Introduction to Geometry

Introduction to Geometry

Building Blocks of Geometry I. Three building blocks of geometry: points, lines, and planes. 1. A point is the most basic building block of geometry. It has no size. It only has location. You represent a point with a dot, and you name it with a capital letter. The point shown is called A. Can lie on a and/or a. 2. A line is a straight, continuous arrangement of infinitely many points. It has infinite length, but no thickness. It extends forever in two directions. How many points name a line? Example: Name the lines: 3. A has length and width, but no thickness. A plane extends in 2 dimensions infinitely. Represented by a shape like. Example: II. Collinear and Coplanar Points A. Collinear Points are points that lie on the same. Example: Which points are collinear? 2

B. Coplanar points are points that lie on the same. Example: Which points are coplanar? III. Segments and Rays A. A line segment consists of two points called the of the segment and all the points between them that are collinear with the two points. You can write line segment AB, using a segment symbol, as AB or BA. There are two ways to write the length of a segment. You can write AB = 2 in., meaning the distance from A to B is 2 inches. You can also use an m for measure in front of the segment name, and write the distance as mab = 2 in. If no measurement units are used for the length of a segment, it is understood that the choice of units is not important or is based on the length of the smallest square in the grid. AB = 2 in., or mab = 2 in. MN = 5 units, or mmn = 5 units Examples: The above example may be symbolized by or. The second example may be symbolized by or. B. A ray begins at a point and extends infinitely in one direction. The initial point is called the endpoint. You need two letters to name a ray. The first letter is the endpoint of the ray, and the second letter is any other point that the ray passes through. 3

Example: Name the example above: Name the example above: IV. Intersections A. Two or more geometric figures if they have one or more points in common. B. The of two or more geometric figures is the set of the figures have in common. C. Name the intersection of the two lines below: The intersection of two different lines is a. D. Name the intersection of the two planes below: The intersection of two different planes is a. V. Postulates or Axioms rules accepted. VI. Theorems rules that are. 4

VII. Segment Addition Postulate: If B is A and C, then. If, then B is A and C. Example 1: DF= Example 2: Suppose M is between L and N. Use the Segment Addition Postulate to solve for the variable. Then find the lengths of LM, MN, and LN. 1. LM = 3x + 8, MN = 2x 5, LN = 23 2. LM 1 w 2, MN 3w 3 2 2, LN 5w 2 VIII. Congruent Segments Line segments that have the same. In the diagram below, you can say the length of AB is equal to the length of EF, or you can say AB is congruent to EF. The symbol means is congruent to. Lengths are equal: AB = EF Segments are congruent: AB CD 5

1. Name a line, segment, and ray in this figure: Skills Practice: 2. What is wrong with the following: AB CD? 3. Name each of the lines in two different ways. a. b. c. 4. Use a ruler to draw each line, ray, or segment. Don t forget to use arrow heads to show that the line extends indefinitely and to label the two points. a. AB b. KL c. XY 5. Name each line segment. a. b. c. 6. Name the ray in two different ways. a. b. c. 6

7. Draw a plane containing four coplanar points P, Q, R, and S, with exactly three collinear points Q, R, and S. For Exercises 8-10, use the figure located on the right. 8. Draw AB where point B has coordinates (2,-6). 9. Draw OM with endpoint (0,0) that goes through point M(2,2). 10. Draw CD through points C(-2,1) and D(-2,-3). 11. Given AC= 38. Find AB and BC. 12. Find the length of MO. 13. Given the following pairs of congruent segments, label the figure below. AB DE, AC DF, and BC EF 7

Midpoint and Distance I. Midpoint - a point that divides, or the segment into two. Example 1: Identify the midpoint of the following line segment below. 9 y X 8 7 6 5 Z 4 3 2 1 Y x 1 2 3 4 5 6 Endpoint X Endpoint Y Midpoint Z Example 2: Find the midpoint of the following line segment below. A 5 4 M 3 2 1 y -3-2 -1 1 2 3 4 5 What are the coordinates of the endpoints and midpoint of line segment AB? Endpoint A Endpoint B Midpoint M B x Looking at the endpoints and middle coordinates, is there another way to find the middle of a line segment? 8

