Vocabulary A point has no dimension, a line has one dimension, and a plane has two dimensions.


 Laurence Richard
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1 Name This is the Honors Geometry Summer work. Each lesson in Ch. 1 of the Geometry book is covered briefly in the following worksheet. Please read through the lesson Study Guide for each section of Chapter 1 and Examples given (Note: all examples are worked out for you with solutions given, you do not have to redo any examples). Complete the exorcise problems as well as the Standardized Practice Test and Algebra 1 Review on this worksheet and turn your work in on the first day of school. You may use a calculator on sections that ask for decimal answers. LESSON 1.1 Study Guide GOAL Name and sketch geometric figures. Vocabulary A point has no dimension, a line has one dimension, and a plane has two dimensions. Collinear points are points that lie on the same line. Coplanar points are points that lie in the same plane. Line AB (written as AB or BA) passes through points A and B. The line segment AB, or segment AB (written as AB points on AB between A and B. or BA), consists of the endpoints A and B and all The ray AB (written as AB ) consists of the endpoints and all points on AB that lie on the same side of A as B. If point C lies on AB between A and B, then CA and CB are opposite rays. Two or more geometric figures intersect if they have one or more points in common. The intersection of the figures is the set of points the figures have in common.
2 EXAMPLE 1 (DO NOT DO THIS PROBLEM, JUST READ THROUGH IT, THE SOLUTIONS ARE GIVEN BELOW) Directions: Name points, lines, planes, segments, and rays a. Give two other names for EF. Give another name for plane A. b. Name three points that are collinear. Name four points that are coplanar. c. Give another name for EF. d. Name a ray with endpoint E that is an opposite ray of EC. Solution a. Other names for EF are FE and line m. Other names for plane A are plane BCD and plane CDE. b. Points C, E, and D lie on the same line, so they are collinear. Points B, C, E, and D lie in the same plane, so they are coplanar. c. Another name for EF is FE. d. ED is a ray with endpoint E that is an opposite ray of EC.
3 LESSON 1.1 Exercises for Example 1 (complete these problems below) Use the diagram in Example Give two other names for CD. 2. Give another name for CE. 3. Name a ray with endpoint F. 4. Name a point that is not collinear with C, E, and D. 5. Name a point that is not coplanar with B,C,E, and D. 6. Give another name for DE. EXAMPLE 2 (DO NOT DO THIS PROBLEM, JUST READ THROUGH IT, THE SOLUTIONS ARE GIVEN BELOW) Sketch intersections of lines and planes Perform the indicated operations. a. Sketch a plane and a line that is in the plane. b. Sketch a plane and a line that does not intersect the plane. c. Sketch a plane and a line that intersects the plane at a point. d. Sketch two planes that intersect in a line. Solution (These are examples of what you could draw, you re solutions may look a little different). a. b. c. d. STEP 1 Draw one plane as if you are facing it. Shade the plane. STEP 2 Draw a second plane that is horizontal. Shade this plane a different color. Use dashed lines to show where one plane is hidden. STEP 3 Draw the line of intersection.
4 Exercises for Example 2 (complete these problems below) Sketch the figure described. 7. Two lines that lie in a plane and intersect at one point. 8. One line that lies in a plane, and two lines that do not lie in the plane
5 LESSON 1.2 GOAL Use segment postulates to identify congruent segments. Vocabulary In Geometry, a rule that is accepted without proof is called a postulate or axiom. Postulate 1 Ruler Postulate: The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point. The distance between points A and B, written as AB, is the absolute value of the difference of the coordinates of A and B. When three points are collinear, you can say that one point is between the other two. Postulate 2 Segment Addition Postulate: If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between, A and C. Line segments that have the same length are called congruent segments. For example if AB and DC have the same length, then AB DC. EXAMPLE 1 Find a length (DO NOT DO THIS PROBLEM, JUST READ THROUGH IT, THE SOLUTIONS ARE GIVEN BELOW) Use the diagram to find BC. Solution Use the Segment Addition Postulate to write an equation. Then solve the equation to find. BC. AC = AB + BC Segment Addition Postulate 32 = 10 + BC Substitute 32 for AC and 10 for AB. 22 = BC Subtract 10 from each side.
6 Exercises for Example 1 (complete these problems below) Use the diagram to find BC
7 LESSON 1.2 EXAMPLE 2 Compare segments for congruence (DO NOT DO THIS PROBLEM, JUST READ THROUGH IT, THE SOLUTIONS ARE GIVEN BELOW) Use the diagram to determine whether AB and CD are congruent. Solution To find the length of a horizontal segment find the absolute value of the difference of the xcoordinates of the endpoints. AB = 1 3 = 4 = 4 Use Ruler Postulate. To find the length of a vertical segment, find the absolute value of the difference of the ycoordinates of the endpoints. CD = 5 1 = 4 Use Ruler Postulate. AB and CD have the same length. So, AB CD.
