MATH STUDENT BOOK. 8th Grade Unit 6


 Lorin Carson
 1 years ago
 Views:
Transcription
1 MATH STUDENT BOOK 8th Grade Unit 6
2 Unit 6 Measurement Math 806 Measurement Introduction 3 1. Angle Measures and Circles 5 Classify and Measure Angles 5 Perpendicular and Parallel Lines, Part 1 12 Perpendicular and Parallel Lines, Part 2 20 Circles 25 SELF TEST 1: Angle Measures and Circles Polygons 37 Classifying Polygons 37 Interior and Exterior Measures of Polygons 42 Classifying Triangles and the Triangle Inequality Theorem 50 The Quadrilateral Family 57 SELF TEST 2: Polygons Indirect Measure 65 Pythagorean Theorem, Part 1 65 Pythagorean Theorem, Part 2 70 SELF TEST 3: Indirect Measure Review 77 LIFEPAC Test is located in the center of the booklet. Please remove before starting the unit. Section 1 1
3 Measurement Unit 6 Author: Glynlyon Staff Editor: Alan Christopherson, M.S. Westover Studios Design Team: Phillip Pettet, Creative Lead Teresa Davis, DTP Lead Nick Castro Andi Graham Jerry Wingo 804 N. 2nd Ave. E. Rock Rapids, IA MMXIV by Alpha Omega Publications a division of Glynlyon, Inc. All rights reserved. LIFEPAC is a registered trademark of Alpha Omega Publications, Inc. All trademarks and/or service marks referenced in this material are the property of their respective owners. Alpha Omega Publications, Inc. makes no claim of ownership to any trademarks and/ or service marks other than their own and their affiliates, and makes no claim of affiliation to any companies whose trademarks may be listed in this material, other than their own. Some clip art images used in this curriculum are from Corel Corporation, 1600 Carling Avenue, Ottawa, Ontario, Canada K1Z 8R7. These images are specifically for viewing purposes only, to enhance the presentation of this educational material. Any duplication, resyndication, or redistribution for any other purpose is strictly prohibited. Other images in this unit are 2009 JupiterImages Corporation 2 Section 1
4 Unit 6 Measurement Measurement Introduction This unit covers basic geometric concepts associated with angles, parallel and perpendicular lines, and circles. The properties of various polygons and their angle measures are used to solve for missing angle measures. The special relationship that exists between the sides and angles of right triangles, including a proof of the Pythagorean theorem, is also discussed. Objectives Read these objectives. The objectives tell you what you will be able to do when you have successfully completed this LIFEPAC. When you have finished this LIFEPAC, you should be able to: z Classify and measure angles and lines. z Identify and find measures of angles created by transversals. z Identify parts of circles and their measures. z Classify polygons and find measures of their interior and exterior angles. z Classify triangles and use the triangle inequality theorem. z Classify quadrilaterals and the relationships among them. z Find side lengths of right triangles using the Pythagorean theorem. Section 1 3
5 Measurement Unit 6 Survey the LIFEPAC. Ask yourself some questions about this study and write your questions here. 4 Section 1
6 Unit 6 Measurement 1. Angle Measures and Circles Classify and Measure Angles Math is part of our everyday world in everything from balancing checkbooks, to shopping, to measuring a room for new carpet. Math is also a key part of many of the games we play. For instance, the game of pool is nothing but a game of angles. Knowing the correct angle needed to hit the ball is essential to winning a game of pool. One of the best skills you can have if you are a pool player is being able to identify the angle and angle measure needed to sink a shot. Objectives Identify angles by their measure. Classify pairs of angles. Find the measure of an angle. Vocabulary acute angle an angle that measures more than 0 but less than 90 adjacent angles two angles that have a common vertex and side but are not overlapping complementary angles two angles whose sum is 90 obtuse angle an angle that measures more than 90 but less than 180 right angle an angle that measures exactly 90 straight angle an angle that measures exactly 180 supplementary angles two angles whose sum is 180 vertex the point where two line segments, lines, or rays meet to form an angle vertical angles angles that are opposite from one another at the intersection of two lines; vertical angles are congruent Section 1 5
7 Measurement Unit 6 Classifying Angles Angles are a part of our everyday life. Just by looking around the room you are in, you should be able to see hundreds of angles! Each angle or pair of angles has a name and set of properties to define it. You may remember the terms acute angle, right angle, obtuse angle, and straight angle. An acute angle is an angle that measures more than 0 but less than 90. A right angle measures exactly 90, while an obtuse angle measures more than 90 but less than 180. A straight angle, or straight line, measures exactly 180. You can easily classify an angle as acute, right, obtuse, or straight just by looking at it. If you remember that a right angle is like a corner on a piece of paper and a straight angle is a straight line, you can tell what type of angle you have without measuring it. If the angle is smaller than the corner of a piece of paper, you have an acute angle. If it is larger than the corner of a piece of paper but smaller than a line, then you have an obtuse angle. Let s look at how to classify pairs of angles. Adjacent angles are two angles that share a common vertex, point, and side. They are next to one another without overlapping. Two types of angles that are often adjacent are complementary angles and supplementary angles. Complementary angles are two angles whose sum is 90. The two angles can either be adjacent angles or separate angles. The following graphic shows two adjacent, complementary angles. Complementary angles: Supplementary angles are two angles whose sum is 180 ; the angles can either be separate or adjacent. The following graphic shows two adjacent, supplementary angles. Supplementary angles: Angles are also created when lines intersect. When the two lines intersected, they created four angles. The four angles all share a common vertex, where the lines intersect. Some of the angles can also be described as being adjacent. Angle 1 is adjacent to both angle 4 and angle 2. Angle 2 is adjacent to angle 1 and angle 3. Angle 3 is adjacent to angles 2 and 4. Finally, angle 4 is adjacent to angles 3 and 1. 6 Section 1
8 Unit 6 Measurement Within these four angles, there are 2 sets of vertical angles. Vertical angles are angles that are opposite one another but share a vertex. Angles 1 and 3 are vertical angles, because they have the same vertex and are opposite one another. The other set of vertical angles are angles 2 and 4. Again, they have the same vertex and are opposite one another. Vertical angles are easily identifiable, because they are found where lines intersect. Look at the following picture. Three lines intersect at one point. The newly created angles are labeled for you. There are four sets of vertical angles: 1 and 3 2 and 4 5 and 7 6 and 8 Let s look at more sets of vertical angles. See if you can identify them before reading the answers. There are 6 sets of vertical angles: 1 and 4 2 and 3 5 and 8 6 and 7 9 and and 11 Measurements of Angles Now that you know how to classify angles and pairs of angles, we can look into how to find the exact measure of an angle. Remember that complementary angles have a sum of 90, and supplementary angles have a measure of 180. Look at the following examples to see how to find the measure of the angle. Section 1 7
9 Measurement Unit 6 x + 39 = 180 Supplementary angles sum to 180. Subtract 39 from x = both sides. Complete the x = 141 subtractions. The first thing you need to determine is if the pair of angles are complementary or supplementary. The angles are complementary. We can tell this because there is a right angle symbol. We now know that the two angles have a sum of 90. We also know that one of the angles measures x, while the other angle measures 72. We have enough information to set up an equation to find the value of x. x + 72 = 90 Complementary angles sum to 90. Subtract 72 from x = both sides. Complete the x = 18 subtractions. The second angle in the complementary pair of angles measures 18. This time we have a pair of supplementary angles, so we need to set the equation equal to 180. Let s move on to vertical angles. You know that vertical angles are angles that are opposite one another and share a vertex. Vertical angles are also congruent, or equal to one another. This means once you know the measure of one, you automatically know the measure of the other. 1 = 82 What are the measures of the other angles? 1 is a vertical angle with 3. Therefore, we now know that 3 equals 82. We can easily find angles 2 and 4. We know that they are also a set of vertical angles. This means that all we have to do is find the measure of one angle, and we ll know the measure of the other. If we look closely at the image, we see that angle 1 and angle 2 are supplementary, because they form a straight angle. Since supplementary angles equal 180, angle 1 equals 82, and angle 2 equals x, you can now set up an equation. 8 Section 1
