Geometry. Chapter 3. Congruent Triangles Ways of Proving Triangles Corresponding Parts of Δ s (CP Δ=) Theorems Based on Δ s

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1 Geometry hapter 3 ongruent Triangles Ways of Proving Triangles orresponding Parts of Δ s (P Δ=) Theorems ased on Δ s

2 Geometry hapter 3 ongruent Triangles Navigation: lick on sheet number to find that sheet. Sheet Numbers: HW HW 65 Mock test

3 Unit III - ongruent Triangles 05 ongruence of figures - ( ) same size and shape E If EF, then the corresponding vertices would be : F E F the corresponding angles would be: E F the corresponding sides would be: E F EF If two figures are congruent, then the corresponding parts are equal. = = F = = E = = In stating congruence, the vertices should be named in a corresponding way: EF 1

4 Proving Triangles ongruent 10 ll triangles have six basic parts, three sides and three angles. Triangles are congruent when all their parts are equal. Most times, if you can show three parts of one triangle are equal to three parts of another that will force the other parts to be equal and the triangles will be congruent. It is your job to learn which three parts force the other three parts to be equal. Postulate - If three sides of one triangle are equal to three corresponding sides of another triangle, then the triangles are congruent. (SSS Postulate) Given: = E; = F; = EF Therefore EF by the SSS Postulate In XYW and ZYW, XY = ZY and XW = ZW. Since there are only two pair of corresponding sides given to be equal, can you find a third pair and substantiate your findings? In : is opposite is included between and is opposite is included between and

5 Postulate - If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent. (SS Postulate) Given: = E; = F; = Therefore: EF by the SS Postulate Postulate - If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent. (S Postulate) Given: = E; = ; = E Therefore: EF by the S Postulate

6 Two other ways of proving triangles congruent are S and HL Theorem - If two angles and a non-included side of one triangle are equal to the corresponding parts of another triangle, then the triangles are congruent. (S) This is easily proved by using the ngle Side ngle postulate. Given: = ; = F; = FE Prove: EF Statements 1. = ; = F; = FE Reasons 1. Given 2. = E 2. if 2 s of one = to 2 s of another, the third s must be = 3. EF 3. S There is one more way of proving triangles congruent. It also involves three parts of one triangle equal to three parts of another but it is used for right triangle only. Theorem - If in two right triangles have equal hypotenuses and an equal leg then the two triangles will be congruent. This is referred to as Hypotenuse Leg (HL) We will not prove this theorem now but we will discuss it in class.

7 ongruence Worksheet 11 You decide whether the two triangles RE congruent. If they are put the postulate that supports your belief.(sss; SS; S) If there is not enough information write nnc, not necessarily congruent X Y Z X Y Z E E

8 X 10. Z E Y E E

9 Geometry Tool ox 16 1) efine congruence. 2) In, which angle is included between sides and? 3) If EF, and is the largest angle of, then what is the largest angle of EF? 4) What are the basic parts of a triangle? 5) In XYZ, which side is included between X and Z? 6) Given MNR UVW, write six equations that are a result of this congruence. Sides ngles T F 7) hexagon can be congruent to a pentagon. T F 8) Two equilateral triangles must be congruent. If XYZ PQR, is it true to say: T F 9) YZX QRP T F 10) ZYX RPQ T F 11) YXZ QPR

10 We must learn how to interpret given information. I will give you information and you turn it into an equality statement. 12) means 13) bisects means ) is the midpoint of means Given the information stated in each exercise, you are to prove. Without doing the proof, state the method you would use to prove them congruent. (SSS, SS, S, S) 15) Given: = ; = 16) Given: = ; = ) Given: 1 & 2 are right angles; = 18) Given: bisects ; 19) Given: & E bisect each other Show: E 1 2 E

11 ongruent Triangle iscoveries 17 You must mark the illustration. 1) Given: and bisect each other Prove: ΔX ΔX What are the three equal parts? 1 2 X What is the reason ΔX ΔX? Example of answer: Given: and bisect each other Prove: ΔX ΔX What are the three equal parts? a 1 = X s X = X s X = X What is the reason ΔX ΔX? SS 2) Given: 1 = 2; = ; K = K Prove: ΔK ΔK 3 1 What are the three equal parts? K 2 4 What is the reason ΔK ΔK? 8

12 3) Given: = ; = ; K = K Prove: ΔK ΔK 3 1 What are the three equal parts? K 2 4 What is the reason ΔK ΔK? 4) Given: VW bisector of XY Prove: ΔVMX ΔVMY V 6 2 What are the three equal parts? X M Y W What is the reason ΔVMX ΔVMY? 9