Midpoint Formula: Example 1: D( -2, 6) and F(3, 4) Example 2: The midpoint of ST is M (2, 4). One endpoint is S(-1, 7). Find the coordinates of T. Let (x, y) be the coordinates of T. 1 Example 3: The midpoint of JK is M 0,. One endpoint is J (2, -2). Find the coordinates of 2 the other endpoint. II. Segment bisector a line, ray, or segment that intersects a segment at its. 9

Perquisite Skills- Simplifying Radicals Simplify. 1. 90 2. 165 3. 375 4. 180 III. Distance You can think of a coordinate plane as a grid of streets with two sets of parallel lines running perpendicular to each other. Every segment in the plane that is not in the x- or y-direction is the hypotenuse of a right triangle whose legs are in the x- and y-directions. So you can use the Pythagorean Theorem to find the distance between any two points on a coordinate plane. In Steps 1 and 2, find the length of each segment by using the segment as the hypotenuse of a right triangle. Simply count the squares on the horizontal and vertical legs, then use the Pythagorean Theorem to find the length of the hypotenuse. Step 1 Use each segment as the hypotenuse of a right triangle. Draw the legs along the grid lines. Find the length of each segment. Step 2 Graph each pair of points, then find the distances between them. a. ( 1, 2), (11, 7) b. ( 9, 6), (3, 10) 10

What if the points are so far apart that it s not practical to plot them? For example, what is the distance between the points A(15, 34) and B(42, 70)? A formula that uses the coordinates of the given points would be helpful. To find this formula, you first need to find the lengths of the legs in terms of the x- and y-coordinates. From your work with slope triangles, you know how to calculate horizontal and vertical distances. Step 3 Write an expression for the length of the horizontal leg using the x-coordinates. Step 4 Write a similar expression for the length of the vertical leg using the y-coordinates. Step 5 Use your expressions from Steps 3 and 4, and the Pythagorean Theorem, to find the distance between points A(15, 34) and B(42, 70). Step 6 Generalize what you have learned about the distance between two points in a coordinate plane. Distance Formula: Recall Area Formulas: Atriangle 1 bh 2 Arec tangle Example 1: Find the distance of each side. Then find the area and the perimeter of the triangle. bh Example 2: A map is placed on a coordinate grid. Jacksonville is located at (5, 4) and Gainesville is located at (2, 3). How far apart are Jacksonville and Gainesville on the map? If each unit represents 10 miles, about how far is it from Jacksonville to Gainesville? 11

Skills Practice: 1. Find the coordinates of the midpoint of the segment with each pair of endpoints. a. (12,-7) and (-6,15) b. (14, -7) and (-3,18) 2. One endpoint of a segment is (12,-8). The midpoint is (3,18). Find the coordinates of the other endpoint. 3. In each figure below, imagine drawing the diagonals AC and BD. a. Find the midpoint of AC and the midpoint of BD in each figure. b. What do you notice about the midpoints for each figure. 4. Find the distance between each pair of points. a. (10,20) and (13,16) b. (-19,-16) and (-3,14) 12

5. Look at the diagram of Isabella s and Kayleigh s locations. Assume each block is approximately 50 meters long. What is the shortest distance, to the nearest meter, from Isabella to Kayleigh? What is the midpoint? 6. Find the perimeter of ABC with vertices A(2,4), B(8,12), and C(24,0). 7. Find the area and the perimeter of the rectangle with vertices: A(6,8), B(9,7), C(7,1), D(4,2) 8. Find the area and the perimeter of the square with vertices: M(-3,5), N(-1,1), O(3,3), P(1,7) 13

Measure and Classify Angles I. Angle consists of that have the same. The rays are the sides of the angle. The endpoint is the of the angle. Example: C A B Denoted as or Measured in Example: Name 3 different angles.,, J P L K M N II. Measuring Angles- to measure an angle, we use a tool called a protractor. 14

