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Name: Unit 2 ~ Congruent Triangles Assignments Advanced Geometry Ch 3 Green Book *All problems refer to the WRITTEN EXERCISES* (Green Textbook) # Class Date Class Activity Homework 1 Day 1 Section 3.4 Angle Relationships • Complete classwork • Read section 3.4 • Become familiar with the following terms about angles: acute, obtuse, right, adjacent, complementary, supplementary, and vertical. • Do p. 106 #5, 7, 13; p. 110 #12, 15, 20; • Do p. 118 #17, 21, 25 (Two column proof not required. Just explain it in your own words.) 2 Day 2 Introduction to Proofs Fill in the Missing Statements & Fill-in Proofs • Complete classwork • Study first group of reasons (from “given” to “vertical angles theorem”). 3 Day 3 Section 3.5 Congruence • Complete classwork • Read section 3.5 • Do p. 122 #1, 3, 7, 12, 15, 17, 21–23 4 Day 4 Section 3.6 More Ways of Proving Congruence • Complete classwork • Read section 3.6 • Do p. 127 #1, 3, 6, 7, 15, 19, 21, 22, 23, 25, 28 5 Day 5 Section 3.7 CPCTC • Complete classwork • Read section 3.7 • Do p. 130 #3, 5, 7, 12, 15, 19, 25, 27, 31 6 Day 6 Section 3.8 Isosceles Triangles • Complete classwork • Read section 3.8 • Do p. 135 #1, 9, 15, 19, 23, 25, 27, 31, 33 7 Day 7 Section 3.9 Geometric Inequalities • Complete classwork (up to “D-Stix” Lab) • Read section 3.9 • Do p. 140 #1, 3, 11, 13, 15, 17–19, 21, 26–28 8 Day 8 Section 3.9 continued • Complete classwork • Do p. 140 #5, 9, 12, 16, 23-25 9 Day 9 Review for Opportunity • Unit 2 Review Sheet • Study for Opportunity!! 10 Day 10 Unit 2 (Chapter 3) Opportunity • Algebra Review worksheet Due Date Extra Help Times: Math Center 7:30-8:15am M-F, Wednesdays 2:50-3:30pm, X Block, and by appointment Use Canvas or email to ask questions as well! Definitions, Theorems, Postulates, Corollaries, and Properties used in Geometry Proofs Add to this list as needed – Must be a reason in the text or proven/given in class. • Given Properties • Reflexive property • Transitive property • Addition property of equality (APE) • Subtraction property of equality (SPE) • Substitution property (Subst.) • Property of Inequalities Segments & Angles: • Definition of perpendicular bisector • Definition of segment bisector • Definition of angle bisector • Definition of midpoint • Definition of right triangle • Definition of perpendicular lines • Angle Addition Postulate ( ∠ + Post.) • Segment Addition Postulate (Seg. + Post.) • Definition of Supplementary • Definition of Complementary • Supplements to Congruent Angles are Congruent (SCAC) • Complements to Congruent Angles are Congruent (CCAC) • Vertical Angles Theorem (VAT) Triangles: • SSS • SAS • ASA • AAS • HL • All right angles are ≅ • All radii of a circle are ≅ • Definition of Congruent Triangles/Corresponding Parts of ≅ Δ ’s are ≅ (CPCTC) • Definition of an Isosceles Triangle • Isosceles Triangle Theorem (ITT) • Converse of Isosceles Triangle Theorem (CITT) • Remote Interior Angles Theorem (RIAT) • Largest Side Opposite Largest Angle (LSOLA) • Exterior Angle Inequality Theorem • Triangle Inequality Theorem ( Δ ≠ Thm) • Triangle Sum Theorem • 3rd Angles Theorem • Definition of Median 1 Unit 2 Intro (Day 1) Green Book: Euclidean Geometry… read together p. 99, paragraph 2-3 Geometry literally means Postulate: Example: Through any two points, there exists exactly one line. Theorem: Example: If two lines intersect, then they intersect in exactly one point. A • Notation B A B A A • 5 40o C B B Bisectors CD bisects AB at P means: C 1. 