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Chapter 4 Congruent Triangles • Identify the corresponding parts of congruent figures • Prove two triangles are congruent • Apply the theorems and corollaries about isosceles triangles 4.1 Congruent Figures Objectives • Identify the corresponding parts of congruent figures What we already know… • Congruent Segments – Same length • Congruent Angles – Same degree measure Congruent Figures Exactly the same size and shape. Don’t ASSume ! C B A F D E Definition of Congruency Two figures are congruent if corresponding vertices can be matched up so that: 1. All corresponding sides are congruent 2. All corresponding angles are congruent. What does corresponding mean again? • Matching • In the same position Volunteer • Draw a large scalene triangle (with a ruler) • Cut out two congruent triangles that are the same • Label the Vertices A, B, C and D, E, F ABC DEF You can slide and rotate the triangles around so that they MATCH up perfectly. A E C B F D ABC DEF The order in which you name the triangles matters ! A E C B F D Based on the definition of congruency…. • Three pairs of corresponding angles • Three pairs of corresponding sides 1. A D 1. AB DE 2. B E 2. BC EF 3. C F 3. CA FD It is not practical to cut out and move the triangles around ABC XYZ • Means that the letters X and A, which appear first, name corresponding vertices and that – X A. • The letters Y and B come next, so – Y B and –XY AB CAUTION !! • If the diagram doesn’t show the markings or • You don’t have a reason – Shared sides, shared angles, vertical angles, parallel lines White Boards • Suppose TIM BER IM ___ White Boards • Suppose TIM BER IM ER , Why ? White Boards • Corresponding Parts of Congruent Triangles are Congruent White Boards • Suppose TIM BER ___ R White Boards • Suppose TIM BER M R, Why? White Boards • Corresponding Parts of Congruent Triangles are Congruent White Boards • Suppose TIM BER MTI ____ White Boards • Suppose TIM BER MTI RBE White Boards • If ABC XYZ m B = 80 m C = 50 Name four congruent angles White Boards • If ABC XYZ m B = 80 m C = 50 A, C , X, Z White Boards • If ABC XYZ Write six congruences that must be correct White Boards • If ABC XYZ 1. A X 1. AB XY 2. B Y 2. BC YZ 3. C Z 3. CA ZX Remote time A. B. C. D. Always Sometimes Never I don’t know A. B. C. D. Always Sometimes Never I don’t know • An acute triangle is __________ congruent to an obtuse triangle. A. B. C. D. Always Sometimes Never I don’t know • A polygon is __________ congruent to itself. A. B. C. D. Always Sometimes Never I don’t know • A right triangle is ___________ congruent to another right triangle. A. B. C. D. Always Sometimes Never I don’t know • If ABC XYZ, A is ____________ congruent to Y. A. B. C. D. Always Sometimes Never I don’t know • If ABC XYZ, B is ____________ congruent to Y. A. B. C. D. Always Sometimes Never I don’t know • If ABC XYZ, AB is ____________ congruent to ZY. 4.2 Some Ways to Prove Triangles Congruent Objectives • Learn about ways to prove triangles are congruent Don’t ASSume • Triangles cannot be assumed to be congruent because they “look” congruent. and • It’s not practical to cut them out and match them up so, We must show 6 congruent pairs • 3 angle pairs and • 3 pairs of sides WOW • That’s a lot of work Spaghetti Experiment • Using a small amount of playdough as your “points” put together a 5 inch, 3 inch and 2.5 inch piece of spaghetti to forma triangle. • Be careful, IT’S SPAGHETTI, and it will break. • Compare your spaghetti triangle to your neighbors • Compare your spaghetti triangle to my spaghetti triangle. We are lucky….. • There is a shortcut – We don’t have to show • ALL pairs of angles are congruent and • ALL pairs of sides are congruent SSS Postulate If three sides of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. E B A C F D Patty Paper Practice 5 inches 3 inches 2.5 inches Volunteer SAS Postulate If two sides and the included angle are congruent to the corresponding parts of another triangle, then the triangles are congruent. E B C F D ASA Postulate