10 Unit 6 Measurement x + 82 = 180 x = x = 98 Supplementary angles sum to 180. Subtract 82 from both sides. Complete the subtractions. x = 180 x = x = 45 Supplementary angles sum to 180. Subtract 135 from both sides. Complete the subtractions. Angle 2 measures 98. That means angle 4 also equals = = = 65 What are the measures of the other angles? The easiest way to solve this problem is to use one point of intersection at a time. Let s begin with the intersection that creates angles 1, 2, 3, and 4. So far, we know that 2 = 135. We also know that angles 2 and 3 are vertical angles, which means that they are equal. Finally, we can see that angles 1 and 2 are supplementary. That means they have a sum of 180. We now know that angle 1 equals 45. Since angle 1 and angle 4 are vertical angles, we can determine that angle 4 is also equal to 45. Let s move to the intersection that creates angles 5, 6, 7, and 8. You were told that 8 = 160. That means angle 5 also equals 160, because 8 and 5 are vertical angles. You can see that angles 5 and 6 are supplementary angles, so you know their sum is equal to 180. x = 180 x = x = 20 Supplementary angles sum to 180. Subtract 160 from both sides. Complete the subtractions. If angle 6 equals 20, then angle 7 also equals 20. Finally, let s look at the intersection that creates angles 9, 10, 11, and 12. You know that angle 11 equals 65, so that means angle 9 also equals 65. Angles 9 and 10 are supplementary to one another. x + 65 = 180 Supplementary angles sum to 180. Subtract 65 from x = both sides. Complete the x = 115 subtractions. Angle 10 measures 115, as well as angle 12. Section 1 9
11 Measurement Unit 6 Let s Review Before going on to the practice problems, make sure you understand the main points of this lesson. Identify the types of angles. Classify pairs of angles. Find the measure of angles. Complete the following activities. 1.1 If angle 1 measures 98 and angle 2 measures 72, then the two angles are supplementary. True False 1.2 Complementary angles have a sum of 90. True False 1.3 All pairs of vertical angles are equal. True False 1.4 Adjacent angles are always supplementary. True False 1.5 If angle 1 and angle 5 are vertical angles and angle 1 equals 55, then angle 5 will equal can t be determined 1.6 If each of two complementary angles has the same measure, then each angle will equal Section 1
12 Unit 6 Measurement 1.7 In the following diagram, angle 7 equals 61. What is the measure of angle 8? If two angles are supplementary and one of the angles measures 38, what is the measure of the larger angle? Use this illustration for the following questions. 1.9 Angle 3 is equal to angle Angle 2 is equal to angle Angle 7 is equal to angle Angle 6 is equal to angle. Section 1 11
13 Measurement Unit 6 Perpendicular and Parallel Lines, Part 1 Pictured is the Hearst Tower in New York. The tower earned the 2006 Emporis Skyscraper Award for the best skyscraper in the world completed that year. It was also the first green building in New York. Take a closer look at the triangular framing design on the outside of the building. Can you identify the different types of angles that are created? Can you also identify the different types of lines that exist? Objectives Identify lines as parallel, intersecting, or perpendicular. Identify a transversal and the angles it creates. Find the measure of angles created by a transversal. Vocabulary alternate exterior angles two outside angles that lie on different sides of a transversal alternate interior angles two inside angles that lie on different sides of a transversal corresponding angles two angles in the same position on different lines exterior angles outside angles interior angles inside angles intersecting lines lines that share one point parallel lines lines that never cross one another and are the same distance apart at all times perpendicular lines lines that intersect and create right angles transversals lines that intersect two or more lines to create angles Classifying Lines Every set of lines can be classified as parallel, perpendicular, or intersecting. They are often easy to tell apart, just by looking at them. distance apart. Intersecting lines are lines that cross one another at one point. Finally, perpendicular lines are lines that cross one another and create four right angles at the point of intersection. Parallel lines are two lines that never intersect. They are also always the same 12 Section 1
14 Unit 6 Measurement Sometimes, another line will cross a set of lines in a different direction. This individual line is called a transversal. A transversal is a line that crosses two or more lines and creates angles. Take a look at some examples of transversals. The transversals are the red lines in each picture. Notice that the transversal can cut horizontally, vertically, or diagonally. The transversal also creates two sets of angles known as alternate exterior angles. Alternate exterior angles are angles that are on the opposite sides of the transversal, opposite line, and on the outside of the lines. The last two sets of angles created by the transversal are alternate interior angles. Alternate interior angles are angles that are on the opposite sides of the transversal, opposite line, and are on the inside of the lines. When a transversal cuts across a set of two lines, it creates eight angles, four angles per intersection. The angles created on the two lines have a relationship with one another. One type of angle that is created is called corresponding angles. Corresponding angles are angles that are in the same position but on different lines. There are four sets of corresponding angles. When a transversal cuts across a set of parallel lines, the angles that are created have a special relationship. Corresponding angles, alternate interior angles, and alternate exterior angles are all congruent within their own pairs. Key point! The sets of angles only become congruent when a transversal cuts across a set of parallel lines! Section 1 13
15 Measurement Unit 6 Identify the two sets of alternate interior angles. Assume the lines are parallel. 2 and 7 3 and 6 Identify the two sets of alternate exterior angles. Assume the lines are parallel. 1 and 3 2 and 4 5 and 7 6 and 8 Let s look at some examples where we can use congruency to find the measures of the eight angles created by a transversal. What are the measures of each angle? 1 and 7 2 and 8 Identify the four sets of corresponding angles. Assume the lines are parallel. line a line b 2 = 70 Let s begin by finding the value of 3. You know that angles 2 are 3 are vertical angles, and that vertical angles are congruent. So, the measure of angle 3 is Section 1
16 Unit 6 Measurement You also know that angle 1 and 2 are supplementary, which means their sum is equal to 180. Since angle 2 measures 70, then angle 1 must measure 110. You can now use either vertical angles or supplementary angles to find the measure of angle 4. Either way, angle 4 will measure 110. Take a look at what you have so far. Finally, use vertical angles to find angle 7. Since angles 6 and 7 are vertical, and angle 6 measures 70, then angle 7 also measures 70. Here is the completed diagram with angle measures. Now, let s use the new angle relationships you just learned about. You can use corresponding angles, alternate interior angles, or alternate exterior angles to move from line a to line b. Let s use corresponding angles this time. Angle 1 is in the upper left of the intersection on line a. Angle 5 is in the upper left of the intersection on line b. Because lines a and b are parallel, you know that angle 1 and angle 5 are congruent. So, angle 5 has a measure of 110. Now you can use vertical angles to find that the measure of angle 8 is also 110. Again, use supplementary angles to find the measure of angle 6. Since angle 5 measures 110, then angle 6 must measure 70. See if you can spot the four sets of corresponding angles, the two sets of alternate interior angles, and the two sets of alternate exterior angles. Is each set congruent? Find the measure of each angle. line x line y 6 = 47 You know that angle 1 is a vertical angle to angle 6; so, it also must have a measure of 47. You also know that angle 1 and angle 2 are supplementary, so they have Section 1 15
