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Transcript

Geometry Definitions, Postulates, and Theorems Key Chapter 4: Congruent Triangles Section 4.1: Apply Triangle Sum Properties Standards: 12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems. 13.0 Students prove relationships between angles in polygons by using properties of complementary, supplementary, vertical, and exterior angles. B Triangle – A polygon formed by three segments joining three non-collinear points. A triangle can be classified by its sides and then by its angles. A C *Classifying Triangles by Sides: Scalene Triangle – A triangle with no congruent sides. Vertex Angle Isosceles Triangle – A triangle with at least two congruent sides. Legs – The congruent sides of an isosceles triangle, when only two sides are congruent. Leg Leg Base Angle Base Base Angle Base – The third side (non-congruent side) of an isosceles triangle. Equilateral Triangle – A triangle with three congruent sides. *Classifying Triangles by Angles: Acute Triangle – A triangle with three acute angles. 45 50 85 Right Triangle – A triangle with one right angle. Leg Hypotenuse Legs – The sides that form the right angle. Hypotenuse – The side opposite the right angle. Leg Obtuse Triangle – A triangle with one obtuse angle. 100 30 50 Equiangular Triangle – A triangle with three congruent acute angles. (over) Ex. Classify the triangles by their sides and angles. a) b) c) 5 3 120 4 Isosceles Equilateral Obtuse Equiangular/Acute Scalene Right Vertex (plural: vertices) – Each of the three points joining the sides of a triangle. B Adjacent Sides of an Angle – Two sides that share a common vertex. A C Opposite Side from an Angle – The side that does not form the angle. Interior angles – When the sides of a triangle are extended, additional angles are formed. The original angles are the interior angles. Exterior angles – When the sides of a triangle are extended, additional angles are formed. The angles that form linear pairs with the interior angles are the exterior angles. ***Theorem 4.1 – Triangle Sum Theorem The sum of the measures of the three interior angles of a triangle is 180 . B 85 350 0 C m A E 670 D 700 B= ***Theorem 4.2 – Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles. 1 F m 1= Corollary To The Triangle Sum Theorem – The acute angles of a right triangle are complementary (add up to equal 90 ). Y X Z (over) Ex. A triangle has the given vertices. Graph the triangle and classify it by its sides. Then determine if it is a right triangle. Yes, right triangle A( 5, 4), B (2, 6), C (4, 1) B A C Isosceles Ex. Find the value of x and y. Then classify the triangle by its angles. Ex. Find the value of x. Then classify the triangle by its angles. x0 600 500 Linear Pair y x Ex. Find the angle measures of the numbered angles. 620 1 2 3 1100 600 4 Ex. Find the values of x and y. y0 x0 680 (2x-18)0 720 Section 4.2: Apply Congruence and Triangles Standards: 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. Two geometric figures are congruent if they have exactly the same size and shape, like placing one figure perfectly onto another figure. A B D C Y Congruent Figures – All the parts of one figure are congruent to the corresponding parts of the other figure. In congruent polygons, the corresponding sides and the corresponding angles are congruent. V X Congruence Statements – When writing a congruence statement for two polygons, always list the corresponding vertices in the same order! W Ex. GIVEN: ABC DEF B F D A Corresponding Angles – B A C E C AB Corresponding Sides – BC CA Ex. Write a congruency statement. T S Q Ex. ABCD L 9 cm A 910 B 1130 R P JKHL. Find the value of x and y. J 4x–3 cm 0 D 86 (5y–12)0 K C H U ***Theorem 4.3 – Third Angle Theorem IF two angles of one triangle are congruent to two angles of another triangle, THEN, the third angles are also congruent. Ex. B F D K Ex. G 55 J 50 50 A C E F 55 I H (over) ***Theorem 4.4 – Properties of Congruent Triangles Reflexive Property of Congruent Triangles – For any triangle ABC: ABC ABC. Symmetric Property of Congruent Triangles – IF ABC DEF, then DEF ABC. Transitive Property of Congruent Triangles – IF