13 5) Given: is midpoint of ; 6 = 5 Prove: Δ ΔE 17 cont. What are the three equal parts? E What is the reason Δ ΔE? 6) Given: HG ll EF ; HG = EF Prove: ΔHMG ΔFME H 1 2 G What are the three equal parts? M 3 6 E 4 5 F What is the reason ΔHMG ΔFME? 10

14 Sides of Polygons 18 We know the sum of any two sides of a triangle must be greater than the third side. I like to say the sum of the two smaller sides must be greater than the largest side. See if you can use this bit of knowledge to answer the following questions. 1) an a triangle have sides of 8, 19 and 25? 2) an a triangle have sides of 35, 19 and 15? 3) In = 8 and = 20 this means must be greater than. 4) In = 8 and = 20 this means must be less than. 5) If a, b and c represent the length of the sides of a triangle and a = 12 and b = 30 what are the restrictions on c? 6) an a quadrilateral have sides of 12, 20, 8 and 39? 7) an a quadrilateral have sides of 12, 20, 8 and 50? 8) If a, b, c and d represent the length of the sides of a quadrilateral and a = 12, b = 18 and c = 20 what are the restrictions on d? 9) If a, b, c and d represent the length of the sides of a quadrilateral and a = 12, b = 18 and c = 40 what are the restrictions on d? If you got the last two correct you are starting to get smart!

15 Using ongruent Triangles 20 y proving that two triangles are congruent, you can deduce information about the other three parts. s you know a triangle has six basic parts and we have to get three parts of one equal to three parts of the other (the right three parts) in order for them to be congruent. This means we get the other three parts as a bonus because when figures are congruent all the parts of one are equal the corresponding parts to the other. The other three parts are equal for the reason corresponding parts of congruent triangles are equal (P Δ= ). 1) Given: RP bisects QRS & QPS Prove: RQ = RS Q P R S Δ Δ because of RQ = RS because 2) Given: and bisect each other Prove: = 1 2 X Δ Δ because of = because 12

16 3) Given: 1 = 2; 3 = 4 Prove: P = Q P M Q Δ Δ because of P = Q because 4) Given: PM = MQ; P = Q Prove: P = Q P M Q Δ Δ because of P = Q because 13

17 20 cont. 5) Given: O = O; O = O Prove: = O 1 2 Δ Δ because of = because 6) Given: O = O; = Prove: = O 1 2 Δ Δ because of = because 14

18 7) Given: 1 = 2; 3 = 4 Prove: KE = KI K 1 2 E I 3 4 T Δ Δ because of = because 8) Given: KT bisects EKI; KE = KI Prove: TK bisects ETI E K 1 2 I 3 4 T Δ Δ because of What angles have to be equal to get TK = because bisects ETI TK bisects ETI because 15

19 9) Given: ; ; M is midpoint of Prove: = 1 M 2 Δ Δ because of = because 10) Given: ll ; M is midpoint of Prove: M is midpoint of 1 M 2 Δ Δ because of What sides have to be equal to get M is midpoint of = because M is midpoint of 16

20 Isosceles Triangle Theorems 25 Parts of an isosceles triangle - legs base base angles vertex angle - the two equal sides the third side they are the angles opposite the equal sides the angle opposite the base Vertex angle ase ase angles Leg ase Vertex angle Theorem - If two sides of a triangle are equal, then the angles opposite those sides are equal. Given: = Prove: = Statements 1. = s 2. raw, bis. of Reasons 1. Given 2. Every angle has exactly one bisector = 2 a 4. = s 3. def. of bisect 4. Reflexive = 5. SS 6. P r =

21 orollary - n equilateral triangle is also equiangular. orollary - n equilateral triangle has three 60 angles. orollary - The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint. Theorem - If two angles of a triangle are equal, then the sides opposite those angles are equal. Given: = Prove: = Statements 1. = a 2. raw, bis. of Reasons 1. Given 2. Every angle has exactly one bisector = 2 a 3.def of bisect 4. = s 4. Reflexive S 6. = 6. P r = orollary - n equiangular triangle is also equilateral.