17 Measurement Unit 6 a sum of 180. Since 47 of the 180 is occupied by angle 1, that leaves angle 2 to measure 133. Angle 5 is a vertical angle to angle 2; so, that means it also has a measure of 133. This time, let s use alternate exterior angles to move between the lines. Angle 5 is an outside angle, so its alternate exterior angle is angle 4. Since the lines are parallel, these two angles are congruent, giving angle 4 a measure of 133. Angle 4 is a vertical angle to angle 7; so, angle 7 also has a measure of 133. That leaves angles 3 and 8 to find. Angle 3 corresponds to angle 1; so that means they have the same measure of 47. Since angle 3 and angle 8 are vertical, angle 8 also has a measure of 47. There is no one correct way to find the measures of the angles, so don t worry if you and your friend have different ways of finding the answers. Let s see what happens when the set of lines being crossed by the transversal aren t parallel. Find the measures of each angle, if m 1 = 61 and m 6 = 102. Be Careful! Remember that transversals always create sets of corresponding angles, alternate interior angles, and alternate exterior angles. However, the sets of angles are only congruent when the transverse set of lines is parallel. Let s start with line m. Angles 1 and 4 are vertical angles; so, angle 4 must also measure 61. Angles 1 and 2 are supplementary angles; so, that means that angle 2 measures 119. Finally, for line m, angles 2 and 3 are vertical angles; so, angle 3 must also measure 119. Even though corresponding angles, alternate exterior angles, and alternate interior angles exist between the two lines, they are not congruent, so we can t equate the measures. If you look back at the original problem, you were given the measure for angle 6. Since angle 6 is a vertical angle with angle 7, you know that the measure of angle 7 is also 102. Angle 6 and angle 8 are supplementary angles; so, that means angle 8 has to have a measure of 78. Angle 8 is a vertical angle with angle 5, which means that angle 5 also has a measure of Section 1
18 Unit 6 Measurement Let s Review Before going on to the practice problems, make sure you understand the main points of this lesson. Identify lines as parallel, intersecting, or perpendicular. Identify a transversal and the angles it creates. Identify the sets of angles created by a transversal. Find the measure of angles created by a transversal. Complete the following activities. Use the following illustration for Assume the lines are parallel Select all pairs of corresponding angles. 1 and 5 2 and 6 2 and 5 3 and 7 3 and Select all pairs of alternate interior angles. 3 and 4 3 and 6 3 and 5 4 and Select all pairs of alternate exterior angles. 1 and 2 1 and 8 1 and 7 2 and 7 4 and 7 4 and 8 4 and 6 5 and 6 2 and 8 7 and 8 1 and 6 none none none Section 1 17
19 Measurement Unit 6 Use the following illustration for Assume the lines are parallel Select all of the angles that have the same measure as angle Select all of the angles that have the same measure as angle can t be determined 8 can t be determined 1.18 The value of x that makes lines a and b parallel is. 18 Section 1
20 Unit 6 Measurement 1.19 The measure of angle 7 is Line is a transversal. Use the following illustration for Assume the lines are parallel Angle 4 corresponds with angle Find the measure of each angle. m 2 =. m 3 =. m 4 =. m 6 =. m 7 =. m 8 =. m 5 =. Section 1 19
21 Measurement Unit 6 Perpendicular and Parallel Lines, Part 2 What is the only angle from which you should deal with a problem? Answer: Q ZKNQFUST! Can you figure out what the answer to the question is? Try using the secret decoder ring to help figure it out. Use the outside circle for the letter in the code. Replace the code letter with the letter on the inner circle. This lesson focuses on finding the measures of angles when there are variables in the angles. It is similar to being written in code, except this code is in algebraic form with variables. Objectives Identify the relationships between angles created by a transversal across parallel lines. Find the measure of the angles created by a transversal across parallel lines. Finding Angle Measures The only angle from which you should deal with a problem is a TRYangle! The study of writing in code is called cryptography. Sometimes governments send coded messages, and, sometimes, you and your friends do, too. Have you ever written a note in code, so that only you and the person you are writing to know what it says? Previously, you learned about vertical angles, complementary angles, supplementary angles, and angles created by transversals. We are going to continue our study of these angles, but this time we are going to have variables and variable expressions as our angle measures. Just like any other time you see a variable in an equation, you need to find the value of the variable to find the angle measure. Let s begin by working with complementary and supplementary angles. Remember that complementary angles have a sum of 90, and supplementary angles have a sum of Section 1
22 Unit 6 Measurement Find the measure of the two angles. We have a pair of supplementary angles. So, we need to set the equation equal to 180. Be careful the value of 15 doesn t tell us either angle measure. It does, however, help us, because we now know the value of x. All we have to do is plug 15 into the expression for each angle. x + 3x = 180 4x = 180 = Supplementary angles sum to 180. Complete the addition on the left. Divide both sides by 4. 4x = 4(15 ) = 60 2x = 2(15 ) = 30 x = 45 Complete the division. We know that x is 45. That means we can also find the other angle of 3x. 3x = 3(45 ) = 135 The two angle measures are 45 and 135. We have a set of supplementary angles, which sum to 180. Find the measure of each angle. This is a set of complementary angles. So, the sum of the angles will be equal to 90. 4x + 2x = 90 6x = 90 = Complementary angles sum to 90. Complete the addition on the left. Divide both sides by 6. x = 15 Complete the division. 2x + (5x + 33 ) = 180 Supplementary angles sum to x + 33 = 180 Add the variables. Subtract 33 from 7x = both sides. = x = 21 Divide both sides by 7. Complete the division. So far, we know that x = 21. But, that doesn t tell us either angle measure. To find the angle measures, we need to plug 21 in for x in each expression. 2x = 2(21 ) = 42 5x + 33 = 5(21 ) + 33 = = 138 Section 1 21
23 Measurement Unit 6 The two angles measure 42 and 138. You can check your work by adding them together to make sure you get a sum of 180. x + 34 = = 51 Now, let s look at what happens when we replace angle measures in vertical angles. Remember that vertical angles are always equal to one another. Find the measure of the shaded angle. Find the value of x and, then, the measure of the vertical angles. Since vertical angles are equal in measure, we need to put each angle on either side of the equal sign. 3x = x + 34 You solve this equation just as you would solve any other equation, by solving for x. 3x = x + 34 Vertical angles are congruent. 3x  x = x  x + 34 Subtract x from both sides. 2x = 34 Simplify. = Divide both sides by 2. x = 17 Complete the division. Now that you know the value of x, plug that value into each angle s expression to make sure that they are equal to one another. 3x = 3(17 ) = 51 We see that the two angles for which we have values are supplementary angles. So, we can add them together to get a sum of 180. (4x + 25 ) + (7x  21 ) = x + 4 = x = x = 176 = x = 16 Supplementary angles sum to 180. Combine the variables and constants on the left side. Subtract 4 from both sides. Complete the subtraction. Divide both sides by 11. Complete the division. Now, plug that value into the expression 7x  21, because it is a vertical angle to the shaded angle. 7x  21 = 7(16 )  21 = = 91 The measure of the shaded angle is Section 1
