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Geometry 2205 Unit 3: Mrs. Bondi Unit 3: Two-Dimensional Shapes Lesson Topics: Part A: Lesson 1: Congruent Figures (PH text 4.1) Lesson 2: Triangle Congruence by SSS and SAS (PH text 4.2) Lesson 3: Triangle Congruence by ASA and AAS (PH text 4.3) Lesson 4: Using Corresponding Parts of Congruent Triangles (PH text 4.4) Lesson 5: Isosceles and Equilateral Triangles (PH text 4.5) Lesson 6: Congruence in Right Triangles (PH text 4.6) Lesson 7: Congruence in Overlapping Triangles (PH text 4.7) Lesson 8: Midsegments of Triangles (PH text 5.1) Lesson 9: Perpendicular and Angle Bisectors (PH text 5.2) Lesson 10: Bisectors in Triangles (PH text 5.3) Lesson 11: Medians and Altitudes (PH text 5.4) Lesson 12: Indirect Proof (PH text 5.5) Lesson 13: Inequalities in One Triangle (PH text 5.6) Lesson 14: Inequalities in Two Triangles (PH text 5.7) Part B: Lesson 15: The Polygon-Angle Sum Theorems (PH text 6.1) Lesson 16: Properties of Parallelograms (PH text 6.2) Lesson 17: Proving that a Quadrilateral is a Parallelogram (PH text 6.3) Lesson 18: Properties of Rhombuses, Rectangles, and Squares (PH text 6.4) Lesson 19: Conditions for Rhombuses, Rectangles, and Squares (PH text 6.5) Lesson 20: Trapezoids and Kites (PH text 6.6) Lesson 21: Polygons in the Coordinate Plane (PH text 6.7) Lesson 22: Applying Coordinate Geometry (PH text 6.8) Lesson 23: Proofs Using Coordinate Geometry (PH 6.9) Lesson 24: Proportions in Triangles (PH text 7.5) Lesson 25: Areas of Parallelograms and Triangles (PH text 10.1) Lesson 26: Areas of Trapezoids, Rhombuses and Kites (PH text 10.2) Lesson 27: Areas of Regular Polygons (PH text 10.3) Lesson 28: Perimeters and Areas of Similar Figures (PH text 10.4) Lesson 29: Trigonometry and Area (PH text 10.5) 1 Geometry 2205 Unit 3: Mrs. Bondi Lesson 1: Congruent Figures (PH text 4.1) Objectives: to recognize congruent figures and their corresponding parts Similar Polygons: same shape, but not necessarily the same size 1. 2. Congruent polygons: exactly the same shape and size 1. 2. When you write a congruency statement, you must list corresponding vertices in the same order. Example 1: BIRD CAGE Name congruent sides. Find the measures of as many angles as you can. B I A R D C Example 2: ACBX PRQY X E List all of the congruent corresponding parts of ACBX and PRQY. B A G 60 Y P R C Q Theorem 4-1 Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. E B If A D, B E , then C F . D A 2 C F Geometry 2205 Unit 3: Mrs. Bondi B E Two-Column Proof for Theorem 4-1 Given: A D and B E Prove: C F A D C Statements 1.) A D and B E F Reasons 1.) Given 2.) mA mB mC 180 and mD mE mF 180 2.) 3.) mA mB mC mD mE mF 3.) 4.) 4.) Substitution 5.) mC mF 5.) 6.) 6.) Def. of Congruent Angles Prove Triangles Congruent: Given: AB ED , side measures indicated on diagram Prove: ABC EDC HW: p.222 #10-31, 33, 36-40 even, one proof from # 32, 44, 45 3 Geometry 2205 Unit 3: Mrs. Bondi 4 Geometry 2205 Unit 3: Mrs. Bondi 5 Geometry 2205 Unit 3: Mrs. Bondi Lesson 2: Triangle Congruence by SSS and SAS (PH text 4.2) Objective: Postulate 4-1 to prove two triangles congruent using the SSS and SAS Postulates Side-Side-Side Postulate (SSS Postulate) If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. GHF PQR Postulate 4-2 Side-Angle-Side Postulate (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 two triangles are congruent. CBA DFE Given: CA RD AR DC Prove: CAR RDC Statement C A D R Reason 6 Geometry 2205 Unit 3: Mrs. Bondi Prove Triangles Congruent: Given: side measures indicated on diagram Prove: ABC EDC Statement Reason Review (frequently): Postulates and Theorems (p.902-910) Properties (p.900) HW: p.230 #8, 11-18, 22, prove two from #28-33 7 Geometry 2205 Unit 3: Mrs. Bondi 8 Geometry 2205 Unit 3: Mrs. Bondi 9 Geometry 2205 Unit 3: Mrs. Bondi Lesson 3: Triangle Congruence by ASA and AAS (PH text 4.3) Objective: to prove two triangles congruent by the ASA Postulate and the AAS Theorem. Postulate 4-3 Angle-Side-Angle (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 two triangles are congruent. GBH KPN B P G Theorem 4-2 H K N Angle-Angle-Side Theorem (AAS Theorem) If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the triangles are congruent. C T M G D X Q Given: XQ TR , XR bisects QT X Prove: XMQ RMT T Statement Reason 10 M R DCM GXT Geometry 2205 Unit 3: Given: Prove: Mrs. Bondi N MNP ONP MPN OPN M MNP ONP O P Statement Reason P Given: