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Chapter 4: Congruent Triangles (page 116) Congruent Figures 4-1: (page 117) CONGRUENT: figures having the same and . A X example: B C Y Z Two triangles are congruent if and only if their vertices can be matched up so that corresponding parts (angles & sides) of the triangles are . Corresponding Vertices: Corresponding Angles Therefore, ∆ ABC ≅ ∆ XYZ Corresponding Sides … also, , , , , Since congruent triangles have the same shape, their corresponding angles are ANGLES determine the . Since congruent triangles have the same size, their corresponding sides are SIDES determine the . . Corresponding Parts of Congruent Triangles are Congruent . examples: Write congruence statements based on the given information. (1) ∆ DEF ≅ ∆ TSR (2) (3) ABCDE ≅ VWXYZ (4) D O S G O R C C E A K T SOCKER is a regular polygon Assignment: Written Exercises, pages 120 & 121: 1-19 odd #’s, 20-23 4-2: Some Ways to Prove Triangles Congruent is opposite ∠A and ∠B is opposite . A ∠A is the included angle between & . AB is the included side between & . B Class Activity (page41) C Draw, as accurately as possible, a triangle based on the following description. (1) ∆ ABC, AB = 3 cm, BC = 5 cm, and AC = 6 cm. (2) ∆ DEF, DE = 3 cm, m∠E = 60º, and EF = 4 cm. (3) ∆ XYZ, m∠X = 30º, XY = 4 cm, and m∠Y = 50º. (4) ∆ UVW, m∠U = 30º, m∠V = 50º, and m∠W = 100º. NOTES: (1) If two triangles are congruent, then you know also congruent. (2) Based on prior exercises, that two triangles are congruent. Postulate 12 pairs of corresponding parts are pairs of congruent corresponding parts will guarantee SSS Postulate If three sides on one triangle are congruent to three sides of another triangle, then the triangles are Postulate 13 . SAS Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are Postulate 14 . ASA Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are . Complete the following proofs. H (1) I Given: HI || GJ ; HG || IJ 1 3 Prove: ∆ GHJ ≅ ∆ IJH 4 2 Proof: G J Statements Reasons 1. 1. 2. ∠ 1 ≅ ∠ 2 ∠3≅ ∠4 2. 3. 3. Reflexive Property 4. 4. O (2) Given: OK bisects ∠ MOT ; OM ≅ OT 1 2 Prove: ∆ MOK ≅ ∆ TOK M Proof: T K Reasons Statements 1. 1. 2. ∠ 1 ≅ ∠ 2 2. 3. 3. Reflexive Property 4. 4. Assignment: Written Exercises, pages 124 to 127: 1-21 odd #’s, 22, 24, 26 Prepare for Quiz on Lessons 4-1 & 4-2 Using Congruent Triangles 4-3: (page 127) CPCTC means . Ways to Prove Triangles Congruent: A Way to Prove Two Segments or Two Angles Congruent: (1) Identify 2 triangles in which the 2 segments or angles are corresponding parts. (2) Prove that the 2 triangles are congruent. (3) State that the 2 parts are congruent, using the reason ____ ____ ____ ____ ____ . NOTE: It will be helpful to plan these proofs by reasoning backward. K (1) R Given: ∠1 ≅ ∠R ∠2 ≅ ∠N 1 M MR ≅ MN 2 Prove: KR ≅ PN N P Proof: Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. (2) Given: AB ≅ CB ; ∠1 ≅ ∠ 2 A Prove: BD bisects AC B 1 2 D Proof: C Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. T (3) Given: RQ ≅ QS ; RT ≅ TS Prove: TQ ⊥ RS Proof: R Statements Q Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. Assignment: Written Exercises, pages 130 to 132: 1, 2, 5, 7, 9, 10 11, 14, 16 Prepare for Quiz on Lessons 4-1 to 4-3 S The Isosceles Triangle Theorems 4-4: (page 134) ISOSCELES TRIANGLE: a triangle with at least sides congruent. LEGS (of an isosceles triangle): the sides. BASE (of an isosceles triangle): the side. A vertex angle: legs: & base angles: & base: B THEOREM 4-1 C THE ISOSCELES TRIANGLE THEOREM If two sides of a triangle are congruent, then the angles those sides are congruent. (ie. Base angles of an isosceles triangle are congruent.) A Given: AB ≅ AC Prove: ∠B ≅ ∠C Proof: B C Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. Corollary 1 An equilateral triangle is also . Corollary 2 An equilateral triangle has three Corollary 3 The bisector of the vertex angle of an isosceles triangle is angles. to the base at its midpoint. example: THEOREM 4-2 If two angles of a triangle are congruent, then the sides those angles are congruent. A Given: ∠B ≅ ∠C Prove: AB ≅ AC Proof: B C Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. Corollary An equiangular triangle is also . examples: Find the value of “x”. (1) 40º 40º 3x-5 x+21 x= (2) xº 70º x= (3) 62º xº x= Assignment: Written Exercises, pages 137 to 139: 1-9 odd #’s, 14, 16, 19, 22, 24, 27, 33 Other Methods of Proving Triangles Congruent 4-5: THEOREM 4-3 (page 140) AAS THEOREM If two angles and a side of one triangle are congruent to the parts of another triangle, then the triangles are congruent. A Given: ∆ ABC & ∆ DEF ∠B ≅ ∠E C ∠C ≅ ∠F B D AC ≅ DF Prove: ∆ ABC ≅ ∆ DEF F E Proof: Statements Reasons 1. 1. 2. 2. 3. 