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Geometry Fall Final Review

Geometry Fall Final Review. 2013-2014. Foundations of Geometry. Types of Geometries Euclidean (studied in this course) Study of flat surfaces The shortest distance between two points is one line The sum of the angles in any triangle equals 180 degrees

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Geometry Fall Final Review

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  1. Geometry Fall Final Review 2013-2014

  2. Foundations of Geometry • Types of Geometries • Euclidean (studied in this course) • Study of flat surfaces • The shortest distance between two points is one line • The sum of the angles in any triangle equals 180 degrees • Euclid’s 5th postulate – the “Parallel Postulate” • Non-Euclidean • Spherical • Study of spherical or curved surfaces • The sum of the angles in any triangle does not have to equal 180 degrees

  3. Foundations of Geometry • Points, Lines, and Planes • Points are most basic figures in geometry • 2 points create a line • 2 lines intersect in a point • 3 points form a plane • Intersection of two planes is a line • Essential Vocabulary • Collinear: points that lie on the same line • Coplanar: points and/or lines that lie in the same plane

  4. Foundations of Geometry • Midpoint • The point that divides a segment into two congruent segments • Formula: • Practice: • Find the midpoint B of a segment with endpoints A (-1, 5) and C (6, -7). A B C

  5. Foundations of Geometry • Distance • The distance between two points in coordinate geometry can be found using two methods: • Distance Formula • Pythagorean Theorem • Practice • Find the distance between A (0, -4) and C (-5, 9) using the distance formula

  6. Foundations of Geometry • Distance • Practice • Find the distance between X (-1, 0) and Y (3, 3) using the Pythagorean Theorem and the coordinate grid below. Spiral Review: What do we call this kind of right triangle?

  7. Foundations of Geometry, Part 2 • Writing Equations Review • Recall that 2 points form a line • Slope is the rise over the run, or you can use the equation: • Once you have found slope, you plug the slop and a point into point-slope form: Remember these do NOT change!

  8. Foundations of Geometry, Part 2 • Practice • Find the equation of the line between the following points: • K (-4, 0) and L (0, 2) • T (2, 4) and S (4 ,-4) Remember these do NOT change!

  9. Foundations of Geometry, Part 2 • Parallel & Perpendicular Lines • Parallel Lines • Never intersect in a plane • Lines have the same slopes • Perpendicular Lines • Intersect in a plane at a 90° angle • Lines have negative reciprocal slopes • Practice • Determine if the lines are parallel or perpendicular and justify your answer with a short sentence:

  10. Foundations of Geometry, Part 2 • Parallel & Perpendicular Lines • Practice • Determine if the lines are parallel, perpendicular, or neither and justify your answer with a short sentence: • 1. • 2. • 3.

  11. Logic and Reasoning • Deductive vs. Inductive Reasoning • Deductive Reasoning: the process of using logic to draw conclusions • Inductive Reasoning: the process of reasoning based on patterns and observations • Examples • There is a myth that the lunar eclipse can only be seen from CRHS on October 18th. Sarah sees the lunar eclipse from the top of Mount Bonnell on October 18th and concludes that the myth is false. • Mrs. Overman visits the Austin Zoo’s lion’s den. She notices that each lion has a mane, if it is a male lion. Mrs. Overman concludes that all male lions in Zimbabwe will have manes.

  12. Logic and Reasoning • Conditional Statements • If ____P______, then ______Q______. • Example: • If Mrs. Navin does not come to school, then she will have a substitute. • Converse, Inverse, and Contrapositive • Converse: If ___Q____, then _____P_____. • Example: If Mrs. Navin has a substitute, then she does not come to school. • Inverse: If ___not P____, then _____not Q_____. • Example: If Mrs. Navin does come to school, then she will not have a substitute. • Contrapositive: If ___not Q____, then _____not P_____. • Example: If Mrs. Navin does not have a substitute, then she will come to school.

  13. Logic and Reasoning • Truth Value • Remember: not all statements are true • Conditional will have same truth value as contrapositive • Converse and Inverse will have the same truth value • Practice • Given the conditional, write the converse, inverse, and contrapositive and determine the truth value of each statement. • Conditional: If a mammal has hair, then it is a human.