22 Isosceles Triangle Worksheet 26 1) 1 = 20 ; 3 = 4 = 4 5 = ) 1 = x ; problems 1 & 2 3 = 4 = 5 = 2 5 3) 1 = 35 ; = 4) 1 = x ; = problems 3 & 4 5) 1 = 23 ; 7 = ) 1 = x ; 7 = 4 6 problems 5 & 6

23 7) is equiangular, = 4x y; = 2x + 3y; = 7 x = y = 8) EF is equilateral, = x + y; E = 2x y x = y = 9) In JKL JK = KL, J = 2x y; K = 2x + 2y; L = x + 2y x = y = Use isosceles triangles to show RU = SV. Mark the illustration and provide the statements that lead to the conclusion. 10) Given: R S ; 1 2 Show: RU = SV R S U 1 2 V T

24 Polygon Questions If you were to place regular congruent hexagons in a side to side circular pattern, you would form a regular hexagon in the middle. See I told you! 2. What figure would be formed if you did the same thing with regular congruent pentagons? This is the way it would start. 3. What about if we started with regular congruent octagons? 4. What about if we started with regular congruent 12-agons? 21

25 Worksheet - Proving Triangles ongruent 40 X Y Z Give the reason YXZ. If they don't have to be congruent, write nnc. 1. = YX; = XZ; = Z 2. = YZ; = X; = Y 3. = Z; = X; = Y 4. = YX; = X = 90 ; = XZ 5. = YZ; = X = 90 ; = XZ 6. = YX; = YZ; = XZ 24

26 Use this figure for proofs 7 & 8 7. Given: ; ; = Prove: = 8. Given: ll Prove: = ; = Statements Reasons Statements Reasons 1. ; 1. Given 1. ll ; 1. Given = = S T H O Z R J K 9. Given: RT = S; RS = T Prove: TS = STR 10. Given: H J; JK J ; JH = K Prove: H = K Statements Reasons Statements Reasons 1. RT = S; 1. Given 1. H J ; JK J 1. Given RS = T JH = K 25

27 Name 46 Geometry Homework Mr. Londino 1) efine congruence. 2) List all the ways to prove triangles congruent. 3) In an isosceles triangle, the sum of the measures of the base angles is 88 less than the measure of the remaining angle. Find the measure of each angle. (show equation) 4) What are the measures of the three angles of an isosceles right triangle? 5) What is the measure of a base angle of an isosceles triangle that has a vertex angle of 85? 6) What is the measure of the vertex angle of an isosceles triangle that has base angles measuring 35 each? 7) If the measure of one base angle of an isosceles triangle is represented as 2x, write an expression in terms of x to represent the vertex angle.

28 8) The vertex angle of an isosceles triangle is (x 4) degrees. Represent one base angle. (simplify the expression) Use isosceles triangles to show 3 = 4. Mark the illustration and provide the statements that lead to the conclusion. 9) Given: P = P; = Show: 3 = 4 P

29 Name 46 cont. 11. = = 6x 6; = 5x + 7 = problems 12, 13 & = 3x 4; = 4x - 7 = = 14. = x ; = x 6 2 = 15. Find each numbered angle if 8 = = 2 = 3 T = 4 = R S 6 V 7 5 = 6 = 7 = 28

30 Medians, ltitudes and Perpendicular isectors 50 Median- ltitude - a segment drawn from any vertex of a triangle to the midpoint of the opposite side a segment drawn from any vertex of a triangle perpendicular to the line that contains the opposite side. In, is the midpoint of, therefore is a median from vertex In, E is perpendicular to, therefore E is an altitude from vertex In EF, H is the midpoint of E, therefore FH is a median from vertex F In EF, FG is perpendicular to E, therefore FG is an altitude from vertex F How many medians does a triangle have? How many altitudes does a triangle have? What is unusual about the altitudes of an obtuse triangle? (like EF above) What is unusual about the altitudes of a right triangle?

31 Theorem - If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment. Given: is the perpendicular bisector of Statements 1. is the bisector of Reasons 1. Given Prove: = 2. raw & 3. 1 = 2 = = 5. = = 2. onstruction 3. definition of 4 definition of bisector 5. Reflexive 6. SS 7. P s r = 1 2 It is also true that = and is isosceles. If you construct an altitude from the vertex angle of an isosceles it will also be a median and vice versa and the vertex angle will be bisected by such a construction. RTS is an isosceles triangle with RT = TS and R = S If TW is an altitude it will also be a median If TW is a median it will also be an altitude In either case RTS the vertex angle will be bisected In the special isosceles triangle, the equilateral triangle all altitudes are medians and all medians are altitudes.