PQ TS P S Prove: PQR STR S R Q Statement T Reason HW: p.238 #8-13, 16-20 11 Geometry 2205 Unit 3: Mrs. Bondi 12 Geometry 2205 Unit 3: Mrs. Bondi 13 Geometry 2205 Unit 3: Mrs. Bondi Lesson 4: Using Corresponding Parts of Congruent Triangles (PH text 4.4) Objective: to use triangle congruence and CPCTC to prove that parts of two triangles are congruent Accepted Fact based on definition of congruent triangles: CPCTC – Corresponding Parts of Congruent Triangles are Congruent We can use this fact in proving congruence of sides or angles. M Given: M P X is midpoint of MP Prove: MN PO X O Statement N Reason 14 P Geometry 2205 Unit 3: Mrs. Bondi Given: D B and DAC BCA Prove: AB CD Statement A D Reason B C Legs of an ironing board bisect each other. Prove that the board is parallel to the floor. (You make the drawing and the given/prove statements.) Given: Prove: Statement Reason HW: p.246 #5-16 15 Geometry 2205 Unit 3: Mrs. Bondi 16 Geometry 2205 Unit 3: Mrs. Bondi 17 Geometry 2205 Unit 3: Mrs. Bondi Lesson 5: Isosceles and Equilateral Triangles (PH text 4.5) Objective: Use and apply properties of isosceles and equilateral triangles. Isosceles Triangle Legs – Base – Vertex Angle – Base Angles – Theorem 4-3 Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent. C A B Theorem 4-4 Converse of the Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent. C A B Write the Isosceles Triangle Theorem and its converse as a biconditional statement. 18 Geometry 2205 Unit 3: Example 1) Mrs. Bondi Explain all angle and side relationships you can. Justify. A B C D E Theorem 4-5 The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base. C A B D Examples: 2) If mL 52 , find the values of x and y. x = ______ M y = ______ x 52 L y O N 3) AB ____ B FD ____ D E CE ____ A F C 19 Geometry 2205 Unit 3: Mrs. Bondi Corollary to Isosceles Triangle Theorem - If a triangle is equilateral, then it is equiangular. Y X Z Corollary to Converse of Isosceles Triangle Theorem - If a triangle is equiangular, then it is equilateral. Y X Z Write a biconditional statement for these two corollaries. Examples: 4) Find the measures of A, B , and ADC . 5) What is the measure of A ? 6) What is the value of x? HW: p.254 # 6-19, 22, 30-32, 37-40 20 Geometry 2205 Unit 3: Mrs. Bondi 21 Geometry 2205 Unit 3: Mrs. Bondi 22 Geometry 2205 Unit 3: Mrs. Bondi Lesson 6: Congruence in Right Triangles (PH text 4.6) Objective: Theorem 4-6 to prove triangles congruent using the Hypotenuse-Leg Theorem Hypotenuse-Leg Theorem (HL Theorem) If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent. PQR XYZ To use the HL Theorem in proofs you must show that these three conditions are met. There are two right triangles There is one pair of congruent hypotenuses There is one pair of congruent legs Proof Given: KJWZ is a rectangle Prove: WJZ KZJ Statement Reason 23 K Z X J W Geometry 2205 Unit 3: Mrs. Bondi Are the two triangles congruent? If so, write the congruence statement. If not, tell why. Practice using the HL Theorem. Given: BD AC AB CB Prove: ADB CDB Statement B A Reason HW: p.261 #1-5, 8-13 (two system of equations problems) 24 D C Geometry 2205 Unit 3: Mrs. Bondi Proof Practice: Given: CD EA .. Prove: CBD EBA A C E B D Statement Given: LP bisects MLN , PM LM , PN LN Prove: PM PN Reason L N M P Statement Reason 25 Geometry 2205 Unit 3: Mrs. Bondi 26 Geometry 2205 Unit 3: Mrs. Bondi 27 Geometry 2205 Unit 3: Mrs. Bondi Lesson 7: Congruence in Overlapping Triangles (PH text 4.7) Objective: to identify congruent overlapping triangles to prove two triangles congruent using other congruent triangles Sometimes triangles overlap one another and share a side or an angle. We can separate and redraw the triangles in order to better see the corresponding parts. What common side does BAE and DEA share? B D C E A What common angle does BAE and DAF share? B D C F E A Given: Prove: BA DE BE DA BEA DAE B D C A E Statement Given: Prove: Reason B 1 2 BD BE 3 4 DAF ECF 1 2 D 3 A E F 4 C 28 HW: p.268 #8-15, 19-21 Geometry 2205 Unit 3: Mrs. Bondi Review 4.1-7 29 Geometry 2205 Unit 3: Mrs. Bondi 30 Geometry 2205 Unit 3: Mrs. Bondi 31 Geometry 2205 Unit 3: Mrs. Bondi Practice for Ch.4Test After ch.4 test HW: p.281 #3-15 32 Geometry 2205 Unit 3: Mrs. Bondi Lesson 8: Midsegments of Triangles (PH text 5.1) Objective: To use properties of midsegments to solve problems Midsegment of a triangle: Theorem 5-1 Triangle Midsegment Theorem If D is the midpoint of CA