Geometry, Definitions, Postulates, And Theorems

Geometry Definitions, Postulates, and TheoremsChapter 4: Congruent TrianglesSection 4.1: Apply Triangle Sum PropertiesStandards: 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 C *Classifying Triangles by Sides: Scalene Triangle – Isosceles Triangle – Legs – Base – Equilateral Triangle – *Classifying Triangles by Angles: Acute Triangle – Right Triangle – Legs – Hypotenuse –Obtuse Triangle – Equiangular Triangle – (over) Ex. Classify the triangles by their sides and angels. a) b) c) 3 5 120° 4B Vertex (plural: vertices) – Adjacent Sides of an Angle – A C Opposite Side from an Angle – Interior angles – Exterior angles – ***Theorem 4.1 – Triangle Sum Theorem B350 C 850 A m  B =***Theorem 4.2 – Exterior Angle TheoremE 670700 1 D F m  1 = Corollary To The Triangle Sum Theorem – YX Z Ex. A triangle has the given vertices. Graph the triangle and classify it by its sides. Then determine if it is a right triangle. A(5,4), B(2,6), C(4,1)Ex. Find the value of x and y. Ex. Find the value of x. Then classify the triangle by its angles. Then classify the triangle by its angles. x0 600500 x y (2x-18)0 720Ex. Find the angle measures of the numbered angles.1 2202 580 3200 4Ex. Find the values of x and y.0 y x0 680 Section 4.2: Apply Congruence and TrianglesStandards: 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.Congruent Figures – Congruence Statements – Ex. GIVEN: ABC  DEFB F D Corresponding Angles – A  B  C  A C E Corresponding Sides – AB  BC  CA  Ex. Write a congruency statement. Ex. ABCD  JKHL. Find the value of x and y.T 9 cm L J Q S A 0 0 B 91 113 4x–3 cmR 0 0 (5y–12) K D 86 CP H U***Theorem 4.3 – Third Angle TheoremB F D K Ex. Ex. G JA C E F I H ***Theorem 4.4 – Properties of Congruent TrianglesReflexive Property of Congruent Triangles –Symmetric Property of Congruent Triangles –Transitive Property of Congruent Triangles –Ex. Find the values of x and y.(8x  2y)0 (6x  y)0 290 1090 Section 4.3: Prove Triangles Congruent by SSSStandards: 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 –B E If Side AB  , Side BC  , and A D F C Side CA  , then   by .Ex. Is the congruence statement true? Ex. Is the congruence statement true? Explain your reasoning. Explain your reasoning. WXY  YZW KJL  MJLX K LY W J M Z Ex. Write a proof. B Given: AD  CD , AB  CB Prove: ABD  CBDA D CStructural Support – Ex. Determine whether the figure is stable. Explain your answer. a) b) c) Section 4.4: Prove Triangles Congruent by SAS and HLStandards: 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 –B E If Side AB  , Angle A  , and A D F C Side AC  , then   by .Ex. Do you have enough information to prove the triangles are congruent by SAS? a) b) A D Ex. Write a proof. C Given: AC  EC , DC  BC Prove: ACB  ECD B EM Ex. Write a proof.Given: AB  PB , MB  AP Prove: MBA  MBP A B P (over) Right Triangles: Legs – Hypotenuse – ***Hypotenuse-Leg (HL) Congruence Theorem –A DC B F EB Ex. Write a proof.Given: AB  BC , BD  AC Prove: ABD  CBD A D C Section 4.5: Prove Triangles Congruent by ASA and AASStandards: 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 –B E If Angle A  , Side AC  , and A C D F Angle C  , then   by .***Angle-Angle-Side (AAS) Congruence Postulate –B E If Angle A  , Angle C  , and A C D 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) Ex. Write a proof. XGiven: WZ bisects XZY and XWY Z W Prove: WZX  WZYYEx. Write a proof. B C Given: C  B , D  F , M is the midpoint of DF Prove: BDM  CFM D M F Section 4.6: Use Congruent TrianglesStandards: 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. 2. The triangles below are congruent by SAS. Since the triangles are congruent, we know that: A X A   C   B C Y Z AC  Ex. Write a proof. H JGiven: HJ II LK , JK II HL Prove: LHJ  JKL L KEx. Write a proof. M R Given: MS II TR , MS  TR A Prove: A is the midpoint of MT S T P Ex. Write a proof.Given: MP bisects LMN , LM  NM Prove: LP  NPN L MEx. Which triangles can you show are congruent in order to prove the statement? What postulate or theorem would you use? a) A  C b) SW  TY B SX Y W A D C TSection 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.Legs – Vertex Angle – Base – Base Angles – ***Theorem 4.7 – Base Angles Theorem***Theorem 4.8 – Converse of Base Angles TheoremCorollary To Base Angles TheoremCorollary to the Converse of Base Angles Theorem Section 4.8: Perform Congruence TransformationsStandards: 22.0 Students know the effect of rigid motions on figures in the coordinate plane and space, including rotations, translations, and reflections.Transformation – Image – *Three Types of Transformations: Translation – Reflection – Rotation – Congruence Transformation – Translate A Figure In The Coordinate Plane – *Coordinate Notation for a Translation 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 unitsEx. Figure ABCD has the vertices A(4,2), B(2,5), C(1,1), and D(3,1) . Sketch ABCD and its image after the translation (x, y)  (x  5, y  2) . 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 ReflectionReflection 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. xEx. 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.90 0 clockwise rotation 60 0 counterclockwise rotation y Ex. Graph PQ and RS . Tell whether RS is a rotation 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) x b) P(4,2), Q(3,3), R(2,4), S(3,3)

Geometry, Definitions, Postulates, And Theorems

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