2. P A 3. B D NOTE: BD bisects ∠ABC means: A D 1. 2. B C NOTE: 2 Day 1 Practice: 1. Two angles are complementary if the sum of their measures is 90°. a.) If ∠A and ∠B are complementary and congruent, find the measure of each angle. b.) ∠1 and ∠2 are complementary. If m∠1 = 3x – 23 and m∠2 = 4x + 1, find x. 2. Two angles are supplementary if the sum of their measures is 180°. If ∠A and ∠B are supplementary and the measure of ∠A is 12 less than twice the measure of ∠B, find the measure of each angle. 3. If ∠A and ∠B are each complementary to ∠C, what can you conclude? Draw a diagram of the situation and explain. 4. If ∠A and ∠B are each supplementary to ∠C, what can you conclude? Draw a diagram of the situation and explain. 3 5. When two lines intersect, as shown below, they form four separate angles, which are labeled here as ∠1, ∠2, ∠3, and ∠4. a.) Suppose m∠1 = 20°. Find the other three angles. Pairs of “opposite” angles such as ∠1 and ∠3 are called vertical angles. b.) Explain why vertical angles are always congruent. 6. In the diagram below, OX bistects ∠AOB. Also m∠XOY = 90°. a.) If m∠1 = 20°, find m∠2, m∠3, and m∠4. b.) If m∠1 = 65°, find m∠2, m∠3, and m∠4. b.) Prove that OY bisects ∠BOC. (You do not need to do a 2-column proof, but give an argument for why this is true no matter what angle measure is given.) 4 End Intro to Proofs (Day 2) Part I: Read each statement and draw a picture that fits the situation being described. Then write the statement and reason that follows directly from the one given. The first problem below is done as an example for you. Statements Reasons Picture ex 1) ∠ 1 and ∠ 2 are supplementary 1) Given 2) m ∠ 1 + m ∠ 2 = 180º (a) 1) F is the midpoint of AB 2) ____________________ 2) Def. of supplementary 1 2 1) Given 2) ____________________ (b) 1) ∠ 1 and ∠ 2 are complementary 1) Given 2) ____________________ (c) 1) OC bisects ∠DOB 2) ____________________ (d) 1) EF bisects AB at point D. 2) ____________________ 2) ____________________ 1) Given 2) ____________________ 1) Given 2) ____________________ (e) 1) Point G is on EF (not necessarily the midpoint) 2) ____________________ (f) 1) m ∠ 1 + m ∠ 2 = m ∠ ABC; m∠1 = m ∠3 2) ____________________ (g) 1) ∠ 4 and ∠ 5 are vertical angles 1) Given 2) ____________________ 1) Given NO PICTURE NEEDED 2) ____________________ 1) Given 2) ____________________ 2) ____________________ (h) 1) AB ⊥ BD 2) ____________________ 1) Given 2) ____________________ 5 Part II: Now let’s reverse the process. The last statement of a proof is given. Again, draw a picture that fits each situation. Write a statement that directly precedes the last statement and write the reason for the last statement. Statements Reasons Picture (a) 4) ____________________ 5) ∠2 and ∠3 are supplementary (b) 