and E is the midpoint of CB , then DE AB and DE = ½ AB. 33 Geometry 2205 Unit 3: Mrs. Bondi Examples: 4. What all can be determined, given that OP is a midsegment of MLN ? M 65o 89o O P N L HW: p.288 #6-26 even, 32-44 even, 48 34 Geometry 2205 Unit 3: Mrs. Bondi 35 Geometry 2205 Unit 3: Mrs. Bondi 36 Geometry 2205 Unit 3: Mrs. Bondi Lesson 9: Perpendicular and Angle Bisectors (PH text 5.2) Objective: Use properties of perpendicular bisectors and angle bisectors Equidistant - Theorem 5-2 Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. P N A B M Theorem 5-3 Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. R U I Biconditional: Examples: 1) What is the length of AB ? 2) 37 What is the length of QR ? Geometry 2205 Unit 3: Mrs. Bondi Distance from a point to a line: Length of the perpendicular segment from the point to the line (the shortest distance) C A Theorem 5-4 B Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. Theorem 5-5 Converse of the Angle Bisector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the angle bisector. Example: 3) What is the length of FB ? HW: p.296 #6-8, 12-22, 29-35 (complete at least two proofs) 38 Geometry 2205 Unit 3: Mrs. Bondi 39 Geometry 2205 Unit 3: Mrs. Bondi 40 Geometry 2205 Unit 3: Mrs. Bondi Lesson 10: Bisectors in Triangles (PH text 5.3) Objective: to identify properties of perpendicular bisectors and angle bisectors, altitudes and medians of a triangle. Concurrent – Point of concurrency – Theorem 5-6 Concurrency of Perpendicular Bisectors Theorem The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. Circumcenter of the triangle – the point of concurrency of the perpendicular bisectors It can be inside, on, or outside the triangle. The circle that has its center as the circumcenter of the triangle is circumscribed about the triangle. The circle contains each of the triangle’s vertices. 41 Geometry 2205 Unit 3: Mrs. Bondi Example: A local park commission wants to build a snack shack at a central point from the tennis courts, ball fields, and playgrounds. How should they determine where the snack shack should be built? BF TC PG Theorem 5-7 Concurrency of Angle Bisectors Theorem The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle. Incenter of the triangle – the point of concurrency of angle bisectors - It is always inside the triangle. The incenter of a triangle is the center of the circle that is inscribed in the triangle (when points on the circle and points on the triangle’s sides are shared). How would you solve this? (think algebraically) HW: p.304 #5, 7-8, 14-18, 23-29, 33-35 42 Geometry 2205 Unit 3: Mrs. Bondi 43 Geometry 2205 Unit 3: Mrs. Bondi Lesson 11: Medians and Altitudes (PH text 5.4) Objective: to identify properties of perpendicular bisectors and angle bisectors, altitudes and medians of a triangle. Median of a triangle – Theorem 5-8 Concurrency of Medians Theorem The medians of a triangle are concurrent at a point that is 2/3 the distance from each vertex to the midpoint of the opposite side. Centroid of the triangle – the point of concurrency of the medians – also called the center of gravity of a triangle because it is the point where a triangular shape will balance. It is always inside the triangle. Example 1: Assume ZA = 9. What is the length of FC ? What is the ratio of ZA to AC? 44 Geometry 2205 Unit 3: Mrs. Bondi Altitude of a triangle – Theorem 5-9 Concurrency of Altitudes Theorem The lines that contain the altitudes of a triangle are concurrent. Orthocenter of the triangle – the point of concurrency of the lines that contain the altitudes It can be inside, on, or outside the triangle. Practice: 45 Geometry 2205 Unit 3: Mrs. Bondi Concept Summary: Special Segments and Lines in Triangles Name and label the point of concurrency for each example. Perpendicular Bisectors Angle Bisectors Medians HW: p.312 #5, 8-14, 17-21, 24-31 (Bring the paper triangles to class.) 46 Altitudes Geometry 2205 Unit 3: Mrs. Bondi 47 Geometry 2206 Unit 3: Mrs. Bondi Lesson 12: Indirect Proof (PH text 5.5) Objective: To write convincing proofs by using indirect reasoning Puzzle using indirect reasoning: Fill in the empty squares with numbers. The numbers one through four must appear exactly once in each row and column. 