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End of Lecture / Start of Lecture mark

End of Lecture / Start of Lecture mark. Warm Up Simplify. 1. 7 – (–3) 2. –1 – (–13) 3. | –7 – 1| Solve each equation. 4. 2 x + 3 = 9 x – 11 5. 3 x = 4 x – 5 6. How many numbers are there between and ?. 10. 12. 8. x=5. x=2. Infinitely many. Vocabulary.

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End of Lecture / Start of Lecture mark

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  1. End of Lecture / Start of Lecture mark

  2. Warm Up Simplify. 1.7 – (–3) 2. –1 – (–13) 3. |–7 – 1| Solve each equation. 4. 2x + 3 = 9x – 11 5. 3x = 4x – 5 6. How many numbers are there between and ? 10 12 8 x=5 x=2 Infinitely many

  3. Vocabulary coordinate midpoint distance bisect length segment bisector construction between congruent segments

  4. A ruler can be used to measure the distance between two points. A point corresponds to one and only one number on a ruler. The number is called a coordinate. The following postulate summarizes this concept.

  5. Postulate • Postulate is an accepted statement of facts.

  6. The distance between any two points is the absolute value of the difference of the coordinates (Corresponding numbers).

  7. If the coordinates of points A and B are a and b, then the distance between A and B is |a – b| or |b – a|. The distance between A and B is also called the length of AB, or AB. A B AB = |a – b| or |b - a| a b

  8. Example 1: Finding the Length of a Segment Find each length. A. BC B. AC BC = |1 – 3| AC = |–2 – 3| = |1 – 3| = |– 5| = 2 = 5

  9. TEACH! Example 1 Find each length. b. XZ a. XY

  10. Congruent segments are segments that have the same length. In the diagram, PQ = RS, so you can write PQRS. This is read as “segment PQ is congruent to segment RS.” Tick marks are used in a figure to show congruent segments.

  11. You can make a sketch or measure and draw a segment. These may not be exact. A construction is a way of creating a figure that is more precise. One way to make a geometric construction is to use a compass and straightedge.

  12. In order for you to say that a point B is between two points A and C, all three points must lie on the same line, and AB + BC = AC.

  13. – 6 –6 Example 2: Using the Segment Addition Postulate G is between F and H, FG = 6, and FH =11. Find GH. FH = FG + GH Seg. Add. Postulate 11= 6 + GH Substitute 6 for FG and 11 for FH. Subtract 6 from both sides. 5 = GH Simplify.

  14. Subtract from both sides. TEACH! Example 2 Y is between X and Z, XZ = 3, and XY = . Find YZ. Seg. Add. Postulate XZ = XY + YZ Substitute the given values.

  15. – 2 – 2 –3x –3x 2 2 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 Simplify. Subtract 2 from both sides. Simplify. 3x + 10 = 5x Subtract 3x from both sides. 10 = 2x Divide both sides by 2. 5 = x

  16. Using the Segment Addition Postulate M is between N and O. Find NO. NO = 5x + 2 = 5(5) + 2 Substitute 5 for x. Simplify. = 27

  17. – 3x – 3x 12 3x = 3 3 TEACH! E is between D and F. Find DF. DE + EF = DF Seg. Add. Postulate (3x – 1) + 13 = 6x Substitute the given values 3x + 12 = 6x Subtract 3x from both sides. Simplify. 12 = 3x Divide both sides by 3. 4 = x

  18. TEACH! Continued E is between D and F. Find DF. DF = 6x = 6(4) Substitute 4 for x. = 24 Simplify.

  19. The midpointM of AB is the point that bisects, or divides, the segment into two congruent segments. If M is the midpoint of AB, then AM = MB. So if AB = 6, then AM = 3 and MB = 3. A M B

  20. D is the midpoint of EF, ED = 4x + 6, and DF = 7x – 9. Find ED, DF, and EF. D is the mdpoint. of EF. –4x –4x +9 + 9 Example 5: Using Midpoints to Find Lengths E D 4x + 6 F 7x – 9 Step 1 Solve for x. ED = DF 4x + 6 = 7x – 9 Substitute 4x + 6 for ED and 7x – 9 for DF. Subtract 4x from both sides. 6 = 3x – 9 Simplify. Add 9 to both sides. 15 = 3x Simplify.

  21. D is the midpoint of EF, ED = 4x + 6, and DF = 7x – 9. Find ED, DF, and EF. 15 3x = 3 3 Example 5 Continued E D 4x + 6 F 7x – 9 Divide both sides by 3. x = 5 Simplify.

  22. D is the midpoint of EF, ED = 4x + 6, and DF = 7x – 9. Find ED, DF, and EF. Example 5 Continued E D 4x + 6 F 7x – 9 Step 2 Find ED, DF, and EF. ED = 4x + 6 DF = 7x – 9 EF = ED + DF = 4(5) + 6 = 7(5) – 9 = 26 + 26 = 52 = 26 = 26

  23. S is the mdpt. of RT. +3x +3x TEACH! Example 5 S is the midpoint of RT, RS = –2x, and ST = –3x – 2. Find RS, ST, and RT. R S T –2x –3x – 2 Step 1 Solve for x. RS = ST Substitute –2x for RS and –3x – 2 for ST. –2x = –3x – 2 Add 3x to both sides. x = –2 Simplify.

