Geometry Unit 4a - Notes Triangle Relationships

Transcription

1 Geometry Unit 4a - Notes Triangle Relationships This unit is broken into two parts, 4a & 4b. test should be given following each part. Triangle - a figure formed by three segments joining three noncollinear points, called vertices. The triangle symbol is. Syllabus Objective The student will classify triangles by sides and/or angles. lassification of Triangles by Sides quilateral triangle - a triangle with three congruent sides. Isosceles triangle - a triangle with at least two congruent sides Scalene triangle - a triangle with no congruent sides. lassification of Triangles by ngles cute triangle - a triangle with three acute angles. Right triangle - a triangle with one right angle. Obtuse triangle - a triangle with one obtuse angle. quiangular triangle a triangle with three congruent angles. djacent sides of a triangle - two sides sharing a common vertex. djacent Sides ommon Vertex Hypotenuse of a right triangle - the side opposite the right angle. Unit 4a Triangle Relationships Page 1 of 23

2 Right Triangle Isosceles Triangle Vertex cute Leg Hypotenuse Vertex l Leg Leg Right l Leg cute l ase ase ase Syllabus Objective The student will solve problems applying the triangle sum and exterior angle theorems. sk the entire class to draw a triangle on a piece of paper, then have each person cut out their triangle. Label the angles 1, 2, and 3 as shown Tear each angle from the triangle and then place them side by side along a straight line. The three angles seem to fill in or form straight line ecause by definition a straight angle is 180º, the experiment might lead students to believe the sum of the interior angles of a triangle is 180º. While that s not a proof, it does provide students with some valuable insights. The fact is, it turns out to be true. Unit 4a Triangle Relationships Page 2 of 23

3 Triangle Sum Theorem: { Sum Th.} The sum of the measures of the interior angles of a triangle is 180º. R S 1 3 Given: Prove: m 1 + m 2+ m 3= 180 The most important part of any proof is the ability to use geometry already learned. In the case, if only the three angles of the triangle were considered, the proof would go nowhere. Using parallel lines, from the previous unit, construct RS parallel to and label the angles formed. Parallel lines being cut by a transversal create angle pairs that will help complete the proof. Statements 1) 1) Given 2) raw RS 2) onstruction 3) 4& S are supp. 3) L.P. Post. 4) m 4 + m S = 180 4) ef of Supp. 5) m S = m 2+ m 5 5) dd. Post. 6) m 4 + m 2 + m 5 = 180 6) Substitution 7) Reasons 1 4, 7) lines cut by trans., alt. int. 's 3 5 8) m 1 + m 2 + m 3= 180 8) Substitution theorem that follows directly from this theorem is one about the relationship between the exterior angle of a triangle and the nonadjacent angles inside the triangle. xterior angle when the sides of a triangle are extended, the angles that are adjacent to the interior angles. Unit 4a Triangle Relationships Page 3 of 23

4 xterior ngle Theorem: The exterior angle of a triangle is equal to the sum of the two remote interior angles. {xt. Th.} Given: Prove: m 1 = m + m Statements Reasons 1) m + m + m 2 = 180 1) Sum Th. 2) 1& 2are supp. 's 2) L.P. Post. 3) m 1 + m 2 = 180 3) ef. of supp. 4) m + m + m 2= m 1+ m 2 4) Substitution 5) m + m = m 1 5) Subtr. Prop. of quality 2 1 xample: ind the measure of Since the sum of the two angles given is 110, 3 must be orollary to Triangle Sum Theorem: The acute angles of a right triangle are complementary. xamples: a) ind the value of x. 30 (2x + 10) 3x The sum of the interior angles is (2x + 10) + (3x) = 180 (combine like terms) 5x + 40 = 180 (subtract 40) 5x = 140 (divide by 5) x = 28 Unit 4a Triangle Relationships Page 4 of 23

