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A point has no dimension.

A point has no dimension.

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A point has no dimension.

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  1. USING UNDEFINED TERMS AND DEFINITIONS Point A A A point has no dimension. It is usually represented by a small dot. •

  2. USING UNDEFINED TERMS AND DEFINITIONS Line or AB A line extends in one dimension. It is usually represented by a straight line with two arrowheads to indicate that the line extends without end in two directions. Collinear points are points that lie on the same line.

  3. USING UNDEFINED TERMS AND DEFINITIONS A plane extends in two dimensions. It is usually represented by a shape that looks like a table or a wall, however you must imagine that the plane extends without end.

  4. USING UNDEFINED TERMS AND DEFINITIONS A C B M Coplanar points are points that lie on the same plane. Plane M or plane ABC

  5. Naming Collinear and Coplanar Points H G F E D • Name three points that are collinear. • Name four points that are coplanar. • Name three points that are not collinear. SOLUTION • Points D, E, F lie on the same line, so they are collinear. • Points D, E, F, and G lie on the same plane, so they are coplanar. Also, D, E, F, and H are coplanar. • There are many correct answers. For instance, points H, E, and G do not lie on the same line.

  6. USING UNDEFINED TERMS AND DEFINITIONS Another undefined concept in geometry is the idea thata point on a line is between two other points on the line. You can use this idea to define other important termsin geometry.

  7. USING UNDEFINED TERMS AND DEFINITIONS Consider the lineAB (symbolized by AB). The line segment or segmentAB (symbolized by AB) consists of the endpointsA and B, and all points on AB that are between A and B.

  8. USING UNDEFINED TERMS AND DEFINITIONS Note that AB is the same as BA, and ABis the same as BA. However, AB and BA are not the same. They have different initial points and extend in different directions. The rayAB (symbolized by AB) consists of the initial pointA and all points on AB that lie on the same side of A as point B.

  9. USING UNDEFINED TERMS AND DEFINITIONS If C is between A and B, then CA and CB are opposite rays. Like points, segments and rays are collinear if they lie on the same plane. So, any two opposite rays are collinear. Segments, rays, and lines are coplanar if they lie on the same plane.

  10. Drawing Lines, Segments, and Rays 1 2 3 4 K Draw JK. J Draw KL. L Draw LJ. Draw three noncollinear points J, K, L. Then draw JK, KL and LJ. SOLUTION Draw J, K, and L

  11. Drawing Opposite Rays So, XM and XN are opposite rays. So, XP and XQ are opposite rays. Draw two lines. Label points on the lines and name two pairs of opposite rays. SOLUTION Points M, N, and X are collinear and X is between M and N. Points P, Q, and X are collinear and X is between P and Q.

  12. SKETCHING INTERSECTIONS OF LINES AND PLANES Two or more geometric figures intersect if they have one or more points in common. The intersection of the figures is the set of points the figures have in common.

  13. Sketching Intersections Sketch a line that intersects a plane at one point. SOLUTION Draw a plane and a line. Emphasize the point where they meet. Dashes indicate where the line is hidden by the plane.

  14. Sketching Intersections Sketch two planes that intersect in a line. SOLUTION Draw two planes. Emphasize the line where they meet. Dashes indicate where one plane is hidden by the other plane

  15. CONGRUENCE OF ANGLES For any angle A, A  A REFLEXIVE If A B, then BA SYMMETRIC If A  B and B  C, then A  C TRANSITIVE THEOREM THEOREM 2.2Properties of Angle Congruence Angle congruence is reflexive, symmetric, and transitive. Here are some examples.

  16. Transitive Property of Angle Congruence C B AB, AC GIVEN PROVE A BC Prove the Transitive Property of Congruence for angles. SOLUTION To prove the Transitive Property of Congruence for angles, begin by drawing three congruent angles. Label the vertices as A, B, and C.

  17. Transitive Property of Angle Congruence 5 4 3 1 2 Statements Reasons A  B, Given AB, AC GIVEN PROVE B  C BC mA = m B Definition of congruent angles mB = m C Definition of congruent angles mA = m C Transitive property of equality A CDefinition of congruent angles

  18. Using the Transitive Property m 3 = 40°, 12, 23 GIVEN m 1 = 40° PROVE 4 1 2 3 Statements Reasons m 3 = 40°, 1 2, Given 2 3 1 3Transitive property of Congruence m 1 = m 3 Definition of congruent angles m 1 = 40° Substitution property of equality This two-column proof uses the Transitive Property.