Example: Use your protractor to measure each angle. Which angle measures more than 90? III. Congruent angles angles that have. IV. A ray is the angle bisector if it contains the vertex and divides the angle into two congruent angles. In the figure at right, CD bisects ACB so that ACD BCD. Example: KL bisects JKM. If JKM 110, then m JKL J L K M 15

Example: RQ bisects PRS. What is m PRQ? P Q x + 40 R 3x-20 S Example: BD bisects ABC. What is the value of x? A (2x+35) D B (5x-22) C V. Angle Addition Postulate If P is in the of RST then m RSP m PST m RST. R S P T Example: m ABC 35 m CBD 15 m ABD B C A D 16

Example: m KLA 87 m BLA 36 m KLB L K B A VII. Angle Classifications 1. Acute angles - 2. Right angles - 3. Obtuse angles - 4. Straight angles - IX. Adjacent Angles Two angles are adjacent if they a common and, but common. Example: Adjacent Angles Example: Non-adjacent angles 17

Angle Relationships I. Vertical Angles Two angles are vertical if their sides form two pairs of opposite. The two angles are. Def. Opposite angles formed when 2 lines cross. Ex. II. Linear Pair two angles whose noncommon sides are opposite rays. Ex. Example 1: 1 3 2 4 Are Are 2 and 3 a linear pair? 3 and 4 a linear pair? Are 1 and 2 vertical angles? Are 1 and 3 vertical angles? 18

III. Complementary angles 2 angles whose sum of their measures is. These angles can be or nonadjacent. Ex of complementary angles: IV. Supplementary angles 2 angles whose sum of their measures is. These angles can be or nonadjacent. Ex of Supplementary angles: Example 2: Given angle A is a complement of angle B and m A 23. What is m B? Example 3: Given angle M is a supplement of angle N and m M 105. What is m N? Example 4: Given angle A is a complement of angle B. Find m A and m B. m A (15x 3) m B (5x 13) Example 5: Given m P and m Qare supplementary. Find m P and m Q. m P (8x 100) m Q (2x 50) 19

1. Name each angle in two different ways. Skills Practice: 2. Draw and label each angle. a. TAN b. BIG c. SML 3. For each figure at right, list the angles that you can name using only the vertex letter. 4. Find the measure of each angle to the nearest degree. a. m AQB c. m AQC e. m XQA b. m ZQY d. m ZQX f. m AQY 20

5. Given the following information, mark the figure below: RA SA, T H, TAR HAS 6. Given the following information, mark the figure below: TA GA, AGT ATG, vertical angles 7. Name the congruent angles in the figure below. 8. Ray AB is the angle bisector of angle CAD. Find the measure of angle CAD and x. 9. Solve for the missing variable below. 21

10. Using the angle addition postulate and m ADC 82, find the missing variable. (2x +21) 11. Classify the following angles (linear pair, acute, obtuse, right, vertical, complementary, and supplementary). a. c. e. 2 and 4 b. d. f. 2 and 1 12. Given that angle A and angle B are supplementary angles and the measure of angle B is 35, what is the measure of angle A? 13. Given that angle C and angle D are complementary angles and the measure of angle D is 35, what is the measure of angle C? 14. Find the missing variables. a. b. 22

Polygons In geometry, a figure that lies in a plane is called a plane figure. A is a closed plane figure with the following properties: 1. It is formed by three of more line segments called. 2. Each side intersects exactly two sides, one at each endpoint, so that no two sides with a common endpoint are collinear. Each endpoint of a side is a of the polygon. The plural of vertex is vertices. A polygon can be named by listing the vertices in consecutive order. For example, ABCDE and EDEAB are both correct names for the polygon at the right. A of a polygon is a line segment that connects two nonconsecutive vertices. A polygon is if no line that contains a side of the polygon contains a point in the interior of the polygon, or all diagonals lie inside the polygon. A polygon that is not convex is called. 23

A polygon is named by the number of its sides. The term n-gon, where n is the number of polygon s sides, can also be used to name a polygon. For example, a polygon with 16 sides is a 16-gon. To name a polygon, be sure to include the name and the vertices in consecutive order. Examples: Name the polygons below. Name: or Name Two polygons are congruent if and only if they are the same and. This means all the corresponding sides are congruent and all the corresponding angles are congruent. 24