5 b) 60 x (2x + 10) xterior angle is equal to the sum of the two remote interior angles. (2x + 10) = x + 60 (subtract x) x + 10 = 60 (subtract 10) x = 50 Third ngle Theorem: If two angles of one triangle are congruent to two angles of another triangle, the third angles are congruent. {3 rd Th.} Syllabus Objective The student will analyze the relationships between congruent figures. ongruence The word congruent is used to describe objects that have the same size and shape. Things that are traced are considered congruent. fter being traced, the objects could be flipped or rotated but they will still maintain their congruence. Stop signs would be examples of congruent shapes. Since a stop sign has 8 sides, they would be congruent octagons. ongruent objects have corresponding or matching sides and angles. In order for objects to be congruent, all their corresponding sides and all their corresponding angles must be congruent to one another. Writing congruence statements: when stating that two polygons are congruent the corresponding angles must be written in order. Unit 4a Triangle Relationships Page 5 of 23

6 Triangles To determine if two triangles are congruent, they must have the same size and shape. They must fit on top of each other, they must coincide. Mathematically, all the sides and angles of one triangle must be congruent to the corresponding sides and angles of another triangle. In other words, it must be shown that angles,, and are congruent ( ) to angles, and, and shown,, and are to,, and respectively. LL six relationships must be shown. Some triangle congruence theorems: Reflexive Property of ongruent Triangles very triangle is congruent to itself. Symmetric Property of ongruent Triangles If, then. Transitive Property of ongruent Triangles - If and JKL, then JKL. Syllabus Objective The student will justify congruence using corresponding parts of congruent triangles. If given three sticks of length 10, 8, and 7 and asked to glue the ends together to make triangles, students would find that something interesting happens. The triangles would all stack on top of each other, they would coincide. ecause they are congruent! Rather than showing all the angles and all the sides of one triangle are congruent to all the sides and all the angles of another triangle (6 relationships), students are now able to determine congruence by just using the 3 sets of corresponding sides. shortcut! That leads to the Side, Side, Side congruence postulate. Unit 4a Triangle Relationships Page 6 of 23

7 Side-Side-Side ongruence Postulate: If three sides of one triangle are congruent, respectively, to three sides of another triangle, then the triangles are congruent. {SSS} Using similar demonstrations, two more congruence postulates can be arrived at. The Side, ngle, Side postulate is abbreviated SS Postulate. Side-ngle-Side ongruence Postulate: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, respectively, then the two triangles are congruent. {SS} third postulate is the ngle, Side, ngle postulate. ngle-side-ngle ongruence Postulate: If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. {S} ombining this information with previous information, students will be able to determine if triangles are congruent. Unit 4a Triangle Relationships Page 7 of 23

8 Syllabus Objective The student will prove that two triangles are congruent. Proofs: ongruent 's To prove triangles are congruent, use the SSS, SS and S congruence postulates. lso, review other theorems that will lead students to more information. ngle-ngle-side ongruence Theorem: If two angles and the non included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. {S} xample: Write a two-column proof. raw and label the two triangles. Given:,, Prove: 1) Statements,, Reasons 1) Given 2) 2) 3 rd Th 3) 3) S Unit 4a Triangle Relationships Page 8 of 23

9 4 ways of proving triangles congruent: SSS, SS, S, and S. When writing proofs: irst, label congruences in the diagram using previous knowledge. Second, label any unspoken information that must be included (like vertical angles or shared sides). Next, try to use one of the four methods (SSS, SS, S, and S) of proving triangles congruent. inally, write those relationships in the body of proof. Given:, bisects Prove: X X X **Students should be reminded to mark their diagrams with given information. They should add any additional information that is found through their investigation. Visualization is very important and helpful in completing proofs! Unit 4a Triangle Relationships Page 9 of 23