  24. TEACH! Example 5 Continued S is the midpoint of RT, RS = –2x, and ST = –3x – 2. Find RS, ST, and RT. R S T –2x –3x – 2 Step 2 Find RS, ST, and RT. RS = –2x ST = –3x – 2 RT = RS + ST = –2(–2) = –3(–2) – 2 = 4 + 4 = 4 = 4 = 8

  25. 2. S is the midpoint of TV, TS = 4x – 7, and SV = 5x – 15. Find TS, SV, and TV. Lesson Quiz: Part I 1. M is between N and O. MO = 15, and MN = 7.6. Find NO. 22.6 25, 25, 50 3. Sketch, draw, and construct a segment congruent to CD. Check students' constructions

  26. K M L Lesson Quiz: Part II 4.LH bisects GK at M. GM =2x + 6, and GK = 24.Find x. 3 5. 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. Always

  27. 1 4 3 2 • An angle is formed by two rays (called sides of the angle), with the same endpoints (called the vertex of the angle).

  28. The name can be the number between the sides of the angle: 3. The name can be the vertex of the angle: G. Finally, the name can be a point on one side, the vertex, and a point on the other side of the angle: AGC,CGA. Name the angle below in four ways.

  29. GEOMETRY LESSON 1-4 Tools of Geometry Measuring Segments and Angles • You can Classify angles based on their measures. • Right Angle • Obtuse Angle • Straight Angle

  30. GEOMETRY LESSON 1-4 Tools of Geometry Measuring Segments and Angles • If an angle measures 90o then it is called Right Angle. 90o degrees

  31. GEOMETRY LESSON 1-4 Tools of Geometry Measuring Segments and Angles • If an angle measures less than 90o degrees then it is called Acute angle. 40o degrees Acute angle

  32. GEOMETRY LESSON 1-4 Tools of Geometry Measuring Segments and Angles • If an angle measures more than 90o degrees, then it is called Obtuse angle. Obtuse Angle

  33. GEOMETRY LESSON 1-4 Tools of Geometry Measuring Segments and Angles • If an angle measures 180o degrees then it is called a straight angle.

  34. 1 4 3 1 2 2 2 1 GEOMETRY LESSON 1-4 Tools of Geometry Measuring Segments and Angles • There are Four Types of Angle Pairs. • Vertical Angles • Adjacent Angles • Complimentary Angles • Supplementary Angles

  35. 1 4 3 2 GEOMETRY LESSON 1-4 Tools of Geometry Measuring Segments and Angles • Vertical angles are two angles whose sides are opposite rays. • 1 and 2 are a pair of Vertical angles • 3 and 4 are also vertical angles

  36. 1 2 GEOMETRY LESSON 1-4 Tools of Geometry Measuring Segments and Angles • Adjacent angles are two Coplanar angles with a common side. • Share the same plane • Common Side • Common Vertex • No interior points in common

  37. 30o 600 GEOMETRY LESSON 1-4 Tools of Geometry Measuring Segments and Angles • Complimentary angles are two angles whose measures add-up to 90o. • They do not have to be adjacent. 1 2

  38. GEOMETRY LESSON 1-4 Tools of Geometry Measuring Segments and Angles • Supplementary angles are angles whose measured sum adds-up to 180o • They can be adjacent linear pair • They can also be non adjacent or share a common side or vertex. 1 2

  39. m 1 + m 2 = m ABCAngle Addition Postulate. 42 + m 2 = 88Substitute 42 for m 1 and 88 for m ABC. m 2 = 46 Subtract 42 from each side. GEOMETRY LESSON 1-4 Tools of Geometry Measuring Segments and Angles Suppose that m 1 = 42 and m ABC = 88. Find m 2. Use the Angle Addition Postulate to solve.

  40. Two angles are supplementary if the sum of their measures is 180. A straight angle has measure 180, and each pair of adjacent angles in the diagram forms a straight angle. So these pairs of angles are supplementary: 1 and 2, 2 and 3, 3 and 4, and 4 and 1. GEOMETRY LESSON 1-4 Tools of Geometry Measuring Segments and Angles Name all pairs of angles in the diagram that are: a. vertical Vertical angles are two angles whose sides are opposite rays. Because all the angles shown are formed by two intersecting lines, 1 and 3 are vertical angles, and 2 and 4 are vertical angles. b. supplementary

  41. GEOMETRY LESSON 1-4 Tools of Geometry Measuring Segments and Angles (continued) c. complementary Two angles are complementary if the sum of their measures is 90. No pair of angles is complementary.

  42. 3 and 5 are not marked as congruent on the diagram. Although they are opposite each other, they are not vertical angles. So you cannot conclude that 3 5. GEOMETRY LESSON 1-4 Tools of Geometry Measuring Segments and Angles Use the diagram below. Can you conclude; 3 is a right angle, 1 and 5 are adjacent, 3 5? You can conclude that 1 and 5 are adjacent because they share a common side, a common vertex, and no common interior points. Although 3 appears to be a right angle, it is not marked with a right angle symbol, so you cannot conclude that 3 is a right angle.

  43. GEOMETRY LESSON 1-4 Tools of Geometry Measuring Segments and Angles Class-Work Use the figure below for Exercises 1–2. 1. Name 2 two different ways. DAB, BAD 2. Measure and classify 1, 2, and BAC. 90°, right; 30°, acute; 120°, obtuse Use the figure below for Exercises 3–4. Sample: 1 and 3, 2 and 4 3. Name a pair of supplementary angles. 4. Can you conclude that there are vertical angles in the diagram? Explain. No; no angle pairs are formed by opposite rays.

  44. RS, TR, ST TO, TP, TR, TS Class-Work • Use the figure below for Exercises 1-3. • Name the segments that form the triangle. • Name the rays that have point T as their endpoint. • Explain how you can tell that no lines in the figure are parallel or skew. The three pairs of lines intersect, so they cannot be parallel or skew.

  45. AC or BD Class-Work Use the figure for Exercises 4 and 5. 4. Name a pair of parallel planes. 5. Name a line that is skew to XW. plane ABCD || plane XWQ

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