For example, if quadrilateral CAMP is congruent to quadrilateral SITE, then their four pairs of corresponding angles and four pairs of corresponding sides are also congruent. When you write a statement of congruence, always write the letters of the corresponding vertices in an order that shows the correspondences. Example: Which polygon is congruent to ABCDE? ABCDE? The of a polygon equals the sum of the lengths of its sides. Perimeter measures the length of the boundary of a two-dimensional figure. Example: Find the perimeter of the quadrilateral below. In an polygon, all sides are congruent. Perimeter: cm In an polygon, all angles in the interior of the polygon are congruent. 25

A polygon is a convex polygon that is both equilateral and equiangular. Guided Practice: Example 1: If the perimeter of the quadrilateral is 20 cm, find x. x = Example 2: What is wrong with this picture of equilateral pentagon ABCDE? Hint: there are two things wrong with this picture. 26

For Exercises 1-3, draw an example of each polygon. Skills Practice: 1. Quadrilateral 2. Dodecagon 3. Octagon For Exercises 4-7, classify each polygon. Assume all sides are straight. For exercises 8-10, give one possible name for each polygon. For exercises 11 and 12, use the information given to name the triangle that is congruent to the first one. 11. 12. 27

13. Indicate if the following figures are polygons. 14. Indicate if the following polygons are concave or convex. 15. In the figure at right, THINK POWER. a. Find the measures a, b, and c. b. If m P 87 and m W 165, which angles in THINK do you know? Write their measures. 16. Each side of a regular dodecagon measures 7 in. Find the perimeter. 17. The perimeter of an equilateral octagon is 42 cm. Find the length of each side. 18. The perimeter of ABCDE is 94 m. Find the lengths of segments AB and CD. 28

Algebraic Proofs Warm Up: Solve for x. 1. 3x 5 17 2. 4x 5 8x 3 3. 2( x 5) 20 0 4. x 8 6 5 Proofs Justifications are used for EVERY step! Here is a list of justifications that CAN be used. There are other justifications that you learned and can use as well. 29

Example 1: Complete a two column proof to prove the following: Given: x 5 2 Prove: x 7 6 Example 2: Complete a two column proof to prove the following: Given: 6x 3 3( x 1) Prove: x 2 30

Example 3: Complete a two column proof to prove the following: Given: m MRK 3 x, m KRW x 6, m MRW 90 Prove: m MRK 63 Example 4: Complete a two column proof to prove the following: Given: AB 5x 1, BC 3x 4, AC 13 Prove: BC 2 31

Guided Practice: Part A- Identify the property that justifies each statement. 1. AB = AB 2. If m 1 m 2 and m 2 m 4, then m 1 m 4 3. If x = y, then y = x. 4. If ST = YZ, and YZ = PR, then ST = PR 5. If KL = PR, then KL-AB = PR AB 6. 412 = 412 7. If b = a and b = 0, then a = 0 8. Figure A = Figure A 9. If m DEF m ABC, then m DEF m GHI m ABC m GHI x y 10. If x = y, then 3 3 11. If AB = CD, then CD = AB x 12. If 7, then x = 14 2 13. If x = 5 and b = 5, then x = b 14. If XY AB = WZ AB, then XY = WZ 15. If m A m B, and m B m C, then m A m C Part B- Use the property to complete the statement. 16. Reflexive Property: = SE 17. Symmetric Property: If =, then m RST m JKL 18. Transitive Property: m F m J and =, then m F m L 19. Addition Property: If RS = TU, then RS +20 = 20. Multiplication Property: If m 1 m 2then 3( m 1) 21. Substitution Property: If a = 20, then 5a = Part C- Complete the two-column proofs below using the appropriate properties. 22. Given:8x 34 6 23. Given:5( x 3) 4( x 2) Prove: x 5 Prove: x 23 Statement Reason Statement Reason 32