24 Unit 6 Measurement Finally, let s take a look at angles that are created by a transversal. Find the measure of each angle. You are given that m 5 = 3x and m 6 = 2x Angles 5 and 6 are supplementary angles; so, we know they have a sum of 180. Therefore, we begin by adding their values and setting it equal to x + (2x + 25 ) = 180 Supplementary angles sum to 180. Combine the 5x + 25 = 180 variables on the left side. Subtract 25 from 5x = both sides. Complete the 5x = 155 subtraction. = x = 31 Divide both sides by 5. Complete the division. This just tells us the value of x, not an angle measure. We need to plug 31 into each of the expressions to get the angle measures. m 5 = 3x = 3(31 ) = 93 Angle 5 measures 93. m 6 = 2x + 25 = 2(31 ) + 25 = = 87 Angle 6 measures 87. Now you can use vertical angles to finish this intersection. Angle 2 is a vertical angle with angle 5; so, it measures 93. Angle 1 is a vertical angle with angle 6; so, it has a measure of 87. If you use corresponding angles, angle 7 corresponds with angle 5; so, angle 7 also measures 93. If angle 7 measures 93, then angle 4 also measures 93. Angle 8 corresponds to angle 6; so, it has a measure of 87. Because angle 3 is a vertical angle with angle 8, it too has a measure of 87. Let s Review Before going on to the practice problems, make sure you understand the main points of this lesson. A transversal across parallel lines creates corresponding, alternate interior, and alternate exterior angles. You can use congruence among angles, complementary relationships, and supplementary relationships to calculate the measures of unknown angles. Section 1 23
25 Measurement Unit 6 Complete the following activities Use the following diagram to answer the questions. Given A B. The value of x is. The measure of 5 is. The measure of 1 is. The measure of 6 is. The measure of 2 is. The measure of 7 is. The measure of 3 is. The measure of 8 is. The measure of 4 is. Angles 2 and 3 are angles Use the diagram and information to match the following angles with their correct measures. Given Z = 2b + 6 and Y = 3b  1. W X Y Z Section 1
26 Unit 6 Measurement Circles Think back to kindergarten when you first began to learn about shapes. You knew that the triangle had three sides, the square had four sides, and that the circle was the round shape without any sides. You could even identify things around the classroom that were those shapes. But, do you know that there is more to these shapes, especially the circle, then what they taught you in kindergarten? This lesson will focus on the different parts of a circle. Objectives Identify the parts of a circle. Classify angles of circles. Find the measures of arcs and angles of circles. Vocabulary arc part of the circumference of a circle between two points central angle an angle that has the center of a circle as its vertex chord a line segment that connects two points on a circle circle a figure with all of its points the same distance from the center circumference of a circle the length around the exterior of a circle; measured in degrees in terms of an arc diameter a line segment that goes through the center of a circle to connect two points on the circle inscribed angle an angle in a circle with a vertex on the circle and sides that are chords of the circle intercepted arc the part of the circumference of a circle that is bound by two endpoints of a chord, radius, secant line or tangent line major arc an arc with a measure greater than 180 degrees minor arc an arc with a measure less than 180 degrees radius a line segment from the center of a circle to any point on the circle secant a line that goes through the circle at two distinct points on the circumference of a circle; the distance between the two points is considered a chord Section 1 25
27 Measurement Unit 6 semicircle an arc with a measure of 180 degrees tangent a line that touches the exterior part of the circumference of a circle at exactly one point You know that a circle is a round shape that doesn t have any sides, but do you know its real definition? A circle is a figure that has all of its points the same distance from the center. A circle is named by its center point. Look at the following figure. Notice the center dot labeled A. Each of the points that surround this point is the same distance as the other points. Now that you know what a circle really is, let s look into the parts of the circle. Sometimes, there are line segments that connect two points on the circle. This line segment might not always go through the center of the circle. If a line segment connects two points on the circle, then it is called a chord. Look at the picture of some examples of chords. Chords do not all have the same measure because they can have different lengths. A chord can sometimes be a diameter but a chord can never be a radius. Inside the Circle The radius is a line segment that connects the center of a circle to any point on the circle. It doesn t matter where you connect the center of the circle to the edge to form the radius, because all the points that make up the circle are the same distance from the center. Therefore, every line segment drawn from the center of the circle to the edge of the circle has the same measure for that circle. A diameter is a special type of chord. Like all other chords it connects two points on a circle. But, for a chord to be a diameter, it must go through the center of the circle. Another way to think of a diameter is two 26 Section 1
28 Unit 6 Measurement radiuses put together to form a straight line. The measure of a diameter is twice that of the radius. travels through the inside of a circle, while the secant does. Did you know! A radius is half the measure of the diameter. So, r = d and d = 2r. Angles of a Circle Outside the Circle So far, we have looked at line segments inside the circle, but what happens when there are lines, too? The first line we will look at is called a secant. A secant is made up of a chord but it continues past the edge of the circle. You know that an angle is two rays or line segments that share a common vertex. The same is true for angles found in a circle. The first kind of angle we are going to look at is called a central angle. A central angle is an angle that has the center point of the circle as its vertex. Each leg of a central angle is a radius of the circle. Look at the example where the central angle is BAD. Central Angle There is another line that is found completely on the outside of the circle. A tangent is a line that touches the circle at exactly one point. It is easy to tell a tangent from a secant, because a tangent line never There is another type of angle found within a circle. It is called an inscribed angle. An inscribed angle is an angle in a circle with a vertex on the circle and with sides that are chords of the circle. This is a little different from the central angle, because Section 1 27
29 Measurement Unit 6 an inscribed angle s vertex is not the center of the circle but a point on the circle. Look closely at the following picture; do you see the difference between the inscribed angle, XYZ, and the central angle shown previously? Inscribed Angle Reminder! A circle has a measure of 360. Arcs can be all different sizes. Similar to angles of triangles, we can classify arcs by their measure. A minor arc is an arc that has a measure less than 180. Arcs of a Circle What is an arc? An arc is part of a circle between two points on the circle. Look at the following examples. This might help! If an arc is half of the circle, then it is a semicircle. If the arc is less than half of the circle, then it is a minor arc. If the arc is more than half of the circle, then it is a major arc. A semicircle is an arc with a measure of exactly 180. Circle A has the following arcs: 28 Section 1
30 Unit 6 Measurement An arc that has a measure greater than 180 is called a major arc. There is also a special arc called an intercepted arc. An intercepted arc is an arc that is created by an inscribed angle. Remember that an inscribed angle is an angle whose vertex is on the circle. Thus, the intercepted arc is the part of the circle between the legs of this angle. Look at the following picture to help you visualize an intercepted arc. intercepted arc created by the angle. For example, if the measure of a central angle is 79, then the measure of the arc created is also 79. It also works the other way. If you know the measure of the arc created by a central angle is 17, then you know that the measure of the central angle is also 17. An inscribed angle is an angle whose vertex is on the circle and whose legs are chords of the circle. The intercepted arc created by the inscribed angle is two times larger than the measure of the inscribed angle. Let s look at some examples. What is the measure of arc BD? Phew! That was a lot of information to digest all at once. Before learning more about the measures of the angles and arcs of a circle, consider reviewing the definition and picture for each part of a circle. The first step is to determine if the angle that creates the arc is a central angle or an inscribed angle. The angle, BAD, is a central angle. That means the arc will have the same measure. The angle measures 63 ; so, arc BD also measures 63. Measures of Angles and Arcs Central angles are the angles whose vertex is the center of the circle and the legs of the angle are radii. The measure of a central angle is equal to the measure of the Section 1 29
