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Midterm Review

Midterm Review. Objective. SWBAT to make connections to material from Unit 1 – Unit 5. What did we cover so far?. Unit 1: 1.2, 1.3, 1.4, 1.5, 1.7, 2.6, 3.1 – 3.4, 1.6, 3.6 Unit 2: 9.1 – 9.6 Unit 3: 3.5, 4.1 - 4.3, 4.5, 4.6, 5.1, 5.4 Unit 4: 6.1 - 6.9 Unit 5: 7.1 - 7.5. Unit 1. 1.2

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Midterm Review

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  1. Midterm Review

  2. Objective SWBAT to make connections to material from Unit 1 – Unit 5.

  3. What did we cover so far? Unit 1: 1.2, 1.3, 1.4, 1.5, 1.7, 2.6, 3.1 – 3.4, 1.6, 3.6 Unit 2: 9.1 – 9.6 Unit 3: 3.5, 4.1 - 4.3, 4.5, 4.6, 5.1, 5.4 Unit 4: 6.1 - 6.9 Unit 5: 7.1 - 7.5

  4. Unit 1 1.2 1.3 1.4 1.5 1.7 2.6 3.1 – 3.4 1.6 3.6

  5. 1.2 Points, Lines, and Planes A point indicates a location and has no size. A line is represented by a straight path that extends in two opposite directions without end and has no thickness. A plane is represented by a flat surface that extends without end and has no thickness. Points that lie on the same line are collinear points. Points and lines in the same plane are coplanar. Segments and rays are part of lines.

  6. 1.2 Example Name all the segments and rays in the figure. D A B C

  7. 1.2 Example Name all the segments and rays in the figure. Segments: AB, AC, BC, and BD Rays: BA, CA, CB, AC, AB, BC, and BD D A B C

  8. 1.3 Measuring Segments The distance between two points is the length of the segment connecting those points. Segments with the same length are congruent segments. A midpoint of a segment divides the segment into two congruent segments.

  9. 1.4 Measuring Angles Two rays with the same endpoint form an angle. The endpoint is the vertex of the angle. You can classify angles as acute, right, obtuse, or straight. Angles with the same measure are congruent angles.

  10. 1-4 Example If m<AOB = 47 and m<BOC = 73 find m<AOC m<AOC = m<AOB + m<BOC = 47+73 =120 B A C O

  11. 1.5 Exploring Angle Pairs Some pairs of angles have special names. • Adjacent angles: coplanar angles with a common side, a common vertex, and no common interior points. • Vertical angles: sides are opposite rays • Complementary angles: measures have a sum of 90 • Supplementary angles: measures have a sum of 180 • Linear Pairs: adjacent angles with non common sides as opposite rays. Angles of a linear pair are supplementary.

  12. 2.6 Proving Angles Congruent A statement that you prove true is a theorem. A proof written as a paragraph is a paragraph proof. In geometry, each statement in a proof is justified by given information, a property, postulate, definition, or theorem.

  13. 2.6 Example Write a paragraph proof. Given <1≈<4. Prove <2≈<3 2 3 1 4

  14. 2.6 Example Write a paragraph proof. Given <1≈<4. Prove <2≈<3 <1≈<4 because it is given. <1≈<2 because vertical angles are congruent. <4 ≈ <2 by the Transitive Property of Congruence. <4≈<3 because vertical angles are congruent. <2 ≈ <3 by the Transitive Property of Congruence. 2 3 1 4

  15. 3.1 Lines and Angles A transversal is a line that intersects two or more coplanar lines a distinct points.

  16. 3.2 Properties of Parallel Lines If two parallel lines are cut by a transversal, then • Corresponding angles, alternate interior angles, and alternate exterior angles are congruent. • Same-side interior angles are supplementary

  17. 3.2 Properties of Parallel Lines Which other angles measure 110? <6 (corresponding angles) <3 (alternate interior angles) <8 (vertical angles)

  18. 3.3 Proving Lines Parallel If two lines and a transversal form • Congruent corresponding angles • Congruent alternate interior angles • Congruent alternate exterior angles, or • Supplementary same-side interior angles, then the two lines are parallel.

  19. 3.3 Proving Lines Parallel What is the value of x for which l || m ? The given angles are alternate interior angles. So l || m if the given angles are congruent. 2x = 106 x = 53

  20. 3.4 Parallel and Perpendicular Lines • Two lines || to the same line are || to each other. • In a plane, two lines _l_ to the same line are ||. • In a plane, if one line is _l_ to one of two || lines, then it is _l_ to both || lines

  21. 3.4 Parallel and Perpendicular Lines What are the pairs of parallel and perpendicular lines in the diagram?

  22. 1.6 Basic Constructions Construction is the process of making geometric figures using a compass and a straightedge. Four basic constructions involve congruent segments, congruent angles, and bisectors of segments and angles.

  23. 3.6 Constructing Parallel and Perpendicular Lines You can use a compass and a straight edge to construct • A line parallel to a given line through a point not on the line • A line perpendicular to given line through a point on the line, or through a point not on the line

  24. Unit 2 Sections Covered were: 9.1 - 9.6

  25. 9.1 Translation A transformation of a geometric figure is a change in its position, shape or size. An isometry is a transformation in which the preimage and the image are congruent. A translation is an isometry that maps all points of a figure the same distance in the same direction. In a composition of transformations, each transformation is performed on the image of the preceding transformation.