  19. Proving Theorem 2.3 1 and 2 are right angles GIVEN 1 2 PROVE THEOREM THEOREM 2.3Right Angle Congruence Theorem All right angles are congruent. You can prove Theorem 2.3 as shown.

  20. Proving Theorem 2.3 1 2 3 4 1 and 2 are right angles GIVEN Statements Reasons 1 2 PROVE 1 and 2 are right anglesGiven m 1 = 90°, m 2 = 90° Definition of right angles m 1 = m 2 Transitive property of equality 1  2Definition of congruent angles

  21. PROPERTIES OF SPECIAL PAIRS OF ANGLES THEOREMS THEOREM 2.4Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles) then they are congruent. 2 1 3

  22. 1 2 3 1 and 3 If m 1 + m 2 = 180° m 2 + m 3 = 180° 1  3 PROPERTIES OF SPECIAL PAIRS OF ANGLES THEOREMS THEOREM 2.4Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles) then they are congruent. then

  23. PROPERTIES OF SPECIAL PAIRS OF ANGLES THEOREMS THEOREM 2.5Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent. 5 6 4

  24. 5 6 4 6 4 and If m 4 + m 5 = 90° m 5 + m 6 = 90° 4  6 PROPERTIES OF SPECIAL PAIRS OF ANGLES THEOREMS THEOREM 2.5Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent. then

  25. Proving Theorem 2.4 1 2 Statements Reasons 1 and 2 are supplements Given 3 and 4 are supplements 1 4 m 1 + m 2 = 180° Definition of supplementary angles m 3 + m 4 = 180° 1 and 2 are supplements GIVEN 3 and 4 are supplements 1 4 2 3 PROVE

  26. Proving Theorem 2.4 3 5 4 m 1 + m 2 = Transitive property of equality m 3 + m 1 m 3 + m 4 m 1 = m 4 Definition of congruent angles m 1 + m 2 = Substitution property of equality 1 and 2 are supplements GIVEN 3 and 4 are supplements 1 4 2 3 PROVE Statements Reasons

  27. Proving Theorem 2.4 6 7 m 2 = m 3 Subtraction property of equality 2 3Definition of congruent angles 1 and 2 are supplements GIVEN 3 and 4 are supplements 1 4 2 3 PROVE Statements Reasons

  28. m 1 + m 2 = 180° PROPERTIES OF SPECIAL PAIRS OF ANGLES POSTULATE POSTULATE 12Linear Pair Postulate If two angles form a linear pair, then they are supplementary.

  29. Proving Theorem 2.6 THEOREM THEOREM 2.6Vertical Angles Theorem Vertical angles are congruent 1 3, 2 4

  30. Proving Theorem 2.6 5 and 6 are a linear pair, GIVEN 6 and 7 are a linear pair 5 7 1 2 3 PROVE Statements Reasons 5 and 6 are a linear pair,Given 6 and 7 are a linear pair 5 and 6 are supplementary,Linear Pair Postulate 6 and 7 are supplementary 5 7 Congruent Supplements Theorem

  31. PROPERTIES OF PARALLEL LINES 1 2 POSTULATE POSTULATE 15Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 1 2

  32. PROPERTIES OF PARALLEL LINES 3 4 THEOREMS ABOUT PARALLEL LINES THEOREM 3.4Alternate Interior Angles If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. 3 4

  33. PROPERTIES OF PARALLEL LINES m 5 +m 6 = 180° THEOREMS ABOUT PARALLEL LINES THEOREM 3.5Consecutive Interior Angles If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. 5 6

  34. PROPERTIES OF PARALLEL LINES 7 8 THEOREMS ABOUT PARALLEL LINES THEOREM 3.6Alternate Exterior Angles If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. 7 8

  35. PROPERTIES OF PARALLEL LINES jk THEOREMS ABOUT PARALLEL LINES THEOREM 3.7Perpendicular Transversal If a transversal is perpendicular to one of two parallellines, then it is perpendicular to the other.

