Name: Unit 4 Congruency and Triangle Proofs

Transcription

1 Name: Unit 4 ongruency and Triangle Proofs 1

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3 Triangle ongruence and Rigid Transformations In the diagram at the right, a transformation has occurred on. escribe a transformation that created image from. Is congruent to? xplain. The vertices of MP are M(-8, 4), (-6, 8) and P(-2, 7). The vertices of M P are M (8, -4), (6, -8) and P (2, -7). Plot M P. Verify that the sides of the triangles are congruent. escribe a rigid motion that can be used to M P iven PQR with P(-4, 2), Q(2, 6) and R(0, 0) is congruent to STR with S(2, -4), T(6, 2) and R(0, 0). Plot STR. escribe a rigid motion which can be used to verify the triangles are congruent. 3

4 iven RST with R(1, 1), S(4, 5) and T(7, 5). Plot the reflection of RST in the y-axis and label it R S T. Is RST congruent to R S T? xplain. Plot the image of R S T under the translation (x, y) (x + 4, y 8). Label the image of R S T. Is R S T congruent to R S T? xplain. Is RST congruent to R S T? xplain. iven F with (1, -1), F(9, 6) and (5,7) and T with (1, 1), (-6, 9) and T(-7, 5). escribe a transformation that will yield T as the image of F. Is T congruent to F? xplain. iven P with (-4, -2), (2, 4) and P(4, 0) and SUN with S(-8, -4), U(4, 8) and N(8, 0). escribe a transformation that will yield SUN as the image of P. c) Is P congruent to SUN? xplain. 4

5 iscovering ongruent Triangles ctivity Part 1 1. Have students put the 3 straws of different lengths together to form a triangle as shown. 2. Form another triangle with the other set of straws. 3. Measure the angles of both triangles using a protractor. Questions: 1. What are the measures of the 3 angles in the first triangle? Part 2 2. What are the measures of the 3 angles in the second triangle? 3. What is the relationship between the angles of each triangle? 4. re the triangles congruent? 5. an the straws be rearranged to form a triangle with different angles? 1. Take 2 of the straws, place them on a piece of paper, and form a 60 degree angle between them. 2. Take the 2 straws of the same length and also form a 60 degree angle between them. 3. raw a line to represent the 3 rd side. Repeat the process for the 2 nd triangle. 4. Measure the length of the 3 rd side and the two remaining angles for each triangle. Questions: 1. What is the length of the 3 rd side? 2. What are the measures of the remaining angles? 3. re the two triangles congruent? 4. Use any two straws and any angle of your choice. o you get the same result? Will you always get the same result? 5

6 Part 3 1. Measure three angles measuring 80, 60, and 40 degrees on the corners of 2 pieces of construction paper or cardstock, cut them out, and label them. 2. On a piece of paper, take one of the straws, and place two of the cut-out angles on each end as shown. Repeat the process for the 2 nd triangle. 3. Using a ruler, draw a segment along each of the angle. The two segments should intersect forming the last angle. Repeat the process for the 2nd triangle. 4. Measure the 3 rd angle and the lengths of the 2 sides in each triangle. Questions: 1. What is the measure of the 3 rd angle for each triangle? 2. What are the measures of the remaining 2 sides for each triangle? 3. re the triangles congruent? 4. What if you used the 5cm straw? The 8cm straw? straw with a different length? Part 4 1. Use two of the angles used in the example above. 2. Use one of the straws and place one of the angles alongside it as shown. raw a long segment like the dashed one in the drawing. Repeat the process for the 2 nd triangle. 3. Place the second angle along this segment so that when a 2 nd segment is drawn, it will connect with the end of the straw. 4. Measure the 3 rd angle and the two remaining sides. Questions: 1. What is the measure of the 3 rd angle for each triangle? 2. What are the measures of the remaining 2 sides for each triangle? 3. re the triangles congruent? 6

7 Part 5 1. Place two of the straws together forming an angle of any degree for one triangle, and repeat the process for the 2 nd triangle. 2. Use one of the pre-cut angles and place alongside the longer of the sides but not as the included angle. 3. raw a segment to connect the 3 rd side to the other two sides. 4. Swing the 8cm straw so that it hits the 3 rd side at a different spot in the 2 nd triangle as in the first. 5. Measure the 3 rd side and the remaining 2 angles in each triangle. Questions 1. What is the measure of the 3 rd side for each triangle? 2. What are the measures of the remaining 2 angles for each triangle? 3. re the two triangles congruent? 4. o you think that you would get different results if you used a different angle? Part 6 1. Place the 3 angles so that they can form a triangle without measuring the sides initially. raw segments connecting the angles. Repeat the process for the second triangle. 2. Measure the 3 sides for each triangle. Questions 1. What are the measures of the 3 sides for each triangle? 2. re the two triangles congruent? 7