24. Given: 4x 7 6x 7 1 25. Given: x y 9 7 Prove: x 7 Prove: y 7x 63 Statement Reason Statement Reason 26. Given:7x 11 4x 19 27. Given:14x 3 19x 23 Prove: x 10 Prove: x 4 28. Given: 4( 2x 11) 76 29. Given:14( x 1) 7( 4 x) Prove: x 4 Prove: x 2 S T U 30. Given: SU LR, TU LN Prove: ST NR Statements 1. SU LR, TU LN 1. Given Reasons L N R 2. 3. ; 4. 5. 6. 7. 2. Definition of Congruent Segments 3. Segment Addition Postulate 4. Substitution Property 5. Substitution Property 6. 7. 33

Skills Practice: Identify the property that justifies each statement. 1. If HJ + 5 = 20, then HJ = 15 2. If XY + 20 = YW and XY + 20 = DT, then YW = DT 3. If m 1 m 2 90 and m 2 m 3, then m 1 m 3 90 4. If 1 AB 1 EF, then AB = EF 2 2 3 5. If 2( x ) 5, then 2x 3 5 2 6. If m 4 35 and m 5 35, then m 4 m 5 7. If 2 AB 2 CD, then 2AB 2CD 3 3 8. If EF = GH and GH = JK, then EF = JK Use the property to complete the statement. 9. Reflexive Property: If AB AB then 10. Symmetric Property: If AB = CD, then CD = 11. Transitive Property: If m E m F and m F m G, then 12. Multiplication Property: If RS = TU, then x(rs) = 13. Division Property: If 3( m 1 ) 3( m 2), then m 1 14. Transitive Property: If a = bc and bc = de, then 15. Substitution Property: If x = 3c and r = 5x + 7, then Create two-column proofs that prove the following statements using properties. 16. Given: 19 2x 9 17. Given: 3x 2 22 Prove: x 5 Prove: x 8 Statements Reasons Statements Reasons 17. Given: 109 3( 5n 5) 4 Prove: n 8 Statements Reasons 34

18. Given: 1 and 2 are supplementary, m 1 4 x, m 2 80 Prove: m 1 100 1 2 Statements Reasons 19. Given: 1 and 2 are vertical angles, m 1 2x 21, m 2 4x Prove: x 10.5 1 2 Statements Reasons 35

Prove Statements about Segments and Angles Writing a two-column proof is a formal way of organizing your reasons to show a statement is true. Each reason in the right-hand column is the explanation for the corresponding statement. Write a two-column proof for the situations below. Example 1: Given: m 1 m 3 Prove: m EBA m DBC Statements m EBA m 3 m 2 m EBA m 1 m 2 m 1 m 2 m DBC Reasons Transitive Property of Equality Example 2: Given: AC = AB + AB Prove: AB = BC Statements Reasons AB + BC = AC AB + AB = AB + BC AB = BC The reasons used in a proof can include definitions, properties, postulates, and theorems. A is a statement that can be proven. Once you have proven a theorem, you can use the theorem as a reason in other proofs. Theorems: Congruence of Segments Segment congruence is reflexive, symmetric, and transitive. o Reflexive For any segment AB, AB AB o Symmetric If AB CD, then CD AB o Transitive If AB CD and CD EF, then AB EF Congruence of Angles Angle congruence is reflexive, symmetric, and transitive. o Reflexive For any angle A, A A o Symmetric If A B then B A o Transitive If A B and B C, then A C 36

Example 3: Prove this property of midpoints. If you know that M is the midpoint of AB, prove that AB is two times AM and AM is one half of AB. Given: M is the midpoint of AB Prove: AB = 2 AM and AM = 1 AB 2 Statements M is the midpoint of AB AM = MB AM + AM = AB 2 AM = AB 1 AM = AB 2 Given Reasons Definition of Midpoint Definition of congruent segments Segment Addition Postulate Example 4: Complete the proof below. Given: SU LR, TU LN Prove: ST NR SU LR, TU LN Statements SU = ST + TU LR = LN + NR ST + TU = LN + NR ST + LN = LN + NR ST = NR Reasons Definition of Congruent Segments 37