8 Exercises for Example 2 (complete these problems below) Use the diagram to determine whether AB and CD are congruent. Show work for full credit
9 LESSON 1.3 GOAL Find lengths of segments in the coordinate plane Vocabulary The midpoint of a segment is the point that divides the segment into two congruent segments. A segment bisector is a point, ray, line, line, segment, or plane that intersects the segment at its midpoint. A midpoint or a segment bisector bisects a segment. The Midpoint Formula: If A(x 1, y 1 ) and B(x 2, y 2 ) are points in a coordinate plane, then the midpoint M of has coordinates. x + x y + y 1 2, The Distance Formula: If A(x 1, y 1 ) and B(x 2, y 2 ) are points in a coordinate plane, then the distance A and B is. EXAMPLE 1 ( ) ( ) ( ) AB = x x + y y Find segment lengths (DO NOT DO THIS PROBLEM, JUST READ THROUGH IT, THE SOLUTIONS ARE GIVEN BELOW) In the diagram, line l bisects AC at point B, and AB = 8 in. Find AC. Solution Point B is the midpoint of AC AC = AB + BC. So, AB = BC = 8 in. Segment Addition Postulate = Substitute 8 for AB and 8 for BC. = 16 in. Add.
10 Exercises for Example 1 (complete these problems below) Line l bisects the segment. Find the indicated length. 1. Find AC if AB = 10 cm. 2. Find DF if EF = 25 mm. 3. Find JM if RM = 37 in. 4. Find WX if WY = 30 cm.
11 LESSON 1.3 EXAMPLE 2 Use the midpoint and distance formulas (DO NOT DO THIS PROBLEM, JUST READ THROUGH IT, THE SOLUTIONS ARE GIVEN BELOW) a The endpoints of AB are A(3, 2) and B(6, 7). Find the coordinates of the midpoint M. b What is the length of AB? Solution a. Use the Midpoint Formula. ( ) = ( ) M ,, The coordinates of the midpoint M are 9, 9. ( 2 2 ) b. Use the Distance Formula. AB = x x + y y = = = ( ) ( ) 2 2 ( ) ( ) = Distance Formula Substitute. Subtract. Evaluate powers. Add. Use a calculator to approximate the square root. The length of AB is approximately equal to 5.83.
12 Exercises for Example 2 (complete these problems below) Find the coordinates of the midpoint of the segment with the given endpoints. 5. A(l, 2) and B(3,6) 6. J( l, 3) and K(9,0) 7. R(4, 2) and G( 12, 8) 8. C( 3, 1) and D(9, 5) 9. S(5, 2) and T( 3, 4) 10. X(7, 4) and Y( 2, 1) Find the length of the segment. Round to the nearest tenth of a unit
13 LESSON 1.4 GOAL Name, measure, and classify angles. Vocabulary An angle consists of two different rays with the same endpoint. The rays are the sides of the angle. The endpoint is the vertex of the angle. An acute angle has measure greater than 0 and less than 90. A right angle has measure equal to 90. An obtuse angle has measure greater than 90 and less than 180. A straight angle has measure equal to 180. Two angles are congruent angles if they have the same measure. An angle bisector is a ray that divides an angle into two angles that are congruent. Postulate 3 Protractor Postulate: Consider OB and a point A on one side of OB. The rays of the form OA can be matched one to one with the real numbers from 0 to 180. The measure of AOB is equal to the absolute value of the difference between the real numbers for OA and OB. Postulate 4 Angle Addition Postulate: If P is in the interior of RST, then m RST = m RSP + m PST.
14 EXAMPLE 1 Name angles (DO NOT DO THIS PROBLEM, JUST READ THROUGH IT, THE SOLUTIONS ARE GIVEN BELOW) Write three names for the angle and name the vertex and sides of the angle. Solution Three names for the angle are ABC, CBA, or B. The vertex of the angle is point B. The sides of the angle are BA and BC. Exercises for Example 1 (complete these problems below) Write three names for the angle and name the vertex and sides of the angle
15 LESSON 1.4 Measure and classify angles (DO NOT DO THIS PROBLEM, JUST READ THROUGH IT, THE SOLUTIONS ARE GIVEN BELOW) In the diagram, m ABD = 90 and m DBC = 45. Find m ABC. Then classify each angle as acute, right, obtuse, or straight. Solution Use the Angle Addition Postulate to find m ABC. m ABC = m ABD + m DBC Angle Addition Postulate m ABC = Substitute angle measures. m ABC = 135 Add. So, m ABC = 135. Because m ABD = 90, ABD is a right angle. Because m DBC = 45, DBC is an acute angle. Because m ABC = 135, ABC is an obtuse angle. Exercises for Example 2 (complete these problems below) Find the indicated angle measure. Then classify each angle in the diagram as acute, right, obtuse, or straight. 4. m JRN = 35 and m NRM = 90. Find m JRM. 5. m ABD = 60 and m ABC = 180. Find m DBC.