8 ongruent Triangles Investigation Part I What does it mean to say two triangles are congruent? List the ways to justify that triangles are similar. xamine the triangles with all side lengths labeled. re they similar? Why? What is the scale factor? : 3 cm 4 cm 2 cm 2 cm T S 4 cm 3 cm R What do we know about the corresponding angles of similar triangles? What does this tell us about the pair of triangles? Part 2: xamine the triangles with two sides lengths and an included angle labeled. re they similar? Why? What is the scale factor? : 1.5 cm I P 4 cm 5 cm 4 cm 5 cm H O Since the triangles are similar, what do we know about P and H? What do we know about I and O? Use the scale factor you gave in part b to determine the length of OH. What does this tell us about the pair of triangles? Part 3: xamine the triangles with two angle pairs marked congruent. re they similar? Why? What is the scale factor? : Since the triangles are similar, what do we now bout K and Y? Use the scale factor you gave in part c to determine the lengths of KL and. YZ What does this tell us about the pair of triangles? 8 K 3.2 cm 2.15 cm L M X Z 4 cm 4 cm Y

9 Think back to the three situations we examined. In #1, we were given 3 pairs of sides of one triangle are congruent to 3 pairs of sides of another triangle. The triangles are congruent by the,, (SSS) Postulate. In #2, we were given 2 pairs of sides and an included angle of one triangle are congruent to 2 pairs of sides and an included angle of another triangle. The triangles are congruent by the 3 cm 4 cm 2 cm 2 cm T S 4 cm 3 cm R,, (SS) Postulate. In #3, we were given 2 pairs of angles and an included side of one triangle are congruent to 2 pairs of angles and an included side of another triangle. The triangles are congruent by the 1.5 cm I P 4 cm 5 cm 5 cm 4 cm H O,, (S) Postulate. Two other Postulates ngle, ngle, Side Postulate (S Theorem) K 3.2 cm 2.15 cm Y How can you change this into one of the Postulates that we already have? Hypotenuse, Leg Theorem (HL Theorem) L M X Z 4 cm 4 cm 5 cm 10 cm 5 cm 10 cm y Pythagorean Theorem, if you know the hypotenuse and 1 leg, you can calculate the 2 nd leg by Theorem and prove congruence by postulate. Note: Postulate Statement which is taken to be true without proof. Theorem Statement that can be demonstrated to be true by accepted mathematical operations and arguments. 9

10 HW: How do you prove triangles are congruent? #1 & 2 Use the given coordinates to determine if F 10

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12 PT ssential question: What can you conclude about two triangles that are congruent? When you know that two triangles are congruent, you can make conclusions about the sides and angles of the triangles. Reflect: If you know that F, what six congruent statement about segments and angles can you write? Why? 12

13 When two triangles are congruent the corresponding parts are the sides and angles that are images of each other. You write congruence statements for two figures by matching the corresponding parts. In other words, the statement F contains the information that corresponds to so that, corresponds to so that, and so on. orresponding Parts of ongruent Triangles are ongruent Theorem (PT) If two triangles are congruent, then the corresponding sides are congruent and the corresponding angles are congruent. onverse of the orresponding Parts of ongruent Triangles are ongruent Theorem (PT) If tow triangles have corresponding sides that are congruent and the corresponding angles that are congruent, then the triangles are congruent. xamples iscuss: 13

14 PT and Naming ongruent Triangles I. raw and label a diagram. Then solve for the variable and the missing measure or length. 1. If T O, and m = 14, m = 29, and m O = 10x + 7. Find the value of x m O. x = m O= 2. If OW PI, and O = 25, W = 18, I = 23, and P = 7x 17. Find the value of x and P. x = P= 3. If F PQR and = 3x 10, QR = 4x 23, and PQ = 2x + 7. Find the value of x and F. x = F = II. Use the given information and triangle congruence statement to complete the following. 4. O, = 4, = 6, and = 8. What is the length of O? How do you know? 5. LUK, m = 52, m = 48, and m = 80. a. What is the largest angle of LUK? b. What is the smallest angle of LUK? 6. SUN HOT. SUN is isosceles. Is there enough information to determine if HOT is isosceles? xplain why or why not. 14

15 III. omplete the congruence statement for each pair of congruent triangles. Then state the reason you are able to determine the triangles are congruent. If you cannot conclude that triangles are congruent, write none in the blanks. 7. F LKM by by by L F H K M T J by by X Y IV. Use the given information to mark the diagram and any additional congruence you can determine from the diagram. Then complete the triangle congruence statement and give the reason for triangle congruence. 4 3 iven: 1 3, 2 4 iven:, by by 14. F iven: is the midpoint of F and iven: 1 3, by by 15