10 X 1) Statements, bisects 1) Given Reasons 2) X X 2) ef. of bisector 3) 3) lines cut by trans., alt. int. 's 4) X X 4) V.. Th. 5) X X 5) S In order to prove these triangles congruent, the definition of a bisector and the subsequent mathematical relationship was necessary. ven though vertical angles were not part of the given information, they are unspoken information and can be seen in the diagram. Therefore, the theorem that all vertical angles are congruent can be used. Right Triangles While the congruence postulates and theorems apply for all triangles, we have postulates and theorems that apply specifically for right triangles. HL Theorem: If the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent. The other congruence theorems for right triangles can be seen as special cases of the other triangle congruence postulates and theorems. Unit 4a Triangle Relationships Page 10 of 23

11 LL Theorem: If two legs of one right triangle are congruent to two legs of another right triangle, the triangles are congruent. Special case of SS and can be considered redundant. H Theorem: If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and acute angle of another right triangle, the triangles are congruent. S and can be considered redundant. L Theorem: If a leg and an acute angle of one right triangle are congruent to a leg and an acute angle of another right triangle, the triangles are congruent. If drawn and labeled, the diagram of the L ongruence Theorem would show that it is derived from either S or S depending on which corresponding set of legs or angles are congruent. Once again this theorem can be considered redundant. s marked in this first set, S. s marked in this second set, S. Proofs: PT When two triangles are congruent, each part of one triangle is congruent to the corresponding part of the other triangle. That s referred to as orresponding Parts of ongruent Triangles are ongruent, thus PT. One way to determine if two line segments or two angles are congruent is by showing they are the corresponding parts of two congruent triangles. Unit 4a Triangle Relationships Page 11 of 23

12 1. Identify two triangles in which the segments or angles are the corresponding parts. 2. Prove the triangles are congruent. 3. State the two parts are congruent, supporting the statement with the reason; corresponding parts of congruent triangles are congruent. Proving two segments congruent: P Given: and bisect each other Prove: The strategy to prove these segments are congruent is to first show the triangles are congruent. ill in the diagram showing the relationships based upon the information given and the other relationships that exist using previously learned definitions, theorems, and postulates. P Using the definition of bisector, it is determined that P P and P P. lso, notice a pair of vertical angles. The diagram has been marked to show these relationships. ven though the vertical angles are marked in the diagram, the statement must be written in the proof. Unit 4a Triangle Relationships Page 12 of 23

13 Statements 1) and bisect each other 1) Given Reasons 2) P P, P P 2) ef. of bisector 3) P P 3) V.. Th. 4) P P 4) SS 5) 5) PT illing in the body of the proof is easier after marking the congruences in the diagram. The strategy to show angles or segments are congruent is to first show the triangles are congruent, then use PT., Given: X X Prove: X X **Mark the picture with the parts that are congruent based on what s given. **Then mark the relationships based upon knowledge of geometry. In this case: X X. The diagram shows three sides of one triangle are congruent to three corresponding sides of another triangle; therefore the triangles are congruent X X If the triangles are congruent, then all the remaining corresponding parts of the triangles are congruent by PT. That means 1 2. Unit 4a Triangle Relationships Page 13 of 23

14 The idea of using PT after proving triangles congruent by SSS, SS, S, and S will allow students to find many more relationships in geometry. Syllabus Objective The student will prove and use the properties of isosceles and/or equilateral triangles. ase ngle Theorem: If two sides of a triangle are congruent, then the angles opposite them are congruent. orollary to the ase ngle Theorem: If a triangle is equilateral, then it is equiangular. onverse of the ase ngle Theorem: If two angles of a triangle are congruent, then the sides opposite them are congruent., Given: Prove: 1 2 X Statements Reasons 1) raw bisector X 1) onstruction 2) 1 2 2) ef. of bisector 3) 3) Given 4) X X 4) Reflexive Prop. 5) X X 5) S 6) 6) PT orollary to the onverse of the ase ngle Theorem: If a triangle is equiangular, then it is equilateral. Unit 4a Triangle Relationships Page 14 of 23