If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. E B A C F D The order of the letters MEAN something • Is SAS the same as SSA or A$$ ? Construction 2 Given an angle, construct a congruent angle. Given: ABC Construct: CDE ABC Steps: Construction 3 Given an angle, construct the bisector of the angle Given: ABC Construct: bisector of ABC Steps: CAUTION !! • If the diagram doesn’t show the markings or • You don’t have a reason – Shared sides, shared angles, vertical angles, parallel lines Remote Time Can the two triangles be proved congruent? If so, what postulate can be used? A. B. C. D. E. SSS Postulate SAS Postulate ASA Postulate Cannot be proved congruent I don’t know A. B. C. D. E. SSS Postulate SAS Postulate ASA Postulate Cannot be proved congruent I don’t know A. B. C. D. E. SSS Postulate SAS Postulate ASA Postulate Cannot be proved congruent I don’t know A. B. C. D. E. SSS Postulate SAS Postulate ASA Postulate Cannot be proved congruent I don’t know A. B. C. D. E. SSS Postulate SAS Postulate ASA Postulate Cannot be proved congruent I don’t know A. B. C. D. E. SSS Postulate SAS Postulate ASA Postulate Cannot be proved congruent I don’t know A. B. C. D. E. SSS Postulate SAS Postulate ASA Postulate Cannot be proved congruent I don’t know A. B. C. D. E. SSS Postulate SAS Postulate ASA Postulate Cannot be proved congruent I don’t know A. B. C. D. E. SSS Postulate SAS Postulate ASA Postulate Cannot be proved congruent I don’t know A. B. C. D. E. SSS Postulate SAS Postulate ASA Postulate Cannot be proved congruent I don’t know White Board • Decide Whether you can deduce by the SSS, SAS, or ASA Postulate that the two triangles are congruent. If so, write the congruence ( ABC _ _ _ ). If not write not congruent. B A D C DBC ABC SSS A B C D No Congruence Construction 7 Given a point outside a line, construct a line parallel to the given line through the point. Given: line Construct: Steps: l with point A to l through A 4.3 Using Congruent Triangles Objectives • Use congruent triangles to prove other things Our Goal • In the last section, our goal was to prove that two triangles are congruent. The Reason • If we can show two triangle are congruent, using the SSS, SAS, ASA postulates, then we can use the definition of Congruent Triangles to say other parts of the triangles are congruent. – Corresponding Parts of Congruent Triangles are Congruent. This is an abbreviated way to refer to the definition of congruency with respect to triangles. C orresponding P arts of C ongruent T riangles are C ongruent Basic Steps 1. Identify two triangles in which the two segments or angles are corresponding parts. 2. Prove that those two triangles are congruent 3. State that the two parts are congruent using the reason CPCTC. Given: m 1 = m 2 m3=m4 Prove: M is the midpoint of JK L 3 4 J 1 2 M K L Given: m 1 = m 2 m3=m4 34 Prove: M is the midpoint of JK J LM = LM 1 2 M Reflexive Property m J = m K If 2 ’s of 1 are to 2 ’s of another , then the third ’s are . K L Given: m 1 = m 2 m3=m4 34 A A S Prove: M is the midpoint of JK J LM = LM A 1A 2 A M Reflexive Property m J = m K If 2 ’s of 1 are to 2 ’s of another , then the third ’s are . A K L Given: m 1 = m 2 m3=m4 34 A A S Prove: M is the midpoint of JK J JLM KLM ASA 1A 2 A M K L Given: m 1 = m 2 m3=m4 34 Prove: M is the midpoint of JK J Statements Reasons 1 2 M K L Given: m 1 = m 2 m3=m4 34 Prove: M is the midpoint of JK J Statements Reasons 1. m 1 = m 2 m3=m4 1. Given 1 2 M K L Given: m 1 = m 2 m3=m4 34 Prove: M is the midpoint of JK J Statements Reasons 1. m 1 = m 2 m3=m4 1. Given 5. M is the midpoint of JK 1 2 M K L Given: m 1 = m 2 m3=m4 34 Prove: M is the midpoint of JK J Statements Reasons 1. m 1 = m 2 m3=m4 1. Given 4. JM = KM 5. M is the midpoint of JK 1 2 M K L Given: m 1 = m 2 m3=m4 34 Prove: M is the midpoint of JK J Statements Reasons 1. m 1 = m 2 m3=m4 1. Given 3. JLM KLM 4. JM = KM 5. M is the midpoint of JK 4. CPCTC 1 2 M K L Given: m 1 = m 2 m3=m4 34 Prove: M is the midpoint of JK J 1 2 M Statements Reasons 1. m 1 = m 2 m3=m4 2. LM = LM 1. Given 3. JLM KLM 3. ASA Postulate 