8) ____________________ 9) OC bisects < JOK 5) ____________________ 9) ____________________ (c) 6) ____________________ 7) AB ⊥ BD 7) ____________________ (d) 5) ____________________ 6) T is the midpoint of JR 6) ____________________ (e) 7) ____________________ 8) AB = 12 AC 8) ____________________ A B 6 C Fill-in-the-Blanks Proofs 1) Given: DE = AF EF bisects AB Prove: DE = 1 AB 2 *Hint: Mark your diagram! Statements Reasons 1) EF bisects AB 1) ____________________ 2) F is the midpoint of AB 1 3) AF = AB 2 2) ____________________ 4) DE = AF 4) ____________________ 5) ____________________ 5) ____________________ 3) ____________________ 2) Given: ∠1 = ∠3 Prove: ∠PXR = ∠QXS Statements Reasons 1) ∠1 = ∠3 1) 2) ∠ 2 = ∠ 2 2) 3) ∠1 + ∠2 = ∠3 + ∠2 3) ____________________ 4) ∠PXR = ∠1 + ∠2 ∠QXS = ∠3 + ∠2 4) ____________________ 5) ____________________ 5) ____________________ 7 3) Given: ∠2 and ∠3 are complementary A Prove: AB ⊥ CD 3 C E 1 D 2 B Statements Reasons 1) ∠2 and ∠3 are complementary 1) ____________________ 2) m∠2 + m∠3 = 90 o 2) ____________________ 3) m∠2 = m∠1 3) ____________________ 4) m∠1 + m∠3 = 90 o 4) ____________________ 5) m∠1 + m∠3 = m∠AED 5) ____________________ 6) ____________________ 6) Substitution property ( 7) ____________________ 7) ____________________ ) 4) Given: m∠BOA = m∠EOD OC bisects ∠DOB Prove: OC bisects ∠AOE Statements Reasons 1) m∠BOA = m∠EOD OC bisects ∠DOB 1) ____________________ 2) m∠BOC = m∠COD 2) ____________________ 3) m∠BOC + m∠BOA = m∠COD + m∠EOD 3) ____________________ 4) m∠BOC + m∠BOA = m∠COA m∠COD + m∠EOD = m∠COE 4) ____________________ 5) ____________________ 5) Substitution property( 6) ____________________ 6) ____________________ ) 8 End 3-5 Congruent Triangles (Day 3) E Congruent Triangles: B D A F C What is the least amount of information we can know about a pair of triangles to determine that they are congruent? 1. Using graph paper and a ruler, draw two triangles using line segments of 2”, 3”, and 4” in each triangle. What conclusion can you make about these two triangles? 2. Choose any 3 lengths and draw 2 more triangles with the same 3 lengths for the sides. Does your conclusion above remain true for these two triangles? 3. On a blank page, use your ruler to draw a 2” segment that lines up with one side of the angle drawn below. 4. Use your ruler to draw a 4” segment that lines up with the other side of the same angle. 5. Connect the two lines that you drew to form a triangle. 6. Repeat this process with the second angle below. What can you conclude about the third connector side of the two triangles? What can you conclude about the two triangles? Are there other combinations that create congruent triangles? Investigate with a partner until you find 2 more combinations that will give you congruent triangles. 