15 **Isosceles Triangle** The Problem: In the Isosceles triangle shown, =. rom, a P has been drawn to meet the opposite side at right angles (altitude from ). Show that 1 1 P = or a = b. 2 2 Solution: In P, a = 90 c. In b = c = ( c ), Hence 1 b = 2 a or a = b. 2 **Two Triangles** The Problem: Two equilateral triangles, of lengths 10 and 7 respectively, are drawn, with their bases touching and in line Unit 4a Triangle Relationships Page 15 of 23

16 Lines are drawn to connect the tops of each to the furthest corner of the other. Is one line longer than the other? an you prove it? Solution: This can be solved by using the Pythagorean Theorem, but is much easier than that Rotate 60 clockwise about. It s congruent to! Therefore, =. Syllabus Objective The student will solve problems related to congruent triangles using algebraic techniques. Students must be able to match corresponding parts of congruent triangles and solve for unknown values and measures. They may have to determine that the triangles are congruent before they can perform the calculations. xample: Given the two congruent triangles, find the values of x and y. orresponding angles: 4x = 52 x = 13. The other acute angle would be (90 52) or 38. So 6y + 2 = 38 6y = 36 y = 6. 4x 52 (6y + 2) Syllabus Objective The student will classify triangles using coordinate geometry. (May need supplemental material) Unit 4a Triangle Relationships Page 16 of 23

17 oordinate proofs use figures in the coordinate plane and algebra to prove geometric concepts. It is helpful to discuss convenient placement of figures in the coordinate plane. When coordinates are not given, care should be taken to place figures in positions that benefit the work to be done. Some helpful hints: (Whenever possible) Use the origin as a vertex or center of the triangle. Place at least one side of a triangle on an axis. Keep the triangle within the first quadrant. Use coordinates that make computations as simple as possible. Isosceles triangle can be placed as so: Right triangles as so: {Increments have been removed to show how coordinate graphs can be illustrated. ssign variable lengths; a, b, c, etc } xample: Name the missing coordinates of isosceles triangle XYZ. Vertex X is positioned at the origin; its coordinates are (0, 0). Vertex Z is on the x-axis, so its y-coordinate is 0. The coordinates of vertex Z are (a, 0). XYZ is isosceles, so the x-coordinate of Y is located halfway between 0 and a, or at 2 a. We cannot write the y-coordinate in terms of a, so call it b. The coordinates of point Y are a, b. 2 O X (?,?) Y (?,?) Z ( a,?) Unit 4a Triangle Relationships Page 17 of 23

18 xample: Write a coordinate proof to show that a line segment joining the midpoints of two sides of a triangle is parallel to the third side. Place a vertex at the origin and label it. Use coordinates that are multiples of 2 because the Midpoint ormula involves dividing the sum of the coordinates by 2. Given:, S is the midpoint of, T is the midpoint of Prove: ST S (2 b, 2 c ) T O (0, 0) (2 a, 0) Proof: y the Midpoint ormula, the coordinates of S are 2b + 0 2c + 0, or ( bc, ) 2 2 and the coordinates of T are 2a + 2b 0 + 2c, or ( a+ bc, ). 2 2 c c y the Slope ormula, the slope of ST is or 0 and the slope of a + b b 0 0 is or 0. Since ST and have the same slope, ST. 2a a Unit 4a Triangle Relationships Page 18 of 23

19 This unit is designed to follow the Nevada State Standards for Geometry, S syllabus and benchmark calendar. It loosely correlates to hapter 4 of Mcougal Littell Geometry 2004, sections The following questions were taken from the 1 st semester common assessment practice and operational exams for and would apply to this unit. Multiple hoice # Practice xam (08-09) Operational xam (08-09) 18. Which is a valid classification for a triangle?. cute and right. Isosceles and scalene. Isosceles and right. Obtuse and equiangular. Isosceles and obtuse 20. In the figures below, In the figure below, RSTUVW. Which is a valid classification for a triangle?. cute and right. Obtuse and equilateral. Isosceles and scalene RSTUV. V W U R R V U T Which side of RSTUVW corresponds to?. RW. SR. UT. UV S T Which side of RSTUV corresponds to?. SR. TS. UT. VU S Unit 4a Triangle Relationships Page 19 of 23