4. JM = KM 4. CPCTC 5. M is the midpoint of JK 2. Reflexive Property K L Given: m 1 = m 2 m3=m4 34 Prove: M is the midpoint of JK J 1 2 M Statements Reasons 1. m 1 = m 2 m3=m4 2. LM = LM 1. Given 3. JLM KLM 3. ASA Postulate 4. JM = KM 4. CPCTC 2. Reflexive Property 5. M is the midpoint of JK 5. Definition of midpoint K L Given: m 1 = m 2 m3=m4 34 A A Prove: M is the midpoint of JK S J A A 1 2 M Statements Reasons 1. m 1 = m 2 A m3=m4 A 2. LM = LM S 1. Given 3. JLM KLM 3. A S A Postulate 4. JM = KM 4. CPCTC 2. Reflexive Property 5. M is the midpoint of JK 5. Definition of midpoint K M Given: MK OK; KJ bisects MKO; Prove: JK bisects MJO J 1 2 S S O A 3 A 4 S 34 Definition of bisector JK JK Reflexive Property MKJ OKJ SAS Postulate K M Given: MK OK; KJ bisects MKO; Prove: JK bisects MJO J 1 2 S S O Statements Reasons A 3 A 4 S K M Given: MK OK; KJ bisects MKO; Prove: JK bisects MJO J 1 2 S S O A 3 A 4 S Statements Reasons 1. MK OK; KJ bisects MKO 2. 3 4 1. Given 3. JK JK 3. Reflexive Property 2. Def of bisector K M Given: MK OK; KJ bisects MKO; Prove: JK bisects MJO 1 2 J S S O A 3 A 4 S Statements Reasons 1. MK OK; KJ bisects MKO 2. 3 4 1. Given 3. JK JK 3. Reflexive Property 6. JK bisects MJO 6. 2. Def of bisector K M Given: MK OK; KJ bisects MKO; Prove: JK bisects MJO 1 2 J S S O A 3 A 4 S Statements Reasons 1. MK OK; KJ bisects MKO 2. 3 4 1. Given 3. JK JK 3. Reflexive Property 5. 1 2 5. CPCTC 6. JK bisects MJO 6. 2. Def of bisector K M Given: MK OK; KJ bisects MKO; Prove: JK bisects MJO 1 2 J S S O A 3 A 4 S Statements Reasons 1. MK OK; KJ bisects MKO 2. 3 4 1. Given 3. JK JK 3. Reflexive Property 4. MKJ OKJ 4. SAS Postulate 5. 1 2 5. CPCTC 6. JK bisects MJO 6. 2. Def of bisector K M Given: MK OK; KJ bisects MKO; Prove: JK bisects MJO J 1 2 S S O A 3 A 4 S Statements Reasons 1. MK OK; S KJ bisects MKO 2. 3 4 A 1. Given 3. JK JK S 3. Reflexive Property 4. MKJ OKJ 4. SAS Postulate 5. 1 2 5. CPCTC 6. JK bisects MJO 6. Def of bisector 2. Def of bisector K 4.4 The Isosceles Triangle Theorem Objectives • Apply the theorems and corollaries about isosceles triangles Isosceles Triangle By definition, it is a triangle with two congruent sides called legs. X Legs Vertex Angle Base Angles Y Z Base Experiment - Goal • Discover Properties of an Isosceles Triangle Supplies • • • • Blank sheet of paper Ruler Pencil Scissors Procedure 1. Fold a sheet of paper in half. Procedure 2. Draw a line with the ruler going from the folded edge (very important) to the corner of the non folded edge. Folded edge Procedure 3. Cut on the red line Cut here Procedure 4. Open and lay flat. You will have a triangle Procedure 5. Label the triangle P S R Q Procedure 6. Since PRQ fits exactly over PSQ (because that’s the way we cut it), P PRQ PSQ S R Q Procedure 7. What conclusions can you make? Be careful not to ASSume anything. P S R Q Conclusions 1. PRS PSR 2. PQ bisects RPS P 3. PQ bisects RS 4. PQ RS at Q 5. PR PS S R Q These conclusions are actually • Theorems and corollaries Theorem The base angles of an isosceles triangle are congruent. B A C Corollary • An equilateral triangle is also equiangular. Corollary • An equilateral triangle has angles that measure 60. Corollary • The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base. Theorem If two angles of a triangle are congruent, then it is isosceles. B A C Corollary • An equiangular triangle is also equilateral. White Board Practice • Find the value of x 30º xº x = 75º White Board Practice • Find the value of x 2x - 4 2x + 2 x+5 x=9 White Board Practice • Find the value of x 41 42 56 º 62 º x x = 42 4.5 Other Methods of Proving Triangles Congruent Objectives • Learn two new ways to prove triangles are congruent Proving Triangles We can already prove triangles are congruent by the ASA, SSS and SAS. There are two other ways to prove them congruent… AAS Theorem If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. E B A C F D The Right Triangle B leg A acute angles right angle leg C HL Theorem If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent. B A E C F D Five Ways to Prove ’s