32 rawing ltitudes & Medians 51 raw the three medians and three altitudes for the following triangles. First draw the three medians G Then draw the three altitudes G H I H I First draw the three medians M Then draw the three altitudes M L L N N First draw the three medians Then draw the three altitudes 32

33 First draw the three medians Q R P Then draw the three altitudes Q R P 33

34 Name 57 Geometry Homework Mr. Londino 1) efine congruence. 2) List all the ways to prove triangles congruent. 3) In a right triangle, what is true about two of the altitudes? 4) n altitude of a triangle (always, sometimes or never) falls outside the triangle. (circle choice) 5) n angle of an isosceles triangle is 42. The other two angles could be & or & 6) base angle of an isosceles triangle is 15 less than the vertex angle. Find the vertex angle. (show equation) 7) The vertex angle of an isosceles triangle is 20 more than twice the measure of a base angle. Find the measure of each angle. (show equation)

35 Z X Y Refer to the above triangles and write the postulate or theorem you would use to prove XYZ. If none apply, write nnc 8) = XY; = X; = Y 9) = XY; = XZ; = X 10) = Z = 90 ; = XZ; = XY 11) = XY; = XZ; = YZ 12) = Z; = Y; = YZ 13) Given: = P; P Show: = 1 2 P

36 hapter Review 65 Suppose RE SUN: 1) E = 2) = 3) RE = 4) E = 5) RE 6) UNS an the two triangles be proved congruent? If so, name postulate or theorem used 7) 8)

37 9) 10) 11) 12) There are two triangles, and RST. If the triangles can be proven to be congruent from the given information, state the reason they are congruent. If the information is not sufficient to prove they are congruent write nnc. = R; = S; = ST = R; = RS; = ST = RT; = RS; = ST = R; = RS = 14; = ST = 12 = R = 90 ; = RS; = ST

38 13) Two sides of a triangle are 12 and 30 the remaining side is k write the restrictions on the length of k. 14) Four sides of a pentagon are 8, 10, 12 and 20 the remaining side is k write the restrictions on the length of k. 15) Three sides of a quadrilateral are 4, 6 and 12 the remaining side is k write the restrictions on the length of k. 16) Given: ; = a) = 4x 3; = 3x + 11 = b) = 4x 8; = 4x 2 = True or False( the correct box) T F 17) The base of an isosceles triangle is opposite a base angle. T F 18) triangle can have sides of 9, 11, and 29. T F 19) If three parts of one triangle are equal to three parts of a second triangle, then the two triangles must be congruent. T F 20) The vertex angle of an isosceles triangle is included between the legs. T F 21) The base angles of an isosceles triangle must be larger than the vertex angle.

39 22) efine congruence. 23) What are the measures of the three angles of an isosceles right triangle? 24) What is the measure of a base angle of an isosceles triangle that has a vertex angle of 92? 25) What is the measure of the vertex angle of an isosceles triangle that has base angles measuring 43 each? 26) If EF, and is the smallest angle of, then what is the shortest side of EF? 27) In, which angle is included between sides and? Proof: 28) Given: ; E is midpoint of Prove: E E 1 2 E 3 6 Statements 1. ; E is midpoint of Reasons 1. Given 4 5

40 Find the value of x in each diagram: 29) x = 30) x = 60 X 6x x + 5

41 31) raw the three altitudes in each of the triangles below. 32) raw the three medians in the triangle below.

42 Mock Test There are two triangles, and RST. If the triangles can be proven congruent from the given information, state the reason they are congruent. If the information is not sufficient to prove they are congruent write nnc. 1) = R; = S; = ST 2) = RT; = RS; = R 3) = R = 90 ; = RS; = ST True or False( the correct box) T F 4) In a triangle if an altitude and a median are the same segment then the triangle is isosceles. T F 5) median of a triangle can fall outside the triangle. T F 6) If all three angles of a triangle are equal, then all three sides are equal. T F 7) If JET PY, is it true to say: ETJ YP 8) Two sides of a triangle are 11 and 45 the remaining side is k write the restrictions on the length of k. 9) The measure of the base angle of an isosceles triangle is 4x 30. Write the measure of one vertex angle in terms of x. (simplify fully) 10) Four sides of a pentagon are 8, 12, 15 and 40 the remaining side is k write the restrictions on the length of k.

43 11) What is the measure of a base angle of an isosceles triangle that has a vertex angle of 126? 12) The measure of the vertex angle of an isosceles triangle is 4x 30. Write the measure of a base angle in terms of x. (simplify fully) For problems 13 &14: = 13) = 4x + 4; = 8x 17 = 14) ngle is five more than two times, find the measure of. = 15) There exists an isosceles triangle RST whose perimeter is 40. If RS = x and RT = 2x find the measure of ST. ST =

44 For problem 16: arefully mark the diagram with the given information and any other information obvious from the illustration. State the three parts that are equal, the triangles that are congruent and the reason they are. State the reason for EJ = GH 16) Given: EH bisects JG ; EJ l l GH Prove: EJ = GH E J F G H FEJ because of EJ = GH because of