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Ch. 1 Highlights Geometry A

Ch. 1 Highlights Geometry A. Ms. Urquhart Mrs. Vander Bee. Coplanar Objects. **Remember: Any 3 non-collinear points determine a plane!. Coplanar objects (points, lines, etc.) are objects that lie on the same plane. The plane does not have to be visible.

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Ch. 1 Highlights Geometry A

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  1. Ch. 1 HighlightsGeometry A Ms. Urquhart Mrs. Vander Bee

  2. Coplanar Objects **Remember: Any 3 non-collinear points determine a plane! Coplanar objects (points, lines, etc.) are objects that lie on the same plane. The plane does not have to be visible. Are the following points coplanar? A, B, C ? Yes A, B, C, F ? No H, G, F, E ? Yes E, H, C, B ? Yes A, G, F ? Yes C, B, F, H ? No Lesson 1-1 Point, Line, Plane

  3. Front Side – True or False Lesson 1-1 Point, Line, Plane

  4. Example 3 Point S is between point R and point T. Use the given information to write an equation in terms of x. Solve the equation. Then find both RS and ST. RS = 3x – 16 ST = 4x – 8 RT = 60 I---------------60 -------------------I 3x-16 I-----------4x-8-----------I Lesson 1-1 Point, Line, Plane

  5. ALGEBRA Point Mis the midpoint of VW. Find the length of VM . EXAMPLE 2 Use algebra with segment lengths Lesson 1-1 Point, Line, Plane

  6. GUIDED PRACTICE linel Identify the segment bisector of . Then find PQ. Lesson 1-1 Point, Line, Plane

  7. MIDPOINT FORMULA The midpoint of two points P(x1, y1) and Q(x2, y2) is M(X,Y) = M(x1 + x2, x2 +y2) 2 2 Think of it as taking the average of the x’s and the average of the y’s to make a new point. Lesson 1-1 Point, Line, Plane

  8. a. FIND MIDPOINTThe endpoints ofRSare R(1,–3) and S(4, 2). Find the coordinates of the midpoint M. EXAMPLE 3 Use the Midpoint Formula Lesson 1-1 Point, Line, Plane

  9. 1 , – , M M = 2 5 2 The coordinates of the midpoint Mare 1 5 – , 2 2 ANSWER 1 + 4 – 3 + 2 2 2 EXAMPLE 3 Use the Midpoint Formula SOLUTION a. FIND MIDPOINTUse the Midpoint Formula. Lesson 1-1 Point, Line, Plane

  10. FIND ENDPOINTLet (x, y) be the coordinates of endpoint K. Use the Midpoint Formula. b. FIND ENDPOINTThe midpoint of JKis M(2, 1). One endpoint is J(1, 4). Find the coordinates of endpoint K. STEP 1 Find x. STEP 2 Find y. 4+ y 1+ x 1 2 = = 2 2 ANSWER The coordinates of endpoint Kare (3, – 2). EXAMPLE 3 Use the Midpoint Formula 4 + y = 2 1 + x = 4 y =–2 x =3 Lesson 1-1 Point, Line, Plane

  11. Distance Formula The distance between two points A and B is Lesson 1-1 Point, Line, Plane

  12. EXAMPLE 4 Standardized Test Practice SOLUTION Use the Distance Formula. You may find it helpful to draw a diagram. Lesson 1-1 Point, Line, Plane

  13. Name the three angles in diagram. Name this one angle in 3 different ways. Naming Angles WXY, WXZ, and YXZ The vertex of the angle What always goes in the middle? Lesson 1-1 Point, Line, Plane

  14. 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 2 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. Lesson 1-1 Point, Line, Plane

  15. 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 2 Find angle measures Lesson 1-1 Point, Line, Plane

  16. 3. Given that KLMis straight angle, find mKLN andm NLM. STEP 1 Write and solve an equation to find the value of x. m KLM + m NLM = 180° (10x – 5)° + (4x +3)° = 180° = 180 14x – 2 = 182 14x x = 13 GUIDED PRACTICE Find the indicated angle measures. SOLUTION Straight angle Substitute angle measures. Combine like terms. Subtract 2 from each side. Divide each side by 14. Lesson 1-1 Point, Line, Plane

  17. STEP 2 Evaluate the given expressions when x = 13. mKLM = (10x– 5)° = (10 13– 5)° = 125° mNLM = (4x+ 3)° = (4 13+ 3)° = 55° mKLM = 125° mNLM = 55° ANSWER GUIDED PRACTICE Lesson 1-1 Point, Line, Plane

  18. In the diagram at the right, YWbisects XYZ, and mXYW = 18. Find m XYZ. o By the Angle Addition Postulate, m XYZ =mXYW + m WYZ. BecauseYW bisects XYZyou know thatXYW WYZ. So, m XYW = m WYZ, and you can write M XYZ = m XYW + m WYZ = 18° + 18° = 36°. ~ EXAMPLE 3 Double an angle measure SOLUTION Lesson 1-1 Point, Line, Plane

  19. Example 4 Lesson 1-1 Point, Line, Plane

  20. 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 a. Given that 1 is a complement of 2 and m1 = 68°, find m2. SOLUTION Lesson 1-1 Point, Line, Plane

  21. Sports 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 Lesson 1-1 Point, Line, Plane

  22. 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. Lesson 1-1 Point, Line, Plane

  23. STEP2 Evaluate: the original expressions when x = 34. m BCE = (4x + 8)° = (4 34 + 8)° = 144° m ECD = (x + 2)° = ( 34 + 2)° = 36° ANSWER The angle measures are144°and36°. EXAMPLE 3 Find angle measures Lesson 1-1 Point, Line, Plane

  24. Angles Formed by the Intersection of 2 Lines  Click Me! Lesson 1-1 Point, Line, Plane

  25. 1 and 4 are a linear pair. 4 and 5 are also a linear pair. 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. ANSWER EXAMPLE 4 Identify angle pairs SOLUTION To find linear pairs, look for adjacent angles whose noncommon sides are opposite rays. To find vertical angles, look or angles formed by intersecting lines. Lesson 1-1 Point, Line, Plane

  26. Example 5 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. Lesson 1-1 Point, Line, Plane

  27. Example 6 Given that m5 = 60 and m3 = 62, use your knowledge of linear pairs and vertical angles to find the missing angles. Lesson 1-1 Point, Line, Plane

  28. a. b. c. d. a. Some segments intersect more than two segments, so it is not a polygon. The figure is a convex polygon. b. Part of the figure is not a segment, so it is not a polygon. c. 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 Lesson 1-1 Point, Line, Plane

  29. 3 4 5 6 7 8 9 10 12 n What is a polygon with 199 sides called? 199-gon Lesson 1-1 Point, Line, Plane

  30. 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 Lesson 1-1 Point, Line, Plane

  31. 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. Lesson 1-1 Point, Line, Plane

  32. 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. Lesson 1-1 Point, Line, Plane

  33. Perimeter/Area Rectangle Square Triangle Circle Lesson 1-1 Point, Line, Plane

  34. Area The area of the triangle is 14 square inches and its height is 7 inches. Find the base of the triangle. Lesson 1-1 Point, Line, Plane

  35. Perimeter The perimeter of a rectangle 84.6 centimeters. The length of the rectangle is twice as long as its width. Find the length and width of the rectangle. Lesson 1-1 Point, Line, Plane

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