3. NOTE: Two-column proofs may be shortened by writing them in form which emphasize the steps in the proof. RIGHT TRIANGLE: a triangle with one angle. HYPOTENUSE: in a right triangle, the side opposite the LEGS (of a right triangle): the other two angle. . A hypotenuse: legs: & C THEOREM 4-4 B HL THEOREM If the hypotenuse and a leg of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. Given: ∆ ABC & ∆ DEF B E ∠C & ∠F are right angles AB ! DE BC ! EF Prove: ∆ ABC ≅ ∆ DEF A Proof: “see paragraph form with key steps” C D F Ways to Prove ANY Two Triangles Congruent: Postulate Postulate Postulate Theorem Ways to Prove Two RIGHT Triangles Congruent (Look at the Classroom Exercises on page 143, #14.): Theorem Method Method Method examples: State which congruence method(s) can be used to prove the triangles congruent. If no method applies, write none. (1) (2) ________________ ________________ (3) (4) Y W X ________________ Z ________________ (5) (6) A P C D E ∠A ≅ ∠C AE = DC B Q ________________ R S ________________ Assignment: Written Exercises, pages 143 to 145: 1, 2, 3, 5, 6, 7, 10 to 18 even #’s Prepare for Quiz on Lessons 4-4 & 4-5 T Using More than One Pair of Congruent Triangles 4-6: Sometimes it is impossible to prove a pair of triangles congruent directly. You may first need to prove another pair of triangles ______________. Use the congruent corresponding parts to prove the original triangles ______________. There is often more than one correct ________________. This depends upon which triangles you choose to prove ________________ first. NOTE: Plan the proof by reasoning backward. Use a paragraph proof that focuses on the key ideas or a key step proof. Given: X is the midpoint of BD AC ⊥ B BD A C X Prove: ∠ABC ≅ ∠ADC D Proof: Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8. 9. 9. Assignment: Written Exercises, pages 148 to 150: 1, 3-10 all #’s, *13* (page 146) 4-7: Medians, Altitudes, and Perpendicular Bisectors MEDIAN (of a triangle): a segment from a vertex to the B A of the opposite side. B C A Y B C Y X (page 152) A C Y Z X Z X Z Name given to point of intersection for the 3 medians: ALTITUDE (of a triangle): the perpendicular segment from a containing the opposite side. B A B C Y A B C Y X to the line Z A C Y X Name given to point of intersection for the 3 altitudes: Z X Z PERPENDICULAR BISECTOR (of a segment): a line, ray, or segment that is perpendicular to the segment at its . B A B C Y A B C A Y X Z C Y X Z X Z Name given to point of intersection for the 3 perpendicular bisectors: BISECTOR of an ANGLE: the ray that divides the angle into two adjacent angles. B A B C Y A B C A Y X Z C Y X Z Name given to point of intersection for the 3 angle bisectors: X Z THEOREM 4-5 If a point lies on the perpendicular bisector of a segment, then the point is from the endpoints of the segment. Given: Line l is the perpendicular bisector of BC . A is on l A Prove: AB = AC B X C l Proof: To prove this theorem, the following triangles must be proven congruent … ∆ ___ ___ ___ ≅ ∆ ___ ___ ___ , by THEOREM 4-6 . If a point is equidistant from the endpoints of a segment, then the point lies on the bisector of the segment. Given: AB = AC Prove: A is on the perpendicular bisector of BC . A B C Proof: To prove this theorem, the following triangles must be proven congruent. ∆ ___ ___ ___ ≅ ∆ ___ ___ ___ , by Theorem 4-6 is the A point is on the ⊥−-bisector of a segment . of Theorem 4-5 and can be combined into a biconditional. it is equidistant from the endpoints of the segment. A DISTANCE from a POINT to a LINEX(or plane): the length of the segment from the point to the line (or plane). Z RP B Y THEOREM 4-7 C t If a point lies on the bisector of an angle, then the point is from the sides of the angle. !!!" Given: BZ bisects !ABC PX ! BA PY ! BC Prove: PX = PY Proof: Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. A THEOREM 4-8 If a point is equidistant from X the sides of an angle, then the point lies on the Given: PX ! BA B PY ! BC of the angle. P Y C PX = PY !!!" Prove: BP bisects !ABC Key Step Proof: Statements Reasons 1. 1. 2. 2. 3. 3. Theorem 4-8 is the of Theorem 4-7 and can be combined into a biconditional. A point is on the bisector of an angle it is equidistant from the sides of the angle. Assignment: Written Exercises, pages 156 & 157: 1-13,19,23 Prepare for Quiz on Lessons 4-6 & 4-7 Prepare for Test on Chapter 4 What’s up with Side-Side-Angle (SSA)? Class Activity You will need a compass, protractor, and pencil. Exercise #1 Draw ∆ ABC so that AB = 6 cm, BC = 8 cm, and m∠C = 40º. Exercise #2 Draw ∆ ABC so that AB = 6 cm, BC = 8 cm, and m∠C = 140º. Exercise #3 Draw ∆ ABC so that AB = 8 cm, BC = 6 cm, and m∠C = 140º. Exercise #4 Draw ∆ ABC so that AB = 8 cm, BC = 6 cm, and m∠C = 40º. SUMMARY: Based on the results from the previous exercises and all the information you recieved from Chapter 4, what can you conclude about SSA? SSA only works if … How many triangles do you see? Directions: Count the number of triangles that you see. Then ask someone else how many triangles they see and take a second count. My 1st count is . My 2nd count is . Actual # of ∆’s is . Honors Geometry Problem Name Worth 10 points plus BONUS Lesson 4-6: Using More than One Pair of Congruent Triangles Page 150: #13 M Q L Given: KL & MN bisect each other @ O. Prove: O is the midpoint of PQ . O K P Proof: Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8. 9. 9. 10. 10. N