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Geometry Final Review

Geometry Final Review. Chapter 1 Tools of Geometry. Collinear & Coplanar. Collinear Two or more points on the same line Coplanar Two or more points, lines, rays or angles on the same plane. A. B. C. Naming the basics of geometry. Line Name with any two points on the line. Example:

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Geometry Final Review

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  1. GeometryFinal Review Chapter 1 Tools of Geometry

  2. Collinear & Coplanar Collinear • Two or more points on the same line Coplanar • Two or more points, lines, rays or angles on the same plane

  3. A B C Naming the basics of geometry Line • Name with any two points on the line. Example: Name the line.

  4. Naming the basics of geometry Line Segment • Name with the endpoints of the line segment. Example: Name the line segment. A B C

  5. Naming the basics of geometry Angle • Name by three points on the angle with the vertex being the middle letter Example: Name angle 1.

  6. Naming the basics of geometry Ray • Name with the starting point and any other point on the ray in the direction of the ray. Example: Name the ray. A B C

  7. Distance & Midpoint Distance d= Midpoint Example: Find the midpoint and distance of the line that passes through (-4, 3), (7, 2).

  8. GeometryFinal Review Chapter 2 Reasoning and Proof

  9. Sequences and Patterns Examples: Describe a pattern in the sequence of numbers. Predict the next three numbers. 3, 7, 11, 15, 19, … 4, 7, 14, 17, 34, … Sketch the next figure.

  10. Inverse, Converse, and Contrapositive Conditional Statement – Original If-Then Statement Inverse – Negation of the conditional Converse – Switch of the conditional Contrapositive – Negation and Switch of the conditional

  11. Example Write the inverse, converse, and contrapositive of the given conditional statement. If your shoes are not tied, then you will trip. Inverse: If your shoes are tied, then you will not trip. Converse: If you trip, then your shoes were not tied. Contrapositive: If you do not trip, then your shoes were tied.

  12. Algebraic Proof Reasons Addition Property: a + b = c + b Subtraction Property: a – b = c – b Multiplication Property: ab = cb Division Property: a/b = c/b if b ≠ 0 Reflexive Property: a = a Symetric Property: If a = b then b = a Transitive Property: If a = b, b = c, then a = c Substitution Property: If a = 3, a + b = 10, then 3+b = 10. Distributive Property: a (b + c) = ab + ac

  13. Algebraic Proofs Solve the equation state a reason for each step. 3(x – 9) = 2x + 7

  14. Geometric Proofs - Segments Reflexive: AB = AB Symmetric: AB = BA Transitive: If AB = BC, BC = CD, then AB = CD Segment Addition: AB + BC = AC Midpoint: If A is the midpoint of MN, then MA ≌ NA

  15. Geometric Proofs - Segments Use the diagram and the given information to complete the missing steps and reasons in the proof. Given: LK = 5, JK = 5, JK ≌ JL Prove: LK ≌ JL K J L

  16. Geometric Proofs - Angles Reflexive: ∠A= ∠A Symmetric: ∠A = ∠B, then ∠B = ∠A Transitive: If ∠A = ∠B, ∠B = ∠C then ∠A = ∠C Angle Addition: ∠ABC + ∠CBD = ∠ABD Angle Bisector: If AB bisects ∠CAD then ∠CAB ≌∠BAD

  17. Geometric Proofs - Angles 1 4 2 3 Use the diagram and the given information to complete the missing steps and reasons in the proof. Given: m ∠3 = 40, ∠1≌ ∠2, ∠2≌ ∠3 Prove: m ∠1 = 40

  18. GeometryFinal Review Chapter 3 Angles & Lines

  19. Angles by size Acute – angle less than 90o Right – angle equal to 900 Obtuse – angle greater than 900 but less than 1800 Straight – angle equal to 1800

  20. Angles Relationships Vertical - Angles created by the intersection of two lines • Across the intersection from each other • Their measures are congruent Linear - two angles that combine to make a line • Sum of their angles is 1800 Complementary • Sum of two angles is 90o Supplementary • Sum of two angles is 180o

  21. Parallel Lines Cut by a Transversal Corresponding Angles • Same Spot, Different Tables • Angle measures congruent Alternate Exterior Angles • Opposite Sides, Outside the parallel lines • Angle measures congruent Alternate Interior Angles • Opposite Sides, Inside the parallel lines • Angle measures congruent Same - Side Interior Angles • Same side, Inside the parallel lines • Angle measures are supplementary

  22. Parallel Lines Proof Use the properties of parallel lines to find the value of x. State the property used for each step. Statements Reasons 146o (2x +5)o

  23. Lines Parallel Lines ∥ • Never Intersect • Go in the same direction • In the same plane Perpendicular Lines ⊥ • Intersect at a right angle • In the same plane Skew Lines • Never Intersect • Not parallel • In different planes

  24. GeometryFinal Review Chapter 4 Triangles

  25. Corresponding Parts • Parts in the same spot on different triangles. Example: Identify all of the corresponding angles and sides if DSQR≌ ΔVTH

  26. Filling in a Proof • State each GIVEN • State all conclusions • Sides • Share a side • Reflexive • Symmetric • If the given states midpoint • Angles • Vertical • Alternate Interior Angles (If parallel lines)

  27. Filling in a Proof • State why the triangles are congruent • Side-Side-Side (SSS) – 3 sides congruent • Side-Angle-Side (SAS) – 2 sides and the included angle • Angle-Side-Angle (ASA) – 2 angles and the included side • Angle-Angle-Side (AAS) – 2 angles and the adjacent side • Hypotenuse-Leg (HL) – hypotenuse and leg of a right triangle only

  28. Filling in a Proof continued • State the corresponding parts of congruent triangles are congruent (CPCTC) • State any further things needed • For example prove the lines parallel

  29. M N P O Proof Example Write a two column proof for the following proof. Given: Prove:

  30. GeometryFinal Review Chapter 6 Quadrilaterals

  31. Angles of Polygons Quadrilaterals • Sum of the angles is 360o Parallelogram • Opposite angles are congruent • Adjacent angles are supplementary Isosceles Trapezoid • Base angles are congruent • Leg angles are supplementary

  32. Angles of Polygons Sum of All Interior Angles • (n – 2) ∙ 180 Measure of Each Interior Angle • [(n – 2) ∙ 180]/n Sum of All Exterior Angles • 360 Measure of Each Exterior Angle • 360/n

  33. Polygon Names SidesName 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 11 Undecagon 12 Dodecagon

  34. Quadrilateral Names Quadrilaterals Trapezoid Parallellogram Kite Isosceles Rhombus Rectangle Trapezoid Square

  35. M N P O Proof Example Write a two column proof for the following proof. Given: MNOP is a parallelogram Prove:

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