31 Measurement Unit 6 What is the measure of arc HT? The arc is formed from an inscribed angle. So, this time the measure of the angle is half of the arc. To find the measure of the arc, multiply the measure of the angle by (2) = 148 What is the value of x? Since the arc is created by a central angle, we can set the two measures equal to one another. 5x + 9 = 144 5x = 135 x = 27 What is the measure of arc RB? This time, you know the angle has a measure of 62, and that it is an inscribed angle. This means that the arc measure is twice the angle s measure. 62 (2) = 124 What is the measure of x? We have an intercepted arc and an inscribed angle. That means that we need to multiply the angle measure by 2 and, then, set it equal to the arc measure. 30 Section 1
32 Unit 6 Measurement 2(3x + 10 ) = 104 6x + 20 = 104 6x = 84 x = 14 Be sure that you understand the relationships that exist between the different angles and the arcs that are created by them. A central angle and its arc have the same measure, while an inscribed angle has half the measure of the intercepted arc. Let s Review Before going on to the practice problems, make sure you understand the main points of this lesson. The parts of a circle include the center, equidistant points, central angles, inscribed angles, arcs, intercepted arcs and chords. The measure of a central angle is equal to the measure of its intercepted arc. The measure of an inscribed angle is half the measure of its intercepted arc. The measure of an intercepted arc is twice the measure of its inscribed angle. Complete the following activities A radius is the diameter A semicircle measures degrees A angle has the same measure as its arc A circle is named by its A diameter is also a If an inscribed angle measures 67, how would you find the intercepted arc? Set the angle and arc measures Multiply the angle measure by equal. two. Divide the angle measure by two A line that intersects a circle at exactly one point is called a. chord tangent secant Section 1 31
33 Measurement Unit If an arc measures 179, what kind of arc is it? minor arc major arc semicircle 1.33 If the measure of a central angle is 28, then to find the measure of its arc, you would need to do which of the following? Nothing, they are equal. Divide the angle measure by 2. Multiply the angle measure by A chord is a diameter. always sometimes never 1.35 If the diameter of the circle is 11 inches, then the radius will be inches If the measure of an intercepted arc is 118, and the measure of the inscribed angle is 2x  1, what is the equation to find the value of x? 2x  1 = 118 2x  1 = 2(118 ) 2(2x  1) = The measure of the arc created by a central angle is 57. What type of arc is this? minor arc major arc semicircle 1.38 The measure of an inscribed angle is 110. What is the measure of the intercepted arc? A secant line contains a chord. always sometimes never Review the material in this section in preparation for the Self Test. The Self Test will check your mastery of this particular section. The items missed on this Self Test will indicate specific area where restudy is needed for mastery. 32 Section 1
34 Unit 6 Measurement SELF TEST 1: Angle Measures and Circles Complete the following activities (6 points, each numbered activity) Match each picture with its correct label. complementary angles acute angle straight angle intersecting lines minor arc inscribed angle secant chord Section 1 33
35 Measurement Unit If a central angle measures 87, then its arc will measure. half as much the same twice as much 1.03 Given 1 = 123 and 8 = 7z What is the value of z, if lines a and b are parallel? What is the measure of x in the diagram? If an intercepted arc measures 124, what is the measure of its inscribed angle? the same, 124 twice the measure, 248 half the measure, Two angles are supplementary. One angle measures 2x, and the other angle measures 3x Which equation can you use to find the measure of each angle? 2x + 3x  15 = 90 2x  3x  15 = 90 2x  3x  15 = 180 2x + 3x  15 = If corresponding angles are on parallel lines, then their measure is the same. always sometimes never 1.08 The measure of a central angle is 3x + 18, and the measure of its arc is 147. Which equation can you use to find the value of x? 2(3x + 18) = 147 3x + 18 = 2(147 ) 3x + 18 = Section 1
36 Unit 6 Measurement 1.09 If two angles are complementary, then the sum of their angles is equal to. each other What is the measure of x? If one angle of a set of alternate interior angles on parallel lines measures 77, then the other angle also equals 77. always sometimes never In the graphic, one pair of vertical angles is In the image, the corresponding angle to angle 1 is angle. Section 1 35
37 Measurement Unit If one angle of a set of supplementary angles measures 77, then the other angle measures If one angle of a set of vertical angles measures 63, the sum of the vertical angles is SCORE TEACHER initials date 36 Section 1
38 MAT0806 May 14 Printing ISBN N. 2nd Ave. E. Rock Rapids, IA
MATH STUDENT BOOK. 7th Grade Unit 1
MATH STUDENT BOOK 7th Grade Unit 1 Unit 1 Integers Math 701 Integers Introduction 3 1. Integers 5 Integers on the Number Line 5 Comparing and Ordering Integers 10 Absolute Value 15 Self Test 1: Integers
More informationMATH STUDENT BOOK. 7th Grade Unit 9
MATH STUDENT BOOK 7th Grade Unit 9 Unit 9 Measurement and Area Math 709 Measurement and Area Introduction 3 1. Perimeter 5 Perimeter 5 Circumference 11 Composite Figures 16 Self Test 1: Perimeter 24 2.
More informationInscribed Angle Theorem and Its Applications
: Student Outcomes Prove the inscribed angle theorem: The measure of a central angle is twice the measure of any inscribed angle that intercepts the same arc as the central angle. Recognize and use different
More informationMATH STUDENT BOOK. 9th Grade Unit 7
MATH STUDENT BOOK 9th Grade Unit 7 Unit 7 Radical Expressions Math 907 Radical Expressions INTRODUCTION. REAL NUMBERS RATIONAL NUMBERS IRRATIONAL NUMBERS COMPLETENESS 9 SELF TEST. OPERATIONS SIMPLIFYING
More informationMATH STUDENT BOOK. 10th Grade Unit 10
MATH STUDENT BOOK 10th Grade Unit 10 MATH 1010 Geometry Review INTRODUCTION 3 1. GEOMETRY, PROOF, AND ANGLES 5 GEOMETRY AS A SYSTEM 5 PROOF 8 ANGLE RELATIONSHIPS AND PARALLELS 14 SELF TEST 1 21 2. TRIANGLES,
More informationConstructions, Definitions, Postulates, and Theorems constructions definitions theorems & postulates
Constructions, Definitions, Postulates, and Theorems constructions definitions theorems & postulates Constructions Altitude Angle Bisector Circumcenter Congruent Line Segments Congruent Triangles Orthocenter
More informationPostulate 17 If two points lie in the same plane, then the line containing them lies in the plane.
Geometry Postulates and Theorems Unit 1: Geometry Basics Postulate 11 Through any two points, there exists exactly one line. Postulate 12 A line contains at least two points. Postulate 13 Two lines
More informationContent Area: GEOMETRY Grade 9 th Quarter 1 st Curso Serie Unidade
Content Area: GEOMETRY Grade 9 th Quarter 1 st Curso Serie Unidade Standards/Content Padrões / Conteúdo Learning Objectives Objetivos de Aprendizado Vocabulary Vocabulário Assessments Avaliações Resources
More informationSTANDARDS OF LEARNING CONTENT REVIEW NOTES GEOMETRY. 3 rd Nine Weeks,
STANDARDS OF LEARNING CONTENT REVIEW NOTES GEOMETRY 3 rd Nine Weeks, 20132014 1 OVERVIEW Geometry Content Review Notes are designed by the High School Mathematics Steering Committee as a resource for
More informationGreenwich Public Schools Mathematics Curriculum Objectives. Geometry
Mathematics Curriculum Objectives Geometry June 30, 2006 NUMERICAL AND PROPORTIONAL REASONING Quantitative relationships can be expressed numerically in multiple ways in order to make connections and simplify
More information101 Circles and Circumferences
Geometry: 101 Circles and Circumferences Read pages 863 686 and answer the following questions. 1. Define a circle: 2. Define radius: 3. Define Chord: 4. Define diameter 5. Answer the following questions
More informationAREA OF POLYGONS AND COMPLEX FIGURES
AREA OF POLYGONS AND COMPLEX FIGURES Area is the number of nonoverlapping square units needed to cover the interior region of a twodimensional figure or the surface area of a threedimensional figure.