9 Congruence Theorems: WCombinations that do NOT work (do not prove Δs ≅ ): X Y Z Ex: A Q B Given: PQ ⊥ AB , ∠A ≅ ∠B , Q is the midpoint of AB . Prove: ∆APQ ≅ ∆BPQ (2-column proof) 1. Given: XW ≅ XZ WY ≅ ZY Prove: ∆XWY ≅ ∆XZY 10 W 21 V X Y it necessarily follow 2. (a) Consider the diagram below. If XY bisects ∠ WXZ, does that ∠ 1 ≅ ∠ 2? Explain. Z End (b) Given: ∠ WYV ≅ ∠ ZYV XY bisects ∠ WXZ Prove: ∆XWY ≅ ∆XZY 3. Draw and label a diagram corresponding to each situation below. Identify what is given and what is to be proved. Then state the theorem you would use to prove it. (a) If a line perpendicular to AB passes through the midpoint of AB, and segments are drawn from any other point on that line to A and B, then two congruent triangles are formed. Diagram? Given? Prove? Theorem to prove it? (b) If pentagon ABCDE has all equal sides and has right angles at B and E, then diagonals AC and AD form congruent triangles. Diagram? Given? Prove? Theorem to prove it? 11 HL postulate: 3-6 More Triangle Congruence (Day 4) We saw before that SSA is not a viable postulate. Why not? Does SSA ever work? If so, when? Ex. 1 Given: ∆XWZ and ∆XYZ are right ∆s; WX ≅ YX . Prove: ∆XWZ ≅ ∆XYZ Ex. 2 Given: XY ⊥ AB; XA ≅ XB . Prove: ∆AYX ≅ ∆BYX 12 Proofs Practice: 1. Given: m ∠ B = m ∠ C = 90 , ∠ BAD ≅ ∠ CDA. Prove: ∆ABD ≅ ∆DCA A B 1 1 A 1 2C D 2 D 2 hint: redraw to separate ∆’s 2. Given: m ∠ B = m ∠ C = 90 , BA ≅ CD . Prove: ∆ABD ≅ ∆DCA 3. Given: ∠ 1 ≅ ∠ 2, ∠ BAD ≅ ∠ CDA. Prove: ∆ABD ≅ ∆DCA 13 4. (a) p. 127 #20 (b) p. 128 #27 Given: ∠1 ≅ ∠4; ∠2 ≅ ∠3. Prove: ΔABD ≅ ΔBAC Given: JK ⊥ KM ; LM ⊥ KM ; ∠4 ≅ ∠1 Prove: ΔJKM ≅ ΔLMK Note: You need to use both the addition property and the angle addition postulate. Note: You need to use the angle addition postulate and the subtraction property. End 5. Draw and label a diagram corresponding to each situation below. Identify what is given and what is to be proved. Then state the theorem you would use to prove it. (a) In an isosceles triangle (i.e., two of its sides are congruent), if the angle between the congruent sides is bisected, then two congruent triangles are formed. Diagram? Given? Prove? Theorem to prove it? (b) In an isosceles triangle, if a segment is drawn from the angle between the congruent sides to the midpoint of the opposite side, then congruent triangles are formed. Diagram? Given? Prove? Theorem to prove it? 14 Def. of ≅ Δ ’s (CPCTC) Vertical A Theorem: A Angles B B C D D 3-7 CPCTC (Day 5) C E E Proof: If you knew that ΔABC ≅ ΔEDC, what could you conclude about AB and DE? Ex. 1. Given: AE & BD bisect each other. Prove: AB ≅ DE 15 GR R 2 P 1 3 X X 4 1. Given: GH ⊥ JK JG ≅ KG Prove:J ∠J ≅ ∠K 2. Given: ∆PRX ≅ ∆PSX Prove: ∆PRQ ≅ ∆PSQ H SS Q Q K 3. Given: ∆QRX ≅ ∆QSX Prove: ∆PRX ≅ ∆PSX Hint: ∠3 and ∠RXQ are supplementary. ∠4 and ∠SXQ are supplementary. 16 1 2 M 4. Given: ∆DEF ≅ ∆DCF; ∠2 ≅ ∠3 . E M 5. Given: MR ≅ PR ; MX ≅ PX R P RX X A B Prove: ∆EFA ≅ ∆CFB S1 2 3 4 3 4 5D6 3 4 End C F P S Prove: ∠3 ≅ ∠4 E 6. Given: RS ⊥ MP ; MR ≅ PR . Prove: MX ≅ PX 17 B If AE = DE, B which angles are congruent? E 3-8 Isosceles Triangles (Day 6) 1 2 C A 1. Draw BD that bisects ∠ ABC. Is this always possible? A A B C C D 2. Draw BD ’ that bisects AC . (This is the median of ∆ABC) Is this always possible? 