16 ongruent Triangle Problems - Honors I. ΔPQR Δ. Find the values of x and y. 1. m R = 5x + 70, m = 24x 25, QR = 4y + 2, = x + y 2. m R = 90 y, m = 13, PR = 3x + y 1, = 32 x 3. PQ = 5x 31, QR = 3y 1, = x + 1, = 9 y 4. m = 15y 3, m P = 43 x, PQ = 11 x, = 3y = 2x + y, PQ = 7, = 11, QR = 4x + y 6. ΔXYZ ΔMNO, m X = x + 10, m M = y + 20, m Y = 3x, and m N = x + 3y. Find m X and m Y. II. Indicate which triangles are congruent. e sure to have the correspondence of the letters correct. a. ΔR b. is the midpoint of TP c. ΔOW Why is R R? ΔSP Why is 1 2? R S O T R T T P 1 W 2 16 Y

17 III. oordinate eometry 1. raph each line on a coordinate plane. Identify two congruent triangle formed by the lines. xplain why the triangles are congruent. x=0, y = 0, x = 4, y = 2x 4 2. onsider two triangles, Δ and ΔF, with vertices = (0, 7), = (-4, 0), = (0, 0), = (2, 3), = (2, -1), and F = (9, -1). raw a diagram and explain why Δ ΔF. IV. Solve. 1. The perimeter of is 85. Find the value of x. Is Δ congruent to Δ? xplain. 4x 6x x + 4 5x iven: ΔNW ΔR N = 11 R = 2x 4y NW = x + y = 4x + y W = 10 raw the triangles, solve for x and y, and find R. 17

18 Introduction to Triangle Proof x 1) iven: Prove: Δ Δ iven iven Reflexive Prop F x 2) iven: < <H is the midpoint of H Prove: ΔF ΔIH H is the midpoint of H I <F <IH x 3) iven: // JK ML JK ML Prove: J L JK // ML JK ML Reflexive Prop J K M L Δ Δ 18

19 x 4) iven: Prove: Δ Δ Reflexive Prop ef of right Use separate paper to complete the following. x 5) iven: // Prove: I x 6) iven: I bisects < FIJ IF IH Prove: F H F H x 7) iven: KL // JM K L // KJ LM Prove: KJ LM J M 19

20 Triangle Proofs Practice omplete the following proofs using separate paper. raw and mark each picture before writing the proof. 1. iven: Prove: 2. iven: is the midpoint of FH F LH L Prove: L I F H 3. iven: bisects Prove: 4. iven: Prove: T 5. iven: PR QS P S T V Prove: TR QV P Q R S V 20

21 6. iven: bisects Prove: 7. iven: is the midpoint of FI F I Prove: F IH F H I 8. iven: F //HI is the midpoint of H Prove: F H F H I 9. iven: JM //LK J L Prove: JK ML J K M L 10. iven: JM //LK JK //LM Prove: J L M J L K 21

22 MOR PRTI WITH PROOF X 1) IVN: MQ PR, <M N <P R RIHT NLS. N IS TH MIPOINT OF MP PROV: <MQN <PRN Q R M N P X 2) IVN: < < PROV: X 3) IVN: <I < <1 <2 JI K PROV: <1 < 2 F 1 K I J 2 H 22

23 X 4) IVN: // < <F PROV: F F 2 1 X 5) IVN: PI I RI I PROV: R P I R X 6) IVN: KM JL M IS TH MIPOINT OF JL PROV: JKM LKM 23

24 Unit 4 ay 4: More Practice with Proof 1. iven: Prove: F 2. iven: K HL L HK Prove: K L H K L 3. iven: Prove: 4. iven: <F and <H are right angles is the midpoint of FH L Prove: L I F H 5. iven: 1 2 F F Prove: F 1 2 F 24

25 6. iven: F Prove: F F 7. iven: // F Prove: F 1 F 2 8. iven: L M L M L and M are right angles Prove: L M 1 L M 2 F 9. iven: FI bisects H H Prove: F HI H I 10. iven: FI and H bisect each other Prove: H F H I 25

26 Isosceles Triangle investigation 1. In the box, draw an angle and label the vertex. This will be your vertex angle. Measure. 2. Using point as center, swing an arc that intersects both sides of 3. Label the points of intersection and. onstruct side. You have constructed isosceles Δ with base. 4. Measure sides and. What is the relationship between and? 5. Use your protractor to measure the base angles ( and ) of isosceles Δ. 6. ompare your results with the rest of the class. What relationship do you notice about the base angles of each isosceles triangle? Isosceles Triangle Theorem: If 2 sides of a triangle are congruent, then. Isosceles Triangle Theorem onverse: If 2 angles of a triangle are congruent, then. 1. If M, M then by. O R 4 2. If 2 3, then by. M 26