More informationMATH STUDENT BOOK. 8th Grade Unit 4
MATH STUDENT BOOK 8th Grade Unit 4 Unit 4 Proportional Reasoning Math 804 Proportional Reasoning Introduction 3 1. Proportions 5 Proportions 5 Applications 11 Direct Variation 16 SELF TEST 1: Proportions
More informationGeometry. Robert Taggart
Geometry Robert Taggart Table of Contents To the Student............................................. v Unit 1: Lines and Angles Lesson 1: Points, Lines, and Dimensions........................ 3 Lesson
More informationPostulates. Postulates. 10 Parallel Postulate. 1 Two Points Determine a Line. 2 Three Points Determine a Plane. 11 Perpendicular Postulate
Postulates 1 Two Points Determine a Line Through any two points there is exactly one line. (p. 14) 2 Three Points Determine a Plane Through any three points not on a line there is exactly one plane. (p.
More informationChapter 6 Notes: Circles
Chapter 6 Notes: Circles IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of the circle. Any line segment
More informationMATH STUDENT BOOK. 9th Grade Unit 10
MATH STUDENT BOOK 9th Grade Unit 10 Unit 10 Quadratic Equations and a Review of Algebra Math 910 Quadratic Equations and a Review of Algebra INTRODUCTION 3 1. QUADRATIC EQUATIONS 5 IDENTIFYING QUADRATIC
More informationGEOMETRY A CURRICULUM OUTLINE OVERVIEW:
GEOMETRY A CURRICULUM OUTLINE 20112012 OVERVIEW: 1. Points, Lines, and Planes 2. Deductive Reasoning 3. Parallel Lines and Planes 4. Congruent Triangles 5. Quadrilaterals 6. Triangle Inequalities 7. Similar
More informationIndicate whether the statement is true or false.
Indicate whether the statement is true or false. 1. The bearing of the airplane with the path shown is N 30 E. 2. If M and N are the midpoints of nonparallel sides and of trapezoid HJKL, then. 3. For a
More informationChapters 6 and 7 Notes: Circles, Locus and Concurrence
Chapters 6 and 7 Notes: Circles, Locus and Concurrence IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of
More informationGeometry MASTER Theorem Notebook List
Geometry MASTER Theorem Notebook List 1.4: Introductions midpoint of a segment (use glossary for this one) Midpoint Formula Distance Formula (use the index in the back of the book to find the correct page)
More informationGeometry Notes PERIMETER AND AREA
Perimeter and Area Page 1 of 17 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter
More informationPrep for LA Geometry EOC Assessment
Prep for LA Geometry EOC Assessment This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence
More informationMATH STUDENT BOOK. 10th Grade Unit 6
MTH STUDENT BOOK 10th Grade Unit 6 Unit 6 Circles MTH 1006 Circles INTRODUCTION 3 1. CIRCLES ND SPHERES 5 CHRCTERISTICS OF CIRCLES 5 CHRCTERISTICS OF SPHERES 9 SELF TEST 1 11 2. TNGENTS, RCS, ND CHORDS
More information1. A student followed the given steps below to complete a construction. Which type of construction is best represented by the steps given above?
1. A student followed the given steps below to complete a construction. Step 1: Place the compass on one endpoint of the line segment. Step 2: Extend the compass from the chosen endpoint so that the width
More informationUse the angle sum property of triangles to find the measure of the missing angle.
ngle Sum Property of Triangles The sum of the measures of the interior angles of a triangle is 180º. This is called the angle sum property of triangles, and it applies to LL triangles, regardless of shape
More informationUnit 5: Angles. Name: Teacher: Period:
Unit 5: Angles Name: Teacher: Period: 1 Aim: SWBAT Identify Complementary, Supplementary, and Vertical angles. DO NOW: Read and answer the following question Angle: Formed by 2 rays with a common endpoint.
More information50 Important Math Concepts You Must Know for the ACT
50 Important Math Concepts You Must Know for the ACT 9142323743; alan@sheptin.com The ACT Mathematics exam provides a comprehensive tour through the foundation courses in secondary school Mathematics.
More information1. Vertical angles are congruent.
1. There exists a unique line through any two points. 2. If A, B, and C are three distinct points lying on the same line, then one and only one of the points is between the other two. 3. If two lines intersect
More informationThe measure of an arc is the measure of the central angle that intercepts it Therefore, the intercepted arc measures
8.1 Name (print first and last) Per Date: 3/24 due 3/25 8.1 Circles: Arcs and Central Angles Geometry Regents 20132014 Ms. Lomac SLO: I can use definitions & theorems about points, lines, and planes to
More informationDoverSherborn High School Mathematics Curriculum Geometry Level 2/CP
Mathematics Curriculum A. DESCRIPTION This is the traditional geometry course with emphasis on the student s understanding of the characteristics and properties of two and threedimensional geometry.
More informationGTPS Curriculum Geometry
3 weeks Topic: 1Points, Lines, Planes, and Angles /Enduring G.CO.1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point,
More informationMath Structures II Chapter 11 Study Guide. Section 1
Math Structures II Chapter 11 Study Guide Section 1 630 1. In what country could we stay that geometry got started? 2. Who came up with an approximation for pi? 3. What kind of geometry skill might be
More informationAngles are what trigonometry is all about. This is where it all started, way back when.
Chapter 1 Tackling Technical Trig In This Chapter Acquainting yourself with angles Identifying angles in triangles Taking apart circles Angles are what trigonometry is all about. This is where it all started,
More informationGeometry Work Sheets. Contents
Geometry Work Sheets The work sheets are grouped according to math skill. Each skill is then arranged in a sequence of work sheets that build from simple to complex. Choose the work sheets that best fit
More informationMath News! 4 th Grade Math Module 4: Angle Measure and Plane Figures. Focus Area Topic A Lines and Angles
Grade 4, Module 4, Topic A Module 4: Angle Measure and Plane Figures This document is created to give parents and students a better understanding of the math concepts found in Eureka Math ( 2013 Common
More informationMath Review Large Print (18 point) Edition Chapter 3: Geometry
GRADUATE RECORD EXAMINATIONS Math Review Large Print (18 point) Edition Chapter 3: Geometry Copyright 2010 by Educational Testing Service. All rights reserved. ETS, the ETS logo, GRADUATE RECORD EXAMINATIONS,
More informationCircle Name: Radius: Diameter: Chord: Secant:
12.1: Tangent Lines Congruent Circles: circles that have the same radius length Diagram of Examples Center of Circle: Circle Name: Radius: Diameter: Chord: Secant: Tangent to A Circle: a line in the plane
More informationDefinitions, Postulates and Theorems
Definitions, s and s Name: Definitions Complementary Angles Two angles whose measures have a sum of 90 o Supplementary Angles Two angles whose measures have a sum of 180 o A statement that can be proven
More informationName Period Date MATHLINKS GRADE 8 STUDENT PACKET 12 LINES, ANGLES, AND TRIANGLES
Name Period Date 812 STUDENT PACKET MATHLINKS GRADE 8 STUDENT PACKET 12 LINES, ANGLES, AND TRIANGLES 12.1 Angles and Triangles Use facts about supplementary, complementary, vertical, and adjacent angles
More informationCurriculum Map by Block Geometry Mapping for Math Block Testing 20072008. August 20 to August 24 Review concepts from previous grades.