Name the property illustrated by the statement. Skills Practice 1. If DG CT, then CT DG Property: 2. VWX VWX Property: 3. If JK MN and MN XY, then JK XY Property: 4. YZ = ZY Property: Use the property complete the statement. 5. Reflexive Property of Congruence: SE 6. Symmetric Property of Congruence: If, then RST JKL 7. Transitive Property of Congruence: If F J and, then F L Complete the proofs below. 8. Given: AB = 5, BC = 6 Prove: AC = 11 AB = 5, BC = 6 Statements Reasons Segment Addition Postulate Substitution Property 9. Given: RT = 5, RS = 5, RT TS Prove: RS TS R Statements Reasons T S RT = TS RS = TS Transitive Property of Equality 38

10. Given: m 1 45 and m 2 45 Prove: AB is the bisector of DAC Statements Reasons m 1 45 and m 2 45 1 2 Substitution Property of Equality 11. Given: FD bisects EFC and FC bisects DFB Prove: EFD CFB Statements Reasons Given Given EFD DFC DFC CFB Transitive Property of Congruence 12. Given: 1 and 2 are complementary, 1 3, and 2 4 Prove: 3 and 4 are complementary Statements Reasons Given 1 3 Given Def. of Congruence Def. of Congruence m 1 m 2 90 Substitution Property Def. of Complementary Angles 39

13. Given: 1 and 2 form a linear pair and m 2 2( m 1) Prove: m 1 60 Statements Given Given Reasons 1 and 2 are supplementary angles Def. of Supplementary angles Substitution Property Division Property of Equality 14. Given: 1 and 2 are complementary and m 2 46 Prove: m 1 44 Statements Reasons Given Given m 1 m 2 90 15. Given: m 1 m 2 180 and m 1 62 Prove: m 2 118 Statements Reasons 40

Prove Angle Pair Relationships When two lines intersect, pairs of vertical angles and linear pairs are formed. Linear Pair Postulate: If two angles form a linear pair, then they are supplementary. 1 and 2 form a linear pair, so 1 and 2 are supplementary and m 1 m 2 180 Vertical Angles Congruence Theorem: Vertical angles are congruent. Prove the Vertical Angles of Congruence Theorem- 1 3, 2 4 Given: 4 and 3 are vertical angles Prove: 4 3 Statements Reasons 2 and 4 are a linear pair 2 and 4 are supplementary m 2 m 3 180 Def. of Linear Pair Linear Pair Postulate Def. of Supplementary Substitution Property 41

Examples: Use the diagram below to answer the following questions. Note that the diagram is not drawn to scale. 1. If m 1 112, find m 2, m 3, and m 4. 2. If m 2 67, find m 1, m 3, and m 4. 3. If m 4 71, find m 1, m 2, and m 3. 4. Multiple Choice: Which equation can be used to find x? A. 32 (3x 1) 90 B. 32 (3x 1) 180 C. 32 3x 1 D. 3x 1 212 5. Solve for x in Example 4 above. 6. Find m TPS in Example 4 above. Guided Practice 1. Describe the relationship between the angle measures of complementary angles, supplementary angles, vertical angles, and linear pairs. 42

2. Identify the pair(s) of congruent angles in the figures below. Explain how you know they are congruent. a. c. b. d. 3. Find the measure of each numbered angle. a. m 2 57 c. m 5 22 e. m 1 38 b. m 13 4x 11, d. 9 and 10 are f. m 2 4x 26 m 14 3x 1 complementary. m 3 3x 4 7 9, m 41 43

Skills Practice Find the value of x in each figure. 1. 2. 3. 4. 5. 6. 7. What is the value of x if PQR and SQT are vertical angles and m PQR 47 and m SQT 3x 2? 8. Find the measure of an angle that is supplementary to B if the measure of B is 58 44

Find the measure of each numbered angle and name the theorems that justify your work. 9. m 1 x 10 10. m 4 2x 5 11. m 6 7x 24 m 2 3x 18 m 5 4x 13 m 5x 14 12. x = m MAT 13. x = m PIR m RIM 12. Write a two-column proof. Given: 1 and 2 form a linear pair and 2 and 3 are supplementary Prove: 1 3 45