3. Can you make a triangle such that one segment bisects both an angle and the opposite side at the same time? Prove the Isosceles Triangle Theorem: 4. Given: AB ≅ CB Prove: 5. Given: AE ≅ DE ; ∠1 ≅ ∠2 . (Hint: need auxiliary line BD) Prove: ∆AEB ≅ ∆DEC. 18 L Q 6. Given: ML ≅ NL ; ∠PNM ≅ ∠QMN P Prove: Try separatingInstead of a median, the draw . . . ≅ ∆MPN ∆NQM triangles. N M 7. The converse of the Isosceles ∆ Theorem is: If , then . Consider the following attempt at its proof. Given ∆ABC with ∠ A ≅ ∠ C. Draw median BD from vertex B to side AC. Thus, AD ≅ CD; and since BD ≅ BD, we have ∆ABD ≅ ∆CBD. So, AB ≅ BC by CPCTC. (a) Explain what is wrong with this proof. (b) Now write a correct proof. 19 8. (a) Do p. 136 #17 (b) Do p. 137 #26 Hint: Use what you just proved. Given: ∠1 ≅ ∠2; ∠3 ≅ ∠4; ∠5 ≅ ∠6. Prove: LK bisects MN (c) Do p. 137 #32 Hint: You need to prove 2 pairs of Given: CX ≅ DX ; AD ≅ BC. Prove: AC ≅ BD ≅ Δ. RS ⊥ PQ; End. ∠PRS ≅ ∠PSR Prove: RQ ≅ SQ Given: 9. Draw any triangle, neatly, using a ruler. Carefully measure your side lengths in mm. Draw the three medians of your triangle. Try this with a couple of different triangles. Explain what you observe. A What if you draw the three angle bisectors instead? 20 B Corollary to the Triangle Sum Theorem 90o Triangle Sum Theorem The 3-9 The sum of the measures of the A 40o triangle isTriangle Sum °. Theorem angles of a right triangle are 7) Geometric Inequalities (Day angles of a Xo C . D Place triangle here. Corollary to the Triangle Sum Theorem Definition of Corollary: Remote Interior Angles Theorem Definition of Exterior Angle: Find the value of x. _______ What can you conclude? 21 The measure of an angle of a triangle is equal to the Inequality Theorem of the measures of the two terior Angles (EAIT) angles. Let’s show this with paper… Place triangle on this line. Now let’s prove it! Consider ∆ABC shown below. Prove that m∠4 = m∠1 + m∠2 . A corollary… 22 arger Side Opposite Larger Angle (LSOLA): P S 1 What is your 2nd to last step? Q 2 End 3 R 3. What’s wrong with the following argument? D-Stix” Lab Here BC > AB. Also ∠ 1 is opposite BC and ∠ 2 is opposite AB. Since the longer side is opposite a larger angle, it follows that m ∠ 1 > m ∠ 2. You have 4 colored sticks. There are 4 possible ways to choose exactly 3 of them. List these choices by color. a.) b.) c.) d.) The lengths of the sticks are: red 12 in. green 7 in. blue 5 in. yellow 4 in. Put these numbers in the lists above. 4. Given m ∠ Q > m ∠ 2 Prove: PR > RQ continued… 3-9 Geometric Inequalities continued (Day 8) 23 each of the Inequality 4 choices attempt to put sticks end to end to form a triangle. In some cases this Triangle Theorem ( Δthe ≠ Thm): S Q ot be done. State yes or no for each choice. P 1 2 b.) c.) d.) Write a sentence or two describing when it can and cannot be done. 3 R o write a theorem regarding a requirement on three numbers that form the sides of a triangle. Then…) nsion: State a set of four numbers that could not be the sides of a quadrilateral. 1. If ∆ABC has AB = 3cm and BC = 7cm, what are the possible lengths of AC? Explain (remember, side lengths do not have to be integers). Proofs Practice: 2. Given: PR > RQ; RS > PS. Prove: m ∠ Q > m ∠ 3 24 Z Q E 3. Given: XZ = XY; ZW = ZY. Prove: ZY > WY 2 X K P Y W 3 1 A End C R 4. Given: QR > QP PR ≅ PQ Prove: m∠P > m∠Q 5. Given: AC ≅ AE AE ≅ KE Prove: m∠1 > m∠2 25