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Geometry Mrs. Kapler Welcome Vitruvian Man Leonardo Geometry Euclidean Flat Earth Geometry • Points, Lines and Planes Geometry • Geos – earth • Metric – study of • Geometry - Based on a small set of intuitively appealing axioms, and deducing from them propositions/theorems with the use of undefined terms • Euclid – Greek Philosopher • Elements – Euclid’s book setting out Geometry system • Postulate = axiom - a logic statement that is assumed to be true. The truth is taken for granted. • Undefined terms: point, line, plane Geometry Undefined terms • Point 0 dimension • Line 1 dimension • Plane 2 dimension Euclid’s Postulates 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. Given a straight line and a point that does not lie on the line, one and only one straight line may be drawn that is parallel to the first line and passes through the point. Euclid’s Postulates Lines • Intersecting Lines • Perpendicular Lines • Parallel Lines • Skew Lines • Coinciding lines Non-Euclidean Geometry Logic - formal systematic study of the principles of valid inference and correct reasoning Geometry relies on a logical progression of thought to determine truth Logic Aristotle And the Chair What does your perfect chair look like? Draw Line Segment Endpoint Ray Opposite Ray Every point in 3-dimensional Euclidean space is determined by three coordinates. 1. 2. 3. 4. 5. The intersection of plane N and plane T. ____ A plane containing E, D, and B. ____ A point on BC. ____ Two opposite rays. ____ and ____ The intersection of and is a ___________. 6. If two points lie in a plane, then the line they determine lies in the plane. True or False 7. If three points lie in the same plane, they are _____________________. 8. If three points exist in the same line, they are _____________________. Construction – precise form of drawing; uses straightedge and compass Congruent Segments segments having the same length. PQ RS Tick marks – indicate congruency Tick Marks and Marks of Parallel Lines indicate Rhombi (Diamond) Homework 1.1 Geometry • Measuring and Constructing Segments Using the Segment Addition Postulate M is between N and O. Find NO. NM + MO = NO Seg. Add. Postulate 17 + (3x – 5) = 5x + 2 Substitute the given values 3x + 12 = 5x + 2 –2 Simplify. –2 Subtract 2 from both sides. 3x + 10 = 5x –3x Simplify. –3x Subtract 3x from both sides. 10 = 2x 2 5=x 2 Divide both sides by 2. Tell whether the statement below is sometimes, always, or never true. Support your answer with a sketch. If M is the midpoint of KL, then M, K, and L are collinear. K M L Always Draw • Two lines intersect. Label the point of intersection. • Two lines intersect at one point in a plane, but only one of the lines lies in the plane. Homework 1.2 Geometry • Measuring and Constructing Angles Angle An angle is a pair of rays that share a common endpoint. The rays are called the sides of the angle. The common endpoint is called the vertex of the angle. Angles ∠SRT What is the measure of angle "c"? Arc of a Circle L = length of the arc r = radius = theta, measure of angle Circumference = 2 r L circumference 2 Angle Measures Classification of Angles 180⁰ 285⁰ Acute Right Obtuse Straight Reflex Protractor – measures angles in degrees Measuring and Classifying Angles Find the measure of each angle. Then classify each as acute, right, or obtuse. A. WXV mWXV = 30° WXV is acute. B. ZXW mZXW = |130° - 30°| = 100° ZXW = is obtuse. Draw AB and AC, where A, B, and C are noncollinear. B A C Finding the Measure of an Angle Example 1 mXWZ = 121° and mXWY = 59°. Find mYWZ. mYWZ = mXWZ – mXWY Add. Post. mYWZ = 121 – 59 Substitute the given values. mYWZ = 62 Subtract. Example 2 Finding the Measure of an Angle KM bisects JKL, mJKM = (4x + 6)°, and mMKL = (7x – 12)°. Find mJKM. Finding the Measure of an Angle Example 2 Continued Step 1 Find x. mJKM = mMKL Def. of bisector (4x + 6)° = (7x – 12)° +12 +12 Substitute the given values. Add 12 to both sides. 4x + 18 –4x = 7x –4x 18 = 3x 6=x Simplify. Subtract 4x from both sides. Divide both sides by 3. Simplify. Finding the Measure of an Angle Continued Step 2 Find mJKM. mJKM = 4x + 6 = 4(6) + 6 Substitute 6 for x. = 30 Simplify. Example 2 Homework 1.3 Geometry • Angle Pairs Paired Identifying Angle Pairs AEB and BED Adjacent Angles Linear Pair DEC and AEB Not Adjacent Angles Complementary? Supplementary? Adjacent Linear Pair Example 1 Angle Pairs Find the measure of each of the following. a. complement of E (90 – x)° 90° – (7x – 12)° = 90° – 7x° + 12° = (102 – 7x)° b. supplement of F (180 – x) 180 – 116.5° = G F Paragraph Proof Assume the measure of Angle A = x. When two adjacent angles form a straight line, they are supplementary. Therefore, the measure of Angle C = 180 − x. Similarly, the measure of Angle D = 180 − x. Both Angle C and Angle D have measures equal to 180 - x and are congruent. Since Angle B is supplementary to both Angles C and D, either of these angle measures may be used to determine the measure of Angle B. Using the measure of either Angle C or Angle D we find the measure of Angle B = 180 - (180 - x) = 180 - 180 + x = x. Therefore, both Angle A and Angle B have measures equal to x and are equal in measure. Homework 1.4 Geometry • Using Formulas in Geometry Formulas 1 Dimension 2 Dimension 3 Dimension Homework 1.5 Geometry • Midpoint, Pythagorean and Distance The Coordinate Plane Midpoint and Distance Example 1 Finding the Coordinates of a Midpoint Find the coordinates of the midpoint of PQ with endpoints P(–8, 3) and Q(–2, 7). = (–5, 5) Jeremiah planted two trees. Trees are at coordinates (0,8) and (12,4). He wants to plant a row of hedges such that any bush in the hedge is the same distance from each of the two trees. Define the line at which the hedge should be planted. mtrees = _____ Point on Line mhedge __________ y=mx+b Trees __________ Hedge __________ Relationship between the two lines _____ Mathematically, how do you know? Example 2 Using the Distance Formula Find FG and JK. Then determine whether FG JK. Step 1 Find the coordinates of each point. F(1, 2), G(5, 5), J(–4, 0), K(–1, –3) Using the Distance Formula Continued Step 2 Use the Distance Formula. Example 2 Pythagorean Theorem Pythagoras of Samos Pythagorean Theorem President Garfield’s Proof 1881 One uses subtraction, the other addition. How can opposite operations prove the same thing? 2 a + 2 b 2 =c Algebraic Proof Area of Large Square A = (a+b) (a+b) Area of Pieces A=c*c A = 4 (1/2) ab Total Area of Pieces A = c2 + 2ab Both areas are equal Therefore, (a+b) (a+b) = c2 + 2ab a2+ + 2ab + b2 = a2+b2 = c2 Given Points R (-4,5) and S (2, -1) Determine the length of Line RS. How high up the wall does the ladder reach? Broken Pole Homework 1.6