20 21. Use the triangles below. Use the triangles below. Which congruence postulate or theorem would prove that these two triangles are congruent?. angle-angle-side. angle-side-angle. side-angle-side. side-side-side 22. In the diagram below, and. Which congruence postulate or theorem would prove these two triangles are congruent?. angle-angle-angle. angle-side-angle. side-angle-side. side-side-side In the diagram below, and bisect each other at. Which congruence postulate or theorem would prove that these two triangles are congruent?. side-side-side. angle-angle-angle. side-angle-side. angle-side-angle 23. Given that RST XYZ, m R= 6n+ 1, m Y = 108, and ( ) ( 9 4) m Z = n, what is the value of n? Which congruence postulate or theorem would prove these two triangles are congruent?. angle-angle-angle. angle-side-angle. side-angle-side. side-side-side Given that RST XYZ, m R= ( 5a), m Y = 65, and m Z = 75, what is the value of a? Unit 4a Triangle Relationships Page 20 of 23

21 24. Given that PQR JKL, PQ = 4x + 12, JK = 7x 6, KL = 2x + 17, and JL = 5x 7, what is the value of x? The statements for a proof are given below. Given: Parallelogram X Y Prove: X Y X Given that PQR JKL, PQ = 9x 45, JK = 6x + 15, KL = 2x, and JL = 5x, what is the value of x? The statements for a proof are given below. Given: Parallelogram X Y Prove: X Y X Proof: Y STTMNTS RSONS 1. Parallelogram, X Y 1. Given X Y What is the reason that the statement in Step 4 is true?. side-angle-side. angle-side-angle. Opposite sides of a parallelogram are congruent.. orresponding angles of congruent triangles are congruent. Proof: Y STTMNTS RSONS 1. Parallelogram, X Y 1. Given X Y X Y 5. What reason makes the statement in Step 4 true?. Side-angle-side congruence theorem.. ngle-side-angle congruence theorem.. Opposite sides of a parallelogram are congruent.. orresponding parts of congruent triangles are congruent. Unit 4a Triangle Relationships Page 21 of 23

22 26. The statements for a proof are given below. Given: Prove: The statements for a proof are given below. Given: Prove: Proof: STTMNTS RSONS Given Given Given orresponding Parts of ongruent Triangles are ongruent What is the missing reason that would complete this proof?. side-side-side. side-angle-side. angle-side-angle. angle-angle-side 27. Given that LMN, m = ( 2x+ 15), m L= 3( x 2), and = 4( x 17), what is LN? In the isosceles triangle below, m H = 137. What is the measure of H? G Proof: STTMNTS RSONS Given Given Given What reason makes the statement in Step 5 true?. ngle-angle-side congruence theorem.. ngle-side-angle congruence theorem.. efinition of congruent segments.. orresponding parts of congruent triangles are congruent. Given that LMN, m = ( x+ 75), m L= ( 3x+ 15), and = 2x 26, what is LN? In the isosceles triangle below, m H = 124. H 124 What is the measure of ? G Unit 4a Triangle Relationships Page 22 of 23

23 Sample ST Question(s): Taken from ollege oard online practice problems. 1. In isosceles triangle above, M and M are the angle bisectors of angle and angle. What is the measure of angle M? () 110 () 115 () 120 () 125 () In the figure above, point lies on side. If 55 < x < 60, what is one possible value of y? sdgrid-in 3. If XYX is equilateral, what is the value of r+ s+ t+ u? Grid-In Unit 4a Triangle Relationships Page 23 of 23