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Ch 1

ESSENTIALS OF GEOMETRY. Ch 1. 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas 1.4 Measure and Classify Angles 1.5 Describe Angle Pair Relationships 1.6 Classify Polygons 1.7 Find Perimeter, Circumference, and Area. Day 1.

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Ch 1

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  1. ESSENTIALS OF GEOMETRY Ch 1 • 1.1 Identify Points, Lines, and Planes • 1.2 Use Segments and Congruence • 1.3 Use Midpoint and Distance Formulas • 1.4 Measure and Classify Angles • 1.5 Describe Angle Pair Relationships • 1.6 Classify Polygons • 1.7 Find Perimeter, Circumference, and Area

  2. Day 1 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence • LEARNING OBJECTIVES • 1. Before, you studied basic concepts of geometry. Now you will name and sketch geometry figures. • After you learn about points, lines, and planes, you will use segment postulates to identify congruent segments. AGENDA 1. Pass Out Books 2. Syllabus / Outlook 3. Section Notes 4. Activity? 5. HW Time (Time Permitting)

  3. Sect 1.1 IDENTIFY POINTS, LINES, & PLANES THREE UNDEFINED TERMS : • _________ • The basic unit of Geometry. • It has no size, is infinitely small, and has only location. • Named with _______________________. • _______ • A straight arrangement of points. • There are infinitely many __________ on a line. • A line has infinite length but no thickness and extends forever in two directions. • You name a line by _______________________________. • __________ • A plane has length and width, but no _______________. • It is a flat surface that ________________________. • A plane is named by ________________________________. POINT A CAPITAL LETTER LINE POINTS TWO POINTS ON THE LINE, or lower case cursive letter PLANE THICKNESS EXTENDS FOREVER THREE POINTS OR capital cursive letter

  4. Sect 1.1 IDENTIFY POINTS, LINES, & PLANES DEF: Collinear means _________________________ DEF: Coplanar means _________________________ ON THE SAME LINE ON THE SAME PLANE

  5. a. Give two other names for PQand for plane R. • Other names for PQare QPand line n. Other names for plane Rare plane SVT and plane PTV. EXAMPLE 1 Name points, lines, and planes b.Name three points that are collinear. Name four points that are coplanar. b. Points S, P, and Tlie on the same line, so they are collinear. Points S, P, T,and Vlie in the same plane, so they are coplanar.

  6. Sect 1.1 IDENTIFY POINTS, LINES, & PLANES DEFINED TERMS of Geometry – Can be described using known words such as point or line: DEF: A line segment (or segment) consists of two points called the endpoints of the segment and all the points between them that are collinear with the two points. EX: Name: DEF: A ray ________________________________________________ ____________________. EX: Name: DEF: Opposite rays _________________________________ __________________________________________________. EX: Name: consists of the endpoint A and all points that line on the same side of A as B. share the same endpoint and go in opposite directions along the same line.

  7. a. Give another name for GH . a. Another name for GHis HG . b. The rays with endpoint Jare JE, JG, JF, and JH . The pairs of opposite rays with endpoint Jare JEand JF, and JGand JH. EXAMPLE 2 Name segments, rays, and opposite rays b. Name all rays with endpoint J . Which of these rays are opposite rays?

  8. c. a. b. EXAMPLE 3 Sketch intersections of lines and planes • Sketch a plane and • a line that is in the • plane. • Sketch a plane and a line that does not intersect the plane. c. Sketch a plane and a line that intersects the plane at a point.

  9. STEP 1 Draw: a vertical plane. Shade the plane. STEP 2 Draw: a second plane that is horizontal. Shade this plane a different color. Use dashed lines to show where one plane is hidden. STEP 3 Draw: the line of intersection. EXAMPLE 4 Sketch intersections of planes Sketch two planes that intersect in a line. SOLUTION

  10. Sect 1.2 Use Segments and Congruence • is used to name a line segment. • means the measure of segment AB and refers to a number. • means the length of or distance from A to B and refers to a number. • is a rule that is accepted without proof. AB + BC = AC B is between A and C.

  11. Measure the length of STto the nearest tenth of a centimeter. Align one mark of a metric ruler with S. Then estimate the coordinate of T. For example, if you align Swith 2, T appears to align with 5.4. ST=5.4 – 2 = 3.4 The length of STis about 3.4 centimeters. ANSWER EXAMPLE 1 Apply the Ruler Postulate SOLUTION Use Ruler Postulate.

  12. Maps The cities shown on the map lie approximately in a straight line. Use the given distances to find the distance from Lubbock, Texas, to St. Louis, Missouri. The distance from Lubbock to St. Louis is about 740 miles. ANSWER EXAMPLE 2 Apply the the Segment Addition Postulate SOLUTION Because Tulsa, Oklahoma, lies between Lubbock and St. Louis, you can apply the Segment Addition Postulate. LS = LT + TS = 380 + 360 = 740

  13. Use the diagram to find GH. FH = FG+ GH 36 21+GH = 15 GH = EXAMPLE 3 Find a length SOLUTION Use the Segment Addition Postulate to write an equation. Then solve the equation to find GH. Segment Addition Postulate. Substitute 36 for FHand 21 for FG. Subtract 21 from each side.