27 Isosceles Triangle Practice 1. In triangle, and m = 124. Find the measure of. 2. In isosceles triangle, =. m = 6x + 10 and m = 3x Find the measure of the exterior angle at the vertex angle. 3. Find the value of x: a. b. c. 27

28 Practice with Isosceles Triangle Theorem and onverse Proofs 1. iven: YX XZ Prove: 3 5 X Y 3 4 Z 5 V 2. iven: KV VZ KO LZ Prove: ΔKVO ΔZVO K O L Z P 3. iven: 1 2 P P Prove: ΔP ΔP

29 Find the value of the variable or question mark x x x x x = x = x = x = x - 8 2x x - 6 x F F H x = x = x = x = omplete the following using the diagram to the right. 9. a. If, then. b. If, then. c. If Δ is an isosceles right triangle with right angle, then the measure of is. 10. iven: PT TR P R I is the midpoint of PR Prove: I I T P I R 11. iven: M is the midpoint of JK 1 2 Prove: J MK 1 J 2 M K 29

30 Use your knowledge of congruent triangle proofs to complete the following flow proofs. 1. iven: is the bisector of Prove: = is the bisector of is the midpoint of Reflexive prop of = Theorem: If a point is on the perpendicular bisect of a segment, it is from the of the segment. 30

31 iven: is on the bisector of, F 3 Prove: = F F F is on the bisector of reflexive prop of Theorem: If a point is on the bisector of an angle, it is from the of the angle. 31

32 Let s Make a Proof! You have spent lots of time in this unit writing proofs that someone else designed. Now it is your turn to make your own! Use the guidelines below to make up and do at least 2 proofs. ach of your 2 proofs should consist of: One of the given diagrams The appropriate given and prove statements to set up your proof The correctly completed proof using the diagram and given and prove statement you wrote t least 3 vocabulary words from the given list (within the completed proof) t least 3 of the rules from the postulates, property, and theorems list (within the completed proof) The second proof you write must use a different diagram, different vocabulary words, and different rules from the first one to meet the minimum requirements. iagram hoices: H P L S M N Q P O I F K V 7. R U T J W P Q Vocabulary Terms: o ongruent segments o ongruent angles o Midpoint o Segment bisector o ngle bisector o Perpendicular lines o Perpendicular bisector o Right angle o Right triangle Rules (Postulates, Properties, Theorems): o Vertical angles are congruent. o Reflexive property of congruence o ll right angles are congruent. o If lines are cut by a transv., corr. s are. o If lines are cut by a transv., alt. int. s are. o If lines are cut by a transv., alt. ext. s are. o SSS o SS o S o S o HL o PT o Isosceles Triangle Theorem o Isosceles Triangle Theorem onverse 32

33 Test Review 1. iven U, find x and U if = 6x+16, U=10x-2 and = 6x+6. x= U= 2. In,, = 6x-5, = 3x+13, and = 4x+7. Find x and the length of the base. x base = 3. In an isosceles triangle, a vertex angle measures 36⁰. What is the measure of each base angle? 4. ecide whether it is possible to prove the triangles are congruent. If yes, then state which congruence postulate you would use. a. b. c. d. 5. If PQR and =2x+2, R=3x-18, find the value of x. 6. XYZ JKL. If Y=14-x, K=2x+50 and L=-4x find the m Z. 7. If LMN, and =4x-y, LM =2x-2y, = x-3y and MN =21 find the value of x and y. 33

34 8. Suppose F. For each of the following, name the corresponding part. : : : In the given triangles, XYZ Which two statements identify corresponding congruent parts for these triangles?. XY and Y. YZ and X. XY and Y. YZ and X 11. OY. Y = 4x+8, = 60, = 80, N = 40. Find x. 12. X STP. = 3x+12, T = 81, and X = 12x 15. Find m. 13. MNO F. alculate the value for x, y, and z. m M = 50 m N = 60 m O = 70 m = (2x 20) m = ( 1 y + 10) z 2 m F = (10 + z) x = y = = 14. The figure shows.. is the midpoint of and. What reason would you use to prove the triangles are congruent, if any? If the triangles are congruent, complete the statement: 34

35 15. If JKM RST, how do you know JK RS?. efinition of a line segment. SSS Postulate. PT. SS Postulate Proving Triangles are ongruent (SSS, SS, S, S, HL) For each of the following, give the reason for triangle congruence P P P Q R Q R Q R P P Q R Q R 21& 22. Solve for x x x + 1 x + 5 2x