Curriculum Map by Geometry Mapping for Math Testing 20072008 Pre s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)
More informationUnit 5 Geometry: Congruence, Constructions, and Parallel Lines. Name:
Unit 5 Geometry: Congruence, Constructions, and Parallel Lines Name: Lesson 5.1 Measuring and Drawing Angles Drawing Angles 1. Construct a ray and label the vertex 2. Place protractor guide over the vertex
More informationGEOMETRY CONCEPT MAP. Suggested Sequence:
CONCEPT MAP GEOMETRY August 2011 Suggested Sequence: 1. Tools of Geometry 2. Reasoning and Proof 3. Parallel and Perpendicular Lines 4. Congruent Triangles 5. Relationships Within Triangles 6. Polygons
More informationGreater Nanticoke Area School District Geometry
Greater Nanticoke Area School District Geometry Geometry emphasizes deductive and inductive reasoning and logic and written analysis of arguments to understand various types of proofs. The course includes
More informationUnits: 10 high school credits UC Requirement Category: c General Description:
Summer 2015 Units: 10 high school credits UC Requirement Category: c General Description: GEOMETRY Grades 912 Geometry is a twosemester course exploring and applying geometric concepts to develop a mathematical
More informationCircle Geometry. Properties of a Circle Circle Theorems:! Angles and chords! Angles! Chords! Tangents! Cyclic Quadrilaterals
ircle Geometry Properties of a ircle ircle Theorems:! ngles and chords! ngles! hords! Tangents! yclic Quadrilaterals http://www.geocities.com/fatmuscle/hs/ 1 Properties of a ircle Radius Major Segment
More information1  GEOMETRY: INTRODUCTION
Geometry Unit Lesson Title Lesson Objectives 1  GEOMETRY: INTRODUCTION Geometry and the World Recognize and describe connections between geometry, the world, and God Nature of Mathematics Restate important
More informationCOURSE: 151 Geometry GRADE LEVEL: 11
COURSE: 151 Geometry GRADE LEVEL: 11 MAIN/GENERAL TOPIC: GEOMETRY (GG #19) COORDINATE GEOMETRY (GG #6274) (GG #1723) SUBTOPIC: ESSENTIAL QUESTIONS: WHAT THE STUDENTS WILL KNOW OR BE ABLE TO DO: Geometry
More information9. Identify and name convex polygons and regular polygons. 10. Find the measures of interior angles and exterior angles of convex polygons. 11. Use in
Objectives After completing Chapter 1: Points, Lines, Planes, and Angles, the student will be able 1. Use the undefined terms point, line, and plane. 2. Use the terms collinear, coplanar, and intersection.
More informationCentroid: The point of intersection of the three medians of a triangle. Centroid
Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:
More informationGrade 7 & 8 Math Circles Math Circles Review April 2/3, 2013
Faculty of Mathematics Waterloo, Ontario NL 3G1 Grade 7 & 8 Math Circles Math Circles Review April /3, 013 Fibonacci & the Golden Ratio The first few terms (or numbers) in the Fibonacci Sequence are 1,
More informationMaterials: Students: The use of GeoGebra dynamic worksheets Teachers: Projection of GeoGebra dynamic worksheets
Circles & Angles: The following series of worksheets were written to help students discover some relationships with angles that are created by tangents, chords, and secants in circles. Lesson: Central
More informationGeometry Seamless Curriculum Guide. Geometry
Geometry QUALITY STANDARD # 1: FUNDAMENTALS OF GEOMETRY 1.1 The student will understand and use the language of Geometry. Undefined terms of point, line, and plane Use appropriate notation Identify, name,
More informationcircumscribed circle Vocabulary Flash Cards Chapter 10 (p. 539) Chapter 10 (p. 530) Chapter 10 (p. 538) Chapter 10 (p. 530)
Vocabulary Flash ards adjacent arcs center of a circle hapter 10 (p. 539) hapter 10 (p. 530) central angle of a circle chord of a circle hapter 10 (p. 538) hapter 10 (p. 530) circle circumscribed angle
More informationAce Your Math Test Reproducible Worksheets
Ace Your Math Test Reproducible Worksheets These worksheets practice math concepts explained in Geometry (ISBN: 9780766037830), written by Rebecca Wingard Nelson. Ace Your Math Test reproducible worksheets
More informationThe Inscribed Angle Alternate A Tangent Angle
Student Outcomes Students use the inscribed angle theorem to prove other theorems in its family (different angle and arc configurations and an arc intercepted by an angle at least one of whose rays is
More informationConjectures. Chapter 2. Chapter 3
Conjectures Chapter 2 C1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C2 Vertical Angles Conjecture If two angles are vertical
More informationAngle Addition Postulate: If ABC. and CBD. are adjacent, then m ABD m ABC m CBD.
Mr. Cheung s Geometry Cheat Sheet Theorem List Version 7.0 Updated 3/16/13 (The following is to be used as a guideline. The rest you need to look up on your own, but hopefully this will help. The original
More informationAlgebra Geometry Glossary. 90 angle
lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:
More informationTopics Covered on Geometry Placement Exam
Topics Covered on Geometry Placement Exam  Use segments and congruence  Use midpoint and distance formulas  Measure and classify angles  Describe angle pair relationships  Use parallel lines and transversals
More informationStudent Name: Teacher: Date: District: MiamiDade County Public Schools. Assessment: 9_12 Mathematics Geometry Exam 1
Student Name: Teacher: Date: District: MiamiDade County Public Schools Assessment: 9_12 Mathematics Geometry Exam 1 Description: GEO Topic 1 Test: Tools of Geometry Form: 201 1. A student followed the
More informationGeometry. Concepts: Practice cooperative learning skills and apply them to logical thought exercises.
Geometry Standards and Inductive Reasoning G 8.2 G 8.3 G 8.1 Practice cooperative learning skills and apply them to logical thought exercises. Apply inductive reasoning to solve problems. Recognize number
More informationThe sum of the lengths of any two sides of a triangle must be greater than the third side. That is: a + b > c a + c > b b + c > a
UNIT 9 GEOMETRY 1. Triangle: A triangle is a figure formed when three non collinear points are connected by segments. Each pair of segments forms an angle of the triangle. The vertex of each angle is
More informationCOURSE OVERVIEW. PearsonSchool.com Copyright 2009 Pearson Education, Inc. or its affiliate(s). All rights reserved
COURSE OVERVIEW The geometry course is centered on the beliefs that The ability to construct a valid argument is the basis of logical communication, in both mathematics and the realworld. There is a need
More informationConjectures for Geometry for Math 70 By I. L. Tse
Conjectures for Geometry for Math 70 By I. L. Tse Chapter Conjectures 1. Linear Pair Conjecture: If two angles form a linear pair, then the measure of the angles add up to 180. Vertical Angle Conjecture:
More informationCircle geometry theorems
Circle geometry theorems http://topdrawer.aamt.edu.au/geometricreasoning/bigideas/circlegeometry/angleandchordproperties Theorem Suggested abbreviation Diagram 1. When two circles intersect, the line
More informationLearning Log Title: CHAPTER 9: ANGLES AND THE PYTHAGOREAN THEOREM. Date: Lesson: Chapter 9: Angles and the Pythagorean Theorem
Chapter 9: Angles and the Pythagorean Theorem CHAPTER 9: ANGLES AND THE PYTHAGOREAN THEOREM Date: Lesson: Learning Log Title: Date: Lesson: Learning Log Title: Chapter 9: Angles and the Pythagorean Theorem
More informationGrade 7 & 8 Math Circles Circles, Circles, Circles March 19/20, 2013
Faculty of Mathematics Waterloo, Ontario N2L 3G Introduction Grade 7 & 8 Math Circles Circles, Circles, Circles March 9/20, 203 The circle is a very important shape. In fact of all shapes, the circle is
More informationGeometry. Patterns and Inductive Reasoning 8.A.4b, 9.C.4b, 9.C.4c Find and describe pattern Use inductive reasoning to make reallife conjectures
Geometry Basics of Geometry Patterns and Inductive Reasoning 8.A.4b, 9.C.4b, 9.C.4c Find and describe pattern Use inductive reasoning to make reallife conjectures Points, Lines, and Planes  9.B.4 Understand
More information1) Definition: Angle. 2) Sides of an angle. 3) Vertex of an angle
Geometry Chapter 3  Note Taking Guidelines Complete each Now try problem after studying the previous example 3.1 Angles and Their Measures 1) Definition: Angle 2) Sides of an angle 3) Vertex of an angle
More informationUnknown Angle Problems with Inscribed Angles in Circles
: Unknown Angle Problems with Inscribed Angles in Circles Student Outcomes Use the inscribed angle theorem to find the measures of unknown angles. Prove relationships between inscribed angles and central
More informationGeometry Chapter 10 Study Guide Name
eometry hapter 10 Study uide Name Terms and Vocabulary: ill in the blank and illustrate. 1. circle is defined as the set of all points in a plane that are equidistant from a fixed point called the center.