  36. Proving the Alternate Interior Angles Theorem 4 3 2 1 p || q GIVEN 1 2 PROVE Statements Reasons 1  3Corresponding Angles Postulate 3  2Vertical Angles Theorem 1  2Transitive property of Congruence Prove the Alternate Interior Angles Theorem. SOLUTION p || q Given

  37. Using Properties of Parallel Lines m 6 = m 5 = 65° Vertical Angles Theorem m 7 =180° – m 5 =115° Linear Pair Postulate m 8 = m 5 = 65° Corresponding Angles Postulate m 9 = m 7 = 115° Alternate Exterior Angles Theorem Given that m 5 = 65°, find each measure. Tell which postulate or theorem you use. SOLUTION

  38. Using Properties of Parallel Lines m 4 =125° Corresponding Angles Postulate m 4 + (x + 15)° =180° Linear Pair Postulate 125° + (x + 15)° =180° Substitute. x =40° Subtract. PROPERTIES OF SPECIAL PAIRS OF ANGLES Use properties of parallel lines to findthe value of x. SOLUTION

  39. Estimating Earth’s Circumference: History Connection 1 m 2 of a circle 50 Over 2000 years ago Eratosthenes estimated Earth’s circumference by using the fact that the Sun’s rays are parallel. When the Sun shone exactly down a vertical well in Syene, he measured the angle the Sun’s rays made with a vertical stick in Alexandria. He discovered that

  40. Estimating Earth’s Circumference: History Connection m 1 = m 2 Using properties of parallel lines, he knew that He reasoned that 1 1 m 2 m 1 of a circle of a circle 50 50

  41. Estimating Earth’s Circumference: History Connection 575 miles of a circle Earth’s circumference Earth’s circumference 50(575 miles) Use cross product property 29,000 miles 1 1 m 1 of a circle 50 50 How did Eratosthenes know that m 1 = m 2 ? The distance from Syene to Alexandria was believed to be 575 miles

  42. Estimating Earth’s Circumference: History Connection Because the Sun’s rays are parallel, Angles 1 and 2 are alternate interior angles, so 1  2 By the definition of congruent angles, How did Eratosthenes know that m 1 = m 2 ? m 1 = m 2 SOLUTION

  43. SSS AND SASCONGRUENCE POSTULATES then If 1.ABDE 4.AD 2.BCEF 5. BE ABCDEF 3.ACDF 6.CF If all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. Sides are congruent Angles are congruent Triangles are congruent and

  44. SSS AND SASCONGRUENCE POSTULATES S S S Side MNQR then MNPQRS Side NPRS Side PMSQ POSTULATE POSTULATE 19Side -Side -Side (SSS) Congruence Postulate If three sides of one triangle are congruent to three sidesof a second triangle, then the two triangles are congruent. If

  45. SSS AND SASCONGRUENCE POSTULATES The SSS Congruence Postulate is a shortcut for provingtwo triangles are congruent without using all six pairsof corresponding parts.

  46. Using the SSS Congruence Postulate Prove that PQWTSW. The marks on the diagram show that PQTS, PWTW, andQWSW. SOLUTION Paragraph Proof So by the SSS Congruence Postulate, you know that PQW TSW.

  47. SSS AND SASCONGRUENCE POSTULATES POSTULATE Side PQWX A S S then PQSWXY Angle QX Side QSXY POSTULATE 20Side-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. If

  48. Using the SAS Congruence Postulate Prove that AEBDEC. 1 2 1 2 Statements Reasons AE  DE, BE  CE Given 1  2Vertical Angles Theorem 3 AEBDEC SAS Congruence Postulate

  49. Proving Triangles Congruent ARCHITECTURE You are designing the window shown in the drawing. You want to make DRAcongruent to DRG. You design the window so that DRAG and RARG. D A G R GIVEN DRAG RARG DRADRG PROVE MODELING A REAL-LIFE SITUATION Can you conclude that DRADRG? SOLUTION

  50. Proving Triangles Congruent GIVEN RARG DRADRG PROVE 1 2 6 3 4 5 Statements Reasons Given DRAG If 2 lines are , then they form 4 right angles. DRA and DRG are right angles. Right Angle Congruence Theorem DRADRG DRAG Given RARG DRDR Reflexive Property of Congruence SAS Congruence Postulate DRADRG D A R G