  14. Sect 1.2 Use Segments and Congruence DEF: Two segments are congruent ( ) segments _____________________________. If they have the same length A B C D

  15. Sect 1.3 USE MIDPOINT AND DISTANCE FORMULA Def: An ________ is two rays that share a common endpoint, provided that the two rays do not lie on the same line. • The common endpoint of the two rays that make an angle is the __ ______of the angle. • The two rays are called the ________ of the angle. ____ is the vertex. ____________are the sides. Name an angle by using __________ ______________________________ ________ or ___________________ ANGLE VERTEX SIDES A A THREE LETTERS, WITH VERTEX IN THE MIDDLE B VERTEX LETTER C

  16. Name the three angles in the diagram. WXY, or YXW YXZ, or ZXY WXZ, or ZXW You should not name any of these angles X because all three angles have Xas their vertex. EXAMPLE 1 Name angles

  17. Sect 1.4 Measure and Classify Angles DEGREES DEF: ___________________ is the unit of measurement we will use to measure an angle. DEF: A ___________________is used to measure angles. PROTRACTOR ACUTE RIGHT OBTUSE STRAIGHT

  18. Sect 1.4 Measure and Classify Angles FAMILY GUY : SHAWSHANK REDEMPTION PARODY (click here)

  19. Use the diagram to find the measure of the indicated angle. Then classify the angle. a.KHJ b.GHK c.GHJ d.GHL EXAMPLE 2 Measure and classify angles SOLUTION A protractor has an inner and an outer scale. When you measure an angle, check to see which scale to use.

  20. o a. HJ is lined up with the 0 on the inner scale of the protractor. HKpasses through 55 on the inner scale. So, mKHJ = 55 . It is an acute angle. o o o b. HG is lined up with the 0 on the outer scale and HKpasses through 125 on the outer scale. So, mGHK = 125 . It is an obtuse angle. o o o c. m GHJ = 180. It is a straight angle. o d. m GHL= 90. It is a right angle. EXAMPLE 2 Measure and classify angles

  21. Sect 1.4 Measure and Classify Angles R S P T

  22. o ALGEBRAGiven that m LKN =145 , find m LKM andm MKN. STEP 1 Write and solve an equation to find the value of x. mLKN = m LKM + mMKN o o o 145 = (2x + 10)+ (4x – 3) EXAMPLE 3 Find angle measures SOLUTION Angle Addition Postulate Substitute angle measures. 145 = 6x + 7 Combine like terms. 138 = 6x Subtract 7 from each side. 23 = x Divide each side by 6.

  23. STEP 2 Evaluate the given expressions when x = 23. mLKM = (2x+ 10)° = (2 23+ 10)° = 56° mMKN = (4x– 3)° = (4 23– 3)° = 89° So, m LKM = 56°and m MKN = 89°. ANSWER EXAMPLE 3 Find angle measures

  24. Sect 1.4 Measure and Classify Angles Congruent DEF: Two angles are ____________ if the have the same measure. A B ANGLE BISECTOR DEF: A ___________________ is a ray that divides an angle into two angles that are congruent.

  25. Ch 1 DAY 3 • 1.1 Identify Points, Lines, and Planes • 1.2 Use Segments and Congruence • 1.3 Use Midpoint and Distance Formulas • 1.4 Measure and Classify Angles • 1.5 Describe Angle Pair Relationships • 1.6 Classify Polygons • 1.7 Find Perimeter, Circumference, and Area

  26. Sect 1.5 DEF: Complementary Angles are _________________ ______________________________________________. DEF: Supplementary Angles are _________________ ______________________________________________. DEF: Adjacent Angles are _________________ ______________________________________________. Describe Angle Pair Relationships two angles whose sum is 90o two angles whose sum is 180o two angles who share a common vertex and side 65o 65o 130o 50o 50o 25o 130o 25o

  27. In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles. Because 32°+ 58° = 90°, BACand RSTare complementary angles. Because 122° + 58° = 180°,CADand RSTare supplementary angles. Because BACand CADshare a common vertex and side, theyare adjacent. EXAMPLE 1 Identify complements and supplements SOLUTION

  28. Given that 1 is a complement of 2 and m1 = 68°, find m2. a.You can draw a diagram with complementary adjacent angles to illustrate the relationship. m 2 = 90° – m 1 = 90° – 68° = 22° EXAMPLE 2 Find measures of a complement and a supplement SOLUTION

  29. m 3 = 180° – m 4 = 180° –56° = 124° b. Given that 3 is a supplement of 4and m 4=56°, find m3. b.You can draw a diagram with supplementary adjacent angles to illustrate the relationship. EXAMPLE 2 Find measures of a complement and a supplement SOLUTION

  30. When viewed from the side, the frame of a ball-return net forms a pair of supplementary angles with the ground. Find mBCEand mECD. EXAMPLE 3 Find angle measures Sports

  31. Use the fact that the sum of the measures of supplementary angles is 180°. STEP1 mBCE+m∠ ECD=180° EXAMPLE 3 Find angle measures SOLUTION Write equation. (4x + 8)°+ (x +2)°=180° Substitute. 5x + 10 = 180 Combine like terms. 5x = 170 Subtract10 from each side. x = 34 Divide each side by 5.