  26. 9.1 Translation What are the coordinates of the image of A(5,-9) for the translation (x,y) (x-2, y+3) ? Substitute 5 for x and -9 for y in the rule. A(5,-9)  (5 -2, -9+3), or A’(3,-6)

  27. 9.2 and 9.3 Reflections and Rotations The diagram shows a reflection across line r. A reflection is an isometry in which a figure and its image have opposite orientations.

  28. 9.2 and 9.3 Reflections and Rotations The diagram shows a rotation of x⁰ about point R. A rotation is an isometry in which a figure and its image have the same orientation.

  29. 9.2 and 9.3 Reflections and Rotations Use points P(1,0), Q(3,-2), and R(4,0). What is the image of PQR reflected across the y-axis?

  30. 9.2 and 9.3 Reflections and Rotations Graph PQR. Find P’, Q’, and R’ such that the y-axis is the perpendicular bisector of PP’, QQ’, and RR’. Draw P’Q’R’.

  31. 9.4 Symmetry A figure has reflection symmetry or line symmetry if there is a reflection for which it is its own image. A figure that has rotational symmetry is its own image for some rotation of 180˚ or less. A figure that has point symmetry has 180˚ rotational symmetry.

  32. 9.4 Symmetry How many lines of symmetry does an equilateral triangle have? An equilateral triangle reflects onto itself across each of its three medians. The triangle has three lines of symmetry.

  33. 9.5 Dilations The diagram shows a dilation with center C and scale factor n. The preimage and image are similar. In the coordinate plane, if the origin is the center of dilation with scale factor n, the P(x,y)  P’(nx, ny).

  34. 9.5 Dilations The blue figure is a dilation image of the black figure. The center of dilation is A. Is the dilation an enlargement or reduction? What is the scale factor? The image is smaller than the preimage, so the dilation is a reduction. The scale factor is: image length = 2 = 2, or 1/3 original length 2 + 4 6

  35. 9.6 Compositions of Reflections The diagram shoes a glide reflection of N. A glide reflection is an isometry in which a figure and its image have opposite orientations. There are exactly four isometries: translation, reflection, rotation, and glide reflection. Every isometry can be expressed as a composition of reflections.

  36. 9.6 Compositions of Reflections Describe the result of reflecting P first across line l and then across line m. A composition of two reflections across intersecting lines is a rotation. The angle of rotation is twice the measure of the acute angle formed by the intersecting lines. P is rotated 100˚ about C.

  37. Home Work Review Unit 3 – Unit 5 Content for Part 2 Review What you do not finish for Class Work 

  38. Class Work 1.5 pg. 21 1-7 1.7 pg. 29 1 -8 2.6 pg. 57 1 – 7 3.1 pg. 61 1 – 8 3.2 pg. 65 1 – 7 9.1 pg. 225 1 – 5 9.2 pg. 229 1-6 9.6 pg. 245 1-6

  39. Unit 3 The Sections Covered were: 3.5 4.1 – 4.3 4.5 4.6 5.1 5.4

  40. 3.5 Parallel Lines and Triangles The sum of the measures of the angles of triangle is 180. The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles.

  41. 3.5 Parallel Lines and Triangles What are the values of x and y? x + 50 = 125 Exterior Angle Theorem x = 75 Simplify x + y + 50 = 180 Triangle Angle Sum Thm 75 + y + 50 = 180 Substitute 75 for x. y = 55 Simplify

  42. 4.1 Congruent Figures Congruent polygons have congruent corresponding parts. When you name congruent polygons, always list corresponding vertices in the same order.

  43. 4.1 Congruent Figures HIJK ≈ PQRS. Write all possible congruence statements. The order of the parts in the congruence statement tells you which parts corresponds. Sides: HI ≈ PQ, IJ≈QR, JK≈RS, KH≈SP Angles: <H≈<P, <I≈<Q, <J≈<R, <K≈<S

  44. 4.2 and 4.3 Triangle Congruence by SSS, SAS, ASA, and AAS You can prove triangles congruent with limited information about their congruent sides and angles.

  45. 4.2 and 4.3 Triangle Congruence by SSS, SAS, ASA, and AAS What postulate would you use to prove the triangles congruent? You know that three sides are congruent. Use SSS.

  46. 4.5 Isosceles and Equilateral Triangles If two sides of a triangle are congruent, then the angles opposite those sides are also congruent by the Isosceles Triangle Theorem. If two angles of a triangle are congruent, then the sides opposite the angle are congruent by the Converse of the Isosceles Triangle Theorem Equilateral triangles are also equiangular.

  47. 4.5 Isosceles and Equilateral Triangles What is the m<G? Since EF ≈ EG, <F≈<G by the Isosceles Triangle Theorem. So m<G = 30.

  48. 4.6 Congruence in Right Triangles If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangle are congruent by the Hypotenuse-Leg (HL) Theorem.

  49. 4.6 Congruence in Right Triangles Which two triangles are congruent? Explain. Since triangle ABC and triangle XYZ are right triangles with congruent legs, and BC ≈ YZ, triangle ABC ≈ triangle XYZ by HL

  50. 5.1 Midsegments of Triangles A midsegment of a triangle is a segment that connects the midpoints of two sides. A midsegment is parallel to the third side and is half as long.

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