1 4 3 2 1 1 2 4 1 4 2 Indirect Reasoning – all possibilities are considered, then all but one are proven false. The remaining possibility must then be true. Indirect Proof – a proof using indirect reasoning – sometimes called a proof by contradiction Often a statement and its negation are the only possibilities. When one of these leads to a conclusion that contradicts a fact you know to be true, you can eliminate that possibility. Explain why the two statements are contradictory. 1) 2) 3) 4) A is acute mA 90 __________________________________________ M is the midpoint of CD CM 5, MD 10 __________________________________________ m forms right angles with n m n __________________________________________ 1 is a complement of 2 m1 m2 180 __________________________________________ 48 Geometry 2206 Unit 3: Mrs. Bondi Writing an Indirect Proof: 1 – State as a temporary assumption the negation of what you want to prove. 2 – Show that this assumption leads to a contradiction. 3 – Conclude that the assumption must be false, so that what you want to prove must be true. Write an indirect proof. P S 1) Given: QS does not bisect PQR Prove: mPQS mSQR Q R Assume ______________________________________________________ Then ______________________________________________________ This contradicts the fact that __________________________________________ Therefore 2) ______________________________________________________ 1 2 3 4 Given: m is not to n Prove: m3 m6 5 6 7 8 m n Assume ______________________________________________________ Then ______________________________________________________ This contradicts the fact that __________________________________________ Therefore ______________________________________________________ 49 Geometry 2206 Unit 3: Mrs. Bondi T 3) Given: TU is not to UW Prove: m1 m2 90 V 1 U 2 W Assume ______________________________________________________ Then ______________________________________________________ This contradicts the fact that __________________________________________ Therefore 4) ______________________________________________________ Given: ABC is scalene. Prove: A, B, and C all have different measures. HW: p.319 #3-21, do at least two proofs from 22-23 & 29-30 p.323 – review and/or practice – you are expected to be able to use and solve linear inequalities 50 Geometry 2206 Unit 3: Mrs. Bondi Lesson 13: Inequalities in One Triangle (PH text 5.6) Objective: Use inequalities involving triangle side lengths and angle measures to solve problems. Comparison Property of Inequality If a = b + c and c > 0, then a > b. Corollary to the Triangle Exterior Angle Theorem The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles If 1 is an exterior angle, then m1 m2 and m1 m3 . Use It: Given the information in the figure, explain why m2 m3 . Theorem 5-10 If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side. 51 Geometry 2206 Unit 3: Mrs. Bondi Theorem 5-11 If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle. B 40 92 48 C A Practice: 52 Geometry 2206 Unit 3: Mrs. Bondi Theorem 5-12 Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Is it possible for a triangle to have sides of the following lengths? 1) 2 in, 5 in, 8 in 2) 4 cm, 6 cm, 9 cm Application: Given the lengths if two sides of a triangle, write an inequality to represent the range of values for the length of the third side, s. a) 15cm, 5 cm b) 3 cm, 10 cm c) a, b (assume a ≥ b) HW: p.328 # 4-38 even 53 Geometry 2206 Unit 3: Mrs. Bondi 54 Geometry 2206 Unit 3: Mrs. Bondi 55 Geometry 2206 Unit 3: Mrs. Bondi Lesson 14: Inequalities in Two Triangles (PH text 5.7) Objective: to apply inequalities in two triangles Think about a set of triangles that has two pairs of congruent sides. What do you know about the relationship of the included angle and the third side? (If it helps, imagine two sets of rulers taped together to form a hinge.) Theorem 5-13 The Hinge Theorem (SAS Inequality Theorem) If two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third side is opposite the larger included angle. Examples: 56 Geometry 2206 Unit 3: Theorem 5-14 Mrs. Bondi Converse of the Hinge Theorem (SSS Inequality Theorem) If two sides of one triangle are congruent to two sides of another triangle, and the third sides are not congruent, then the larger included angle is opposite the longer side. Use an indirect proof to prove the Converse of the Hinge Theorem Examples: 57 Geometry 2206 Unit 3: Mrs. Bondi HW: p.336 #4-18, 23 or 25 58 Geometry 2206 Unit 3: Mrs. Bondi 59 Geometry 2206 Unit 3: Mrs. Bondi 60 Geometry 2206 Unit 3: Mrs. Bondi Review: 61 Geometry 2206 Unit 3: Mrs. Bondi Lessons 5-5 through 5-7 62