  32. STEP2 Evaluate: the original expressions when x = 34. m ECD = (x + 2)° = ( 34 + 2)° = 36° m BCE = (4x + 8)° = (4 34 + 8)° = 144° ANSWER The angle measures are144°and36°. EXAMPLE 3 Find angle measures SOLUTION

  33. Sect 1.5 Describe Angle Pair Relationships 3 5 6 9 10 7 8 4 1 2 DEF: In your own words, write a definition of a Linear Pair of angles:___________________________________________ _________________________________________________. DEF: In your own words, write a definition of a Vertical Pair of angles:_________________________________________ _________________________________________________. Two adjacent and supplementary angles whose noncommon sides are opposite rays. Two angles whose sides form two pairs of opposite rays.

  34. Identify all of the linear pairs and all of the vertical angles in the figure at the right. ANSWER 1 and 5 are vertical angles. 1 and 4 are a linear pair. 4 and 5 are also a linear pair. ANSWER EXAMPLE 4 Identify angle pairs SOLUTION To find vertical angles, look or angles formed by intersecting lines. To find linear pairs, look for adjacent angles whose noncommon sides are opposite rays.

  35. ALGEBRA Two angles form a linear pair. The measure of one angle is 5 times the measure of the other. Find the measure of each angle. Let x° be the measure of one angle. The measure of the other angle is 5x°. Then use the fact that the angles of a linear pair are supplementary to write an equation. EXAMPLE 5 Find angle measures in a linear pair SOLUTION

  36. The measures of the angles are 30oand 5(30)o = 150o. ANSWER EXAMPLE 5 Find angle measures in a linear pair xo+ 5xo = 180o Write an equation. 6x = 180 Combine like terms. x = 30o Divide each side by 6.

  37. Sect 1.6 Classify Polygons • DEF:A __________ is a closed figure in a plane, formed by connecting at least three line segments endpoint to endpoint with each segment intersecting exactly two others. • Each line seg. is called a _____. • Each endpoint where the sides meet is called the _________. • DEF: A __________ is a segment connecting two nonconsecutive vertices. polygon side vertex diagonal

  38. Sect 1.6 Classify Polygons • 2 types of polygons: • Polygon - a polygon in which no segment connecting two vertices is outside the polygon. • In a convex polygon: • Consecutive Vertices share a common side • Consecutive Sides share a common vertex • Consecutive Angles share a common side • Polygon- a polygon in which at least one segment connecting two vertices is outside the polygon. Convex Concave

  39. b. c. d. a. a. Some segments intersect more than two segments, so it is not a polygon. b. The figure is a convex polygon. c. Part of the figure is not a segment, so it is not a polygon. d. The figure is a concave polygon. EXAMPLE 1 Identify polygons Tell whether the figure is a polygon and whether it is convex or concave. SOLUTION

  40. Sect 1.6 Classify Polygons triangle octagon quadrilateral nonagon pentagon decagon hexagon dodecagon heptagon n-gon equilateral DEF: All sides are congruent in an ___________ polygon. All angles are congruent in an ___________ polygon. DEF: A _________ polygon is a convex polygon that is both equilateral and equiangular. equiangular regular

  41. a. b. a. The polygon has 6 sides. It is equilateral and equiangular, so it is a regular hexagon. b. The polygon has 4 sides, so it is a quadrilateral. It is not equilateral or equiangular, so it is not regular. EXAMPLE 2 Classify polygons Classify the polygon by the number of sides. Tell whether the polygon is equilateral, equiangular, or regular. Explain your reasoning. SOLUTION

  42. The polygon has 12 sides, so it is a dodecagon. The sides are congruent, so it is equilateral. The polygon is not convex, so it is not regular. c. c. EXAMPLE 2 Classify polygons Classify the polygon by the number of sides. Tell whether the polygon is equilateral, equiangular, or regular. Explain your reasoning. SOLUTION

  43. A table is shaped like a regular hexagon.The expressions shown represent side lengths of the hexagonal table. Find the length of a side. ALGEBRA 3x + 6 4x – 2 = 6 x – 2 = 8 x = EXAMPLE 3 Find side lengths SOLUTION First, write and solve an equation to find the value of x. Use the fact that the sides of a regular hexagon are congruent. Write equation. Subtract 3x from each side. Add 2 to each side.

  44. 3x + 6 3(8) + 6 = = 30 The length of a side of the table is 30inches. ANSWER EXAMPLE 3 Find side lengths Then find a side length. Evaluate one of the expressions when x = 8.

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