ABC DEF, and DEF JKL, then ABC Ex. Find the values of x and y. (8 x 2 y)0 (6 x y)0 290 1090 JKL. Section 4.3: Prove Triangles Congruent by SSS Standards: 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. ***Side-Side-Side (SSS) Congruence Postulate – IF three sides of one triangle are congruent to three sides of a second triangle, THEN the two triangles are congruent. *Information about the angles is not needed. B E A C If Side AB D F , Side BC , and Side CA , then by Ex. Is the congruence statement true? Explain your reasoning. WXY . Ex. Is the congruence statement true? Explain your reasoning. YZW KJL X MJL L K Y W M J Z B Ex. Write a proof. Given: AD CD , AB CB Prove: ABD CBD A Statements 1. Given 2. 2. Reflexive ABD CBD C Reasons 1. 3. D 3. (over) Structural Support – A diagonal support added to a figure helps make the figure stable. The diagonal support forms triangles with fixed side lengths. By the SSS Congruence Postulate, these triangles cannot change shape and so the figure is stable. Ex. Determine whether the figure is stable. Explain your answer. a) b) B c) X Given: AE FC, BE BF, AB BC Given: WX YX Z is the midpoint of WY Prove: AFB CEB W A F E 1. AE FC, BE BF, AB Z Y Prove: WXZ YXZ C BC 1. Given 1. WX YX 1. Given Z is the midpoint of WY 2. EF = EF 2. ______________ 2. _____________ 2. ____________ 3. AE + EF = AF 3. _____________ 3. WZ ZY 3. _____________ EF + FC = EC 4. FC + EF = AF _______________ 4. Substitution _______________ 4. ______________ 4. ____________ 5. AF = EC 5. ______________ 6. ______________ 6. Def. of Congruent Segments 7. _______________ 7. SSS Section 4.4: Prove Triangles Congruent by SAS and HL Standards: 4.0 Students prove basic theorems involving congruence and similarity. 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. ***Side-Angle-Side (SAS) Congruence Postulate – IF two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, THEN the two triangles are congruent. B E If Side AB Angle A C D F , A , and Side AC , then by . Ex. Do you have enough information to prove the triangles are congruent by SAS? a) b) A Ex. Write a proof. Given: AC EC , DC ECD Prove: ACB D C BC B Statements Reasons 1. 1. Given 2. 2. Vertical Angles are Congruent 3. ACB ECD E 3. (over) Ex. Write a proof. M Given: AB PB , MB AP Prove: MBA MBP A Statements Reasons 1. 1. Given 2. 2. 3. MBA MBP P lines form rights angles 3. 4. 5. B 4. Reflexive Property MBA MBP 5. Right Triangles: Legs – The sides adjacent to the right angle. Hypotenuse – The side opposite the right angle. ***Hypotenuse-Leg (HL) Congruence Theorem – IF the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, THEN the two triangles are congruent. A D C B F E B Ex. Write a proof. Given: AB BC , BD CBD Prove: ABD AC A Statements Reasons 1. 1. Given 2. 2. 3. BDA BDC 4. 5. lines form rights angles 3. 4. Reflexive Property ABD CBD 5. D C Section 4.5: Prove Triangles Congruent by ASA and AAS Standards: 4.0 Students prove basic theorems involving congruence and similarity. 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. ***Angle-Side-Angle (ASA) Congruence Postulate – IF two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, THEN the two triangles are congruent. B E If Angle A Side AC A C D F Angle , , and C , then by . ***Angle-Angle-Side (AAS) Congruence Postulate – IF two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, THEN the two triangles are congruent. B A E C D If Angle A , Angle C , and F Side BC , then by . Ex. Is it possible to prove the triangles are congruent? If so, state the postulate or theorem used. a) / b) / c) / d) / (over) X Ex. Write a proof. Given: WZ bisects XZY and Prove: WZX WZY XWY Z W Y Statements Reasons 1. 1. Given 2. 2. Definition of an angle bisector 3. ZW 4. ZW WZX 3. WZY 4. Ex. Write a proof. Given: Prove: C B, D BDM C FM F , M is the midpoint of D F Statements Reasons 1. 1. Given 2. 2. Definition of a midpoint 3. BDM C B C FM 3. D M F Section 4.6: Use Congruent Triangles Standards: 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. ***Corresponding Parts of Congruent Triangles are Congruent (C.P.C.T.C.) – 1. Prove two triangles are congruent with SSS, SAS, HL, ASA, or AAS. 2. Then, conclude that the corresponding parts of these congruent triangles are congruent as well. The triangles below are congruent by SAS. Since the triangles are congruent, we know that: A X A C B C Y Z Ex. Write a proof. AC H Given: HJ II LK , JK II HL Prove: LHJ JKL J L Statements K Reasons 1. 