point of concurrency of the three medians of the triangle., FG, and H F HI are medians. Point is the centroid. ltitude of Triangle The perpendicular segment from one vertex of the triangle to the line that contains the opposite side. M N K O Orthocenter The point of concurrency of the three altitudes of a triangle. J P JK, LM, and NP are altitudes. Point O is the orthocenter. L Student Notes Geometry hapter 5 Relationships within Triangles KY Page #9

10 Theorems: entroid Theorem The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side. O ; O F; O O F Show: x 1: In HJK, P is the centroid J P a. If JP 1, find PT and JT. H T b. If K JP x and PT x, then x =?. JP ( JT ) 3 1 ( JT ) 3 18 JT PT JT JP PT x x x x 4x 4x 0 xx ( 4) 0 x0, x 4 x : Show that the orthocenter can be inside, on, or outside the triangle. Inside = cute Triangle On = Right Triangle Outside = Obtuse triangle Student Notes Geometry hapter 5 Relationships within Triangles KY Page #10

11 x 3: ecide whether YW is a perpendicular bisector, angle bisector, medians, and/or altitude. Name LL terms that apply. a. YW XZ ltitude b. XYW ZYW angle bisector c. XW ZW median d. YW XZ and XW ZW perpendicular bisector, angle bisector, medians, & altitude e. XYW ZYW perpendicular bisector, angle bisector, medians, & altitude f. YW XZ and XY ZY perpendicular bisector, angle bisector, medians, & altitude losure: Name the four points of concurrency of a triangle and describe how each is formed. 1. ircumcenter-intersection of the perpendicular bisectors. Incenter-intersection of the angle bisectors 3. entroid-intersection of the medians 4. Orthocenter-intersection of the altitudes Student Notes Geometry hapter 5 Relationships within Triangles KY Page #11

12 Section: ssential Question 5 5 Use Inequalities in a Triangle How do you find the possible lengths of the third side of a triangle if you know the lengths of two sides? Warm Up: Theorems: 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. If m m then one angle of a triangle is larger than another angle, the side opposite the larger angle is longer than the side opposite the smaller angle. m m The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Triangle Inequality Theorem Student Notes Geometry hapter 5 Relationships within Triangles KY Page #1

13 x 1: Three wooden beams will be nailed together to form a brace for a wall. The bottom edge of the brace is about 8 feet. One of the angles measures about 86 and the other measure about 35. What is the angle measure opposite the largest side of the brace? x : triangle has one side of length 11 and another of length 6. escribe the possible lengths of the third side Greater than 5 and less than 17 x 3: triangle has one side of length 11 and another of length 15. escribe the possible lengths of the third side Greater than 4 and less than 6 Student Notes Geometry hapter 5 Relationships within Triangles KY Page #13

14 Section: ssential Question 5 6 Inequalities in Two Triangles and Indirect Proof How do you write an indirect proof? Warm Up: Theorems: Z X Y If two sides of 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, Hinge Theorem (SS Inequality Theorem) then the third side of the first is longer than the third side of the second. XZ, XY, N m m X YZ onverse of the Hinge Theorem (SSS Inequality Theorem) If two sides of 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. XZ, XY, N YZ m m X Student Notes Geometry hapter 5 Relationships within Triangles KY Page #14

15 Show: x 1: Given that, how does compare to? 16 ft 15 ft by the SSS Inequality (Hinge onverse) Theorem x : a. If PR = PS and mqpr m QPS, which is longer: SQ or RQ? RQ by the SS Inequality (Hinge) Theorem b. If PR = PS and RQ < SQ, which is larger: RPQ or SPQ? SPQ by the SSS Inequality (Hinge onverse) Theorem x 3: Two runners start together and run in opposite directions. ach one goes 1.5 miles, changes direction, and goes.4 miles. The first runner starts due north and runs. 100 towards the east. The other runner starts due south and turns 130 towards the west. oth runners return to the starting point. Which runner ran farther? xplain. Runner Runner ach triangle has side lengths 1.5 mi and.4 mi, and the angles between those sides are 80 and 50. The Hinge Theorem, the third side of the triangle for Runner 1 is longer, so Runner 1 ran further. 130 Student Notes Geometry hapter 5 Relationships within Triangles KY Page #15

16 Key Vocab: Indirect Proof proof in which you prove that a statement is true by first assuming that its opposite is true. If this assumption leads to an impossibility or contradiction, then you have proved that the original statement is true. Key oncept: How to Write an Indirect Proof: 1. Identify the statement you want to prove. ssume temporarily that this statement is false by assuming that its opposite is true.. Reason logically until you reach a contradiction 3. Point out that the desired conclusion must be true because the contradiction proves the temporary assumption false. x 4: Write an indirect proof of the Parallel Postulate: Through a point not on a line, there is exactly one line parallel to a given line. Temporarily assume that m and n are both parallel to k through point P. y the alternate interior angles theorem, 3 1 and 3. Then, by the transitive property, 1. ut this statement contradicts that fact that m1 m. Therefore, the temporary assumption must be false. Only one line through point P may be parallel to k m 3 P 1 n k x 5: Write an indirect proof of the orollary of the Triangle Sum Theorem. The acute angles of a right triangle are complementary. Temporarily assume that m1 m 90. This presents two cases: o ase 1: m1 m 90 If m1 m 90, then m 90 because of the Triangle Sum Theorem. This contradicts the fact that is right angle. o ase : m1 m 90 If m1 m 90, then m 90 because of the Triangle Sum Theorem. This contradicts the fact that is a right angle. Therefore, the assumption must be false. m1 m 90 losure: escribe the difference between the Hinge Theorem and its onverse. The Hinge Theorem is also called the SS Inequality Theorem and it makes a conclusion about the third side of a triangle. The onverse of the Hinge Theorem is also called the SSS Inequality Theorem and it makes a conclusion about the included angle of a triangle. Student Notes Geometry hapter 5 Relationships within Triangles KY Page #16

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