All Triangles: ASA SSS SAS AAS Right Triangles Only: HL White Board Practice • State which of the congruence methods can be used to prove the triangles congruent. You may choose more than one answer. SSS Postulate SAS Postulate ASA Postulate AAS Theorem HL Theorem SSS Postulate SAS Postulate ASA Postulate AAS Theorem HL Theorem SSS Postulate SAS Postulate ASA Postulate AAS Theorem HL Theorem SSS Postulate SAS Postulate ASA Postulate AAS Theorem HL Theorem SSS Postulate SAS Postulate ASA Postulate AAS Theorem HL Theorem SSS Postulate SAS Postulate ASA Postulate AAS Theorem HL Theorem SSS Postulate SAS Postulate ASA Postulate AAS Theorem HL Theorem SSS Postulate SAS Postulate ASA Postulate AAS Theorem HL Theorem SSS Postulate SAS Postulate ASA Postulate AAS Theorem HL Theorem SSS Postulate SAS Postulate ASA Postulate AAS Theorem HL Theorem SSS Postulate SAS Postulate ASA Postulate AAS Theorem HL Theorem SSS Postulate SAS Postulate ASA Postulate AAS Theorem HL Theorem SSS Postulate SAS Postulate ASA Postulate AAS Theorem HL Theorem 4.6 Using More than One Pair of Congruent Triangles Objectives • Construct a proof using more than one pair of congruent triangles. • Sometimes two triangles that you want to prove congruent have common parts with two other triangles that you can easily prove congruent. More Than One Pair of ’s Given: X is the midpt of AF & CD Prove: X is the midpt of BE D A X E B F C Lecture 7 (4-7) Objectives • Define altitudes, medians and perpendicular bisectors. Median of a Triangle A segment connecting a vertex to the midpoint of the opposite side. midpoint vertex Median of a Triangle Each triangle has three Medians vertex midpoint Median of a Triangle Each triangle has three Medians vertex midpoint Median of a Triangle • Notice that the three medians will meet at one point. If they do not meet, then you are not drawing the segments well. Altitude of a Triangle A segment drawn from a vertex perpendicular to the opposite side. vertex perpendicular Altitude of a Triangle Each Triangle has three altitudes perpendicular vertex Altitude of a Triangle Each triangle has three altitudes perpendicular vertex Altitude of a Triangle Notice that the three altitudes will meet at one point. If they do not meet, then you are not drawing the segments well. Special Cases - Altitudes Obtuse Triangles: Two of the altitudes are drawn outside the triangle. Extend the sides of the triangle Special Cases - Altitudes Right Triangles: Two of the altitudes are already drawn for you. Perpendicular Bisector A segment (line or ray) that is perpendicular to and passes through the midpoint of another segment. Must put the perpendicular and congruent markings ! Angle Bisector A ray that cuts an angle into two congruent angles. Theorem If a point lies on the perpendicular bisector of a segment of a segment, then the point is equidistant from the endpoints. Theorem If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment. Remember • When you measure distance from a point to a line, you have to make a perpendicular line. Theorem If a point lies on the bisector of an angle then the point is equidistant from the sides of the angle. Construction 10 Given a triangle, circumscribe a circle about the triangle. Given: ABC Construct: R Steps: circumscribed about ABC Construction 11 Given a triangle, inscribe a circle within the triangle. Given: ABC Construct: Steps: R inscribed within ABC Remote Time A. B. C. D. Always Sometimes Never I don’t know A. B. C. D. Always Sometimes Never I don’t know • An altitude is _____________ perpendicular to the opposite side. A. B. C. D. Always Sometimes Never I don’t know • A median is ___________ perpendicular to the opposite side. A. B. C. D. Always Sometimes Never I don’t know • An altitude is ______________ a perpendicular bisector. A. B. C. D. Always Sometimes Never I don’t know • An angle bisector is _______________ perpendicular to the opposite side. A. B. C. D. Always Sometimes Never I don’t know • A perpendicular bisector of a segment is ___________ equidistant from the endpoints of the segment.