  14. Logic and Reasoning • Algebraic Expressions • Inductive reasoning can be used to find patterns and formulate expressions for those patterns • Example • Given the table, find the missing terms and the nth term: • Hint: If a function is not linear, it may be quadratic (i.e. n2)

  15. Parallel Lines & Angle Pairs • The Basics Complementary angles: angles that have a sum of 90° Supplementary angles: angles that have a sum of 180°

  16. Parallel Lines & Angle Pairs • The Basics • Linear pair: angles that are adjacent and supplementary • Vertical angles: the nonadjacent angles formed by two intersecting lines

  17. Parallel Lines & Angle Pairs • Transversal • A line that interesects two coplanar lines at two different points • Creates angles that have relationships

  18. Parallel Lines & Angle Pairs • Angle Relationships • Corresponding • Alternate Interior • Alternate Exterior • Same side Interior • Same side Exterior • Parallel Lines cut by a Transversal • Which angles are congruent? • Which angles are supplementary?

  19. Proving Lines Parallel • Complete a two-column proof given the following information: Given: Angle 1 is congruent to Angle 8. Prove: Line AB is parallel to line CD.

  20. Proving Lines Parallel • Using the diagram below, can two lines be proven parallel given the following information? Justify your reasoning. 1. Angle 5 is congruent to Angle 8. 2. Angle 4 is congruent to Angle 5. 3. Angle 1 is supplementary to Angle 6. 4. Angle 2 and Angle 8 are supplementary.

  21. Properties of Triangles • Classifying Triangles • By Sides • Equilateral • Isosceles • Scalene • What are the similarities and differences between equilateral and isosceles triangles? • By Angles • Acute • Right • Obtuse • Equiangular • Is there a special relationship between equilateral and equiangular triangles?

  22. Properties of Triangles • Important Theorems • Triangle Sum Theorem • The sum of the three angles of a triangle is 180° • Exterior Angle Theorem • The sum of the remote interior angles is equal to the measure of the exterior angle

  23. Properties of Triangles • Practice • Triangle Sum Theorem • Given the two angles of a triangle, classify the triangle by angles and find the third angle. • 54° and 67° • 90° and 45°

  24. Properties of Triangles • Important Theorems • Pythagorean Theorem • Only applies to right triangles • “C” is always the hypotenuse, which is across from the right angle • Practice • Find the missing sides, rounding to the nearest tenth if necessary: 7 9.8 3.6 24

  25. Properties of Triangles • Pythagorean Triples • What are the common triples? • 3, 4, ___ • 5, 12, ___ • 8, 15, ___ • 7, 24, ___ • Find the missing sides using Pythagorean Triples: 17 8 13 5

  26. Properties of Triangles • More Important Theorems • Triangle Inequality Theorem • The sum of two sides of a triangle must be greater than the third side • Converse of the Pythagorean Theorem • The sum of the two legs squared is equal to/less than/greater than the longest side squared • Practice • Determine if the lengths form a triangle. If so, classify the triangle as acute, obtuse, or right. • 4, 4, 7 • 5, 8, 9 • 2, 3, 6

  27. Properties of Triangles • Scalene Triangles • We can order the sides and order the angles in scalene triangles using inequalities • Remember: the smallest side is across from the smallest angle, and the largest side is across from the largest angle • The converse is true, also! • Practice • Order the sides or angles from smallest to largest, solving for missing angles if necessary: 29 C X Z 21 18 A 62° 48° B Y

  28. Triangle Congruence • Triangle Congruence Theorems • Side-side-side (SSS) • Three sides of one triangle are congruent to three sides of another triangle

  29. Triangle Congruence • Triangle Congruence Theorems • Side-angle-side (SAS) • Two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle

  30. Triangle Congruence • Triangle Congruence Theorems • Angle-side-angle (ASA) • Two angles and the included side of one triangle are congruent to two angles and the included side of another triangle

  31. Triangle Congruence • Triangle Congruence Theorems • Angle-angle-side (AAS) • Two angles and the nonincluded side of one triangle are congruent to two angles and the nonincluded side of another triangle

  32. Triangle Congruence • Triangle Congruence Theorems • Hypotenuse-Leg (HL) • The hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of another right triangle

  33. Triangle Congruence and CPCTC • Once we have proved two triangles congruent, what comes next? • CPCTC – Corresponding Parts of Congruent Triangles are Congruent

  34. Work Cited • Images courtesy of: • http://www.wyzant.com/Images • http://www.analyzemath.com/Geometry/ • Definitions and Postulates: • Holt McDougal Online • http://my.hrw.com

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