1. Given 2. 2. Alternate Interior Angles are Congruent 3. JL JL 3. 4. LHJ JKL 4. 5. LHJ JKL 5. Ex. Write a proof. M Given: MS II TR , MS TR R A Prove: A is the midpoint of MT S Statements T Reasons 1. 1. Given 2. 2. Alternate Interior Angles are Congruent 3. MAS TAR 3. 4. 4. CPCTC 5. A is the midpoint of MT 5. (over) P Ex. Write a proof. Given: MP bisects Prove: LP LMN , LM NM NP L N M Statements Reasons 1. 2. 1. Given NMP LMP 2. 3. 4. 3. Reflexive Property NMP 5. LP LMP 4. NP 5. Ex. Which triangles can you show are congruent in order to prove the statement? What postulate or theorem would you use? a) A b) SW C B A D TY S C W X Y T Section 4.7: Use Isosceles and Equilateral Triangles Standards: 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. 12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems. Vertex Angle Leg Leg Legs – The two congruent sides in an isosceles triangle. Vertex Angle – The angle formed by the legs in an isosceles triangle. Base – The third side of the isosceles triangle. Base Angle Base Base Angles – The two angles adjacent to the base in an isosceles triangle. Base Angle ***Theorem 4.7 – Base Angles Theorem IF two sides of a triangle are congruent, THEN the angles opposite them are congruent. 30 x ***Theorem 4.8 – Converse of Base Angles Theorem IF two angles of a triangle are congruent, THEN the sides opposite them are congruent. 2x-4 2x+2 x+5 Corollary To Base Angles Theorem IF a triangle is equilateral, THEN it is equiangular. Corollary to the Converse of Base Angles Theorem IF a triangle is equiangular, THEN it is equilateral. Section 4.8: Perform Congruence Transformations Standards: 22.0 Students know the effect of rigid motions on figures in the coordinate plane and space, including rotations, translations, and reflections. Transformation – An operation that moves or changes a geometric figure in some way to produce a new figure. Image – The new figure produced. *Three Types of Transformations: A B Pre-Image Image B' B C A C' A' Translation – Moves every point of a figure the same distance in the same direction. B' C A' Translate down 6 Translate right 10 C' Reflection – Uses a line of reflection to create a mirror image of the original figure. Rotation – Turns a figure about a fixed point, called the center of rotation. Rotate 90 degrees clockwise Congruence Transformation – Changes the position of the figure without changing its size or shape. Translate A Figure In The Coordinate Plane – Moving an object a given distance right or left and up or down. y *Coordinate Notation for a Translation B You can describe a translation by the notation ( x, y) ( x a, y b) which shows that each point ( x, y) of a figure is translated horizontally a units and vertically b units Ex. Figure ABC D has the vertices A( 4, 2), B ( 2, 5), C ( 1,1), and D ( 3, 1) . Sketch ABC D and its image after the translation ( x, y) ( x 5, y 2) . Right 5 Down 2 A C B' A' x C' D D' (over) Usually Reflect A Figure In The Coordinate Plane – The line of reflection is always the x-axis or the y-axis. y *Coordinate Notation for a Reflection Reflection in the x-axis: ( x, y) ( x, y) Multiply the y-coordinate by -1. Reflection in the y-axis: ( x, y) ( x, y) Multiply the x-coordinate by -1. x Ex. Use a reflection in the x-axis to draw the other half of the figure. Rotate A Figure In The Coordinate Plane – The center of rotation is the origin. The direction of rotation can be either clockwise or counterclockwise. The angle of rotation is formed by rays drawn from the center of rotation through corresponding points on the original figure and its image. 900 clockwise rotation 600 counterclockwise rotation Ex. Graph PQ and RS . Tell whether RS is a rotation y P of PQ about the origin. If so, give the angle and direction of rotation. a) P (2, 6), Q (5,1), R(6, 1), S (1, 2) b) P (4, 2), Q (3, 3), R( 2, 4), S ( 3, 3) Q S x R