More information10.1: Areas of Parallelograms and Triangles
10.1: Areas of Parallelograms and Triangles Important Vocabulary: By the end of this lesson, you should be able to define these terms: Base of a Parallelogram, Altitude of a Parallelogram, Height of a
More informationMastery of SAT Math. Curriculum (396 topics + 99 additional topics)
Mastery of SAT Math This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular
More information2006 Geometry Form A Page 1
2006 Geometry Form Page 1 1. he hypotenuse of a right triangle is 12" long, and one of the acute angles measures 30 degrees. he length of the shorter leg must be: () 4 3 inches () 6 3 inches () 5 inches
More informationGrade 4  Module 4: Angle Measure and Plane Figures
Grade 4  Module 4: Angle Measure and Plane Figures Acute angle (angle with a measure of less than 90 degrees) Angle (union of two different rays sharing a common vertex) Complementary angles (two angles
More informationAdvanced Geometry. Segment and their Measures 7.A.4b, 9.B.4 Use segment postulates Use the distance formula to measure distances
Advanced Geometry Basics of Geometry Patterns and Inductive Reasoning 8.A.4b, 9.C.4a, 9.C.4b, 9.C.4c Find and describe pattern Use inductive reasoning to make reallife conjectures Points, Lines, and Planes
More informationAdditional Topics in Math
Chapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are
More informationGeometry Unit 7 (Textbook Chapter 9) Solving a right triangle: Find all missing sides and all missing angles
Geometry Unit 7 (Textbook Chapter 9) Name Objective 1: Right Triangles and Pythagorean Theorem In many geometry problems, it is necessary to find a missing side or a missing angle of a right triangle.
More informationGeometry Chapter 1 Vocabulary. coordinate  The real number that corresponds to a point on a line.
Chapter 1 Vocabulary coordinate  The real number that corresponds to a point on a line. point  Has no dimension. It is usually represented by a small dot. bisect  To divide into two congruent parts.
More informationAngles that are between parallel lines, but on opposite sides of a transversal.
GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,
More informationGeometry. G.1.1 Find the lengths and midpoints of line segments in one or twodimensional coordinate systems.
Standard 1: Point, Lines, Angles, and Planes Students find lengths and midpoints of line segments. They describe and use parallel and perpendicular lines. They find slopes and equations of lines. G.1.1
More informationGeometry and Measurement
The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for
More informationGEOMETRY. Constructions OBJECTIVE #: G.CO.12
GEOMETRY Constructions OBJECTIVE #: G.CO.12 OBJECTIVE Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic
More informationof one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.
2901 Clint Moore Road #319, Boca Raton, FL 33496 Office: (561) 4592058 Mobile: (949) 5108153 Email: HappyFunMathTutor@gmail.com www.happyfunmathtutor.com GEOMETRY THEORUMS AND POSTULATES GEOMETRY POSTULATES:
More informationCCGPS UNIT 3 Semester 1 ANALYTIC GEOMETRY Page 1 of 32. Circles and Volumes Name:
GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 1 of 3 ircles and Volumes Name: ate: Understand and apply theorems about circles M91.G..1 Prove that all circles are similar. M91.G.. Identify and describe relationships
More informationHarbor Creek School District. Advanced Geometry. Concepts Timeframe Skills Assessment Standards Undefined Terms Segments, Rays, and Distance
Advanced Geometry Points, Lines and Planes Deductive Reasoning Concepts Timeframe Skills Standards Undefined Terms Segments, Rays, and Distance Angles Postulates and Theorems If Then Statements Converses
More information1.) Consider a geometry with undefined terms point, line, and contains (or belongs) and the following axioms.
Systems of Geometry Test File Spring 2010 Test 1 1.) Consider a geometry with undefined terms point, line, and contains (or belongs) and the following axioms. Axiom 1. Axiom 2. Axiom 3. Axiom 4. Given
More informationContent Skill CCLS Standard Assessments
GEOMETRY 3 CURRICULUM MAP Month September Essential Questions How do we find the measures of resulting angles, arcs, and segments when lines intersect a circle, or within a circle? NYS Standard Content
More informationGeometry SOL G.11 G.12 Circles Study Guide
Geometry SOL G.11 G.1 Circles Study Guide Name Date Block Circles Review and Study Guide Things to Know Use your notes, homework, checkpoint, and other materials as well as flashcards at quizlet.com (http://quizlet.com/4776937/chapter10circlesflashcardsflashcards/).
More informationPreAlgebra IA Grade Level 8
PreAlgebra IA PreAlgebra IA introduces students to the following concepts and functions: number notation decimals operational symbols inverse operations of multiplication and division rules for solving
More informationMATH STUDENT BOOK. 6th Grade Unit 4
MATH STUDENT BOOK th Grade Unit Unit Fractions MATH 0 Fractions 1. FACTORS AND FRACTIONS DIVISIBILITY AND PRIME FACTORIZATION GREATEST COMMON FACTOR 10 FRACTIONS 1 EQUIVALENT FRACTIONS 0 SELF TEST 1: FACTORS
More informationSCOPE AND SEQUENCE. Grade Level: 7 th Curriculum: Saxon Math 8/7 Publisher: 2004 by Saxon Publishers, Inc.
Grade Level: 7 th Curriculum: Saxon Math 8/7 Publisher: 2004 by Saxon Publishers, Inc. SCOPE AND SEQUENCE NUMBERS AND OPERATIONS Numeration Digits Reading and writing numbers Ordinal numbers Denominate
More informationLesson 1: Introducing Circles
IRLES N VOLUME Lesson 1: Introducing ircles ommon ore Georgia Performance Standards M9 12.G..1 M9 12.G..2 Essential Questions 1. Why are all circles similar? 2. What are the relationships among inscribed
More informationChapter 1: Essentials of Geometry
Section Section Title 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas Chapter 1: Essentials of Geometry Learning Targets I Can 1. Identify,
More informationCircle Geometry. The BEST thing about Circle Geometry is. you are given the Question and (mostly) the Answer as well.
Circle Geometry The BEST thing about Circle Geometry is. you are given the Question and (mostly) the Answer as well. You just need to fill in the gaps in between! Topic: Circle Geometry Page 1 There are
More informationSTANDARDS OF LEARNING CONTENT REVIEW NOTES GEOMETRY. 1 st Nine Weeks,
STANDARDS OF LEARNING CONTENT REVIEW NOTES GEOMETRY 1 st Nine Weeks, 20162017 OVERVIEW 2 3 4 Algebra Review Solving Equations You will solve an equation to find all of the possible values for the variable.
More information1. Line A line consists of an uncountable number of points in a straight line and it has length.
1 Grade 7 Maths Term 1 MATHS Grade 7 CONCEPT : GEOMETRY (LINES AND ANGLES) The world around us is made up of many twodimensional (2D) shapes and three dimensional (3D) objects. Architecture, for example,
More informationGeometry Notes PERIMETER AND AREA
Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter
More information104 Inscribed Angles. Find each measure. 1.
Find each measure. 1. 3. 2. intercepted arc. 30 Here, is a semicircle. So, intercepted arc. So, 66 4. SCIENCE The diagram shows how light bends in a raindrop to make the colors of the rainbow. If, what
More information