16 LESSON 1.5 GOAL Use special angle relationships to find angle measures. Vocabulary Two angles are complementary if the sum of their measures is 90. Two angles are supplementary if the sum of their measures is 180. Adjacent angles are two angles that share a common vertex and side, but have no common interior points. Two adjacent angles are a linear pair if their noncommon sides are opposite rays. Two angles are vertical angles if their sides form two pairs of opposite rays. EXAMPLE 1 Identify complements and supplements (DO NOT DO THIS PROBLEM, JUST READ THROUGH IT, THE SOLUTIONS ARE GIVEN BELOW) In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles. Solution Because = 90, ABD and DBC are complementary angles. Because = 180, ABD and MNO are supplementary angles. Because ABD and DBC share a common vertex and side, they are adjacent angles.
17 Exercise for Example 1 (complete these problems below) 1. In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles. EXAMPLE 2 Find measures of a complement and a supplement (DO NOT DO THIS PROBLEM, JUST READ THROUGH IT, THE SOLUTIONS ARE GIVEN BELOW) a. Given that < 1 is a complement of < 2 and m < 1 = 50, find m < 2. b. Given that < 3 is a supplement of < 4 and m < 3 = 105, find m < 4. Solution a. You can draw a diagram with complementary adjacent angles to illustrate the relationship. m < 2 = 90 m < 1 = = 40 b. You can draw a diagram with supplementary adjacent angles to illustrate the relationship. m < 4= 180 m < 3 = = 75
18 LESSON 1.5 Exercises for Example 2 (complete these problems below) 2. Given that < 1 is a complement of < 2 and m < 1 = 55, find m < Given that < 3 is a supplement of < 4 and m < 3 = 80, find m < 4. EXAMPLE 3 Identify angle pairs (DO NOT DO THIS PROBLEM, JUST READ THROUGH IT, THE SOLUTIONS ARE GIVEN BELOW) Identify all of the linear pairs and all of the vertical angles in the figure at the right. Solution To find linear pairs, look for adjacent angles whose noncommon sides are opposite rays. <1 and < 2 are a linear pair. < 2 and < 3 are also a linear pair. < 3 and < 4 are also a linear pair. < 1 and < 4 are also a linear pair. To find vertical angles, look for angles formed by intersecting lines. < 1 and < 3 are vertical angles.< 2 and < 4 are also vertical angles.
19 EXAMPLE 4 Find angle measures in a linear pair (DO NOT DO THIS PROBLEM, JUST READ THROUGH IT, THE SOLUTIONS ARE GIVEN BELOW) Solve for x in the diagram at the right. Then find the measure of each angle. Solution The two angles form a linear pair. Use the fact that the angles of a linear pair are supplementary to write an equation. x + 2x = 180 Write an equation. 3x = 180 Combine like terms. x = 60 Divide each side by 3. The measures of the angles are 60 and 2(60 ) = 120. Exercises for Examples 3 and 4 (complete these problems below) 4. Identify all of the linear pairs and all of the vertical angles in the figure at the right. Solve for x in the diagram. Then find the measure of each angle
20 LESSON 1.6 GOAL Classify polygons. Vocabulary A polygon is a closed plane figure formed by three or more line segments called sides. 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 vertex of the polygon. A polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon. A polygon that is not convex is called nonconvex or concave. The term ngon, where n is the number of a polygon s sides, can be used to name a polygon. In an equilateral polygon, all sides are congruent. In an equiangular polygon, all angles in the interior of the polygon are congruent. A polygon is regular if all sides and all angles are congruent. EXAMPLE 1 Identify polygons (DO NOT DO THIS PROBLEM, JUST READ THROUGH IT, THE SOLUTIONS ARE GIVEN BELOW) Tell whether the figure is a polygon and whether it is convex or concave. a. b. c. d.
21 Solution a. The figure is a concave polygon. b. Part of the figure is not a segment, so it is not a polygon. c. Some segments intersect more than two segments, so it is not a polygon. d. The figure is a convex polygon. Exercises for Example 1 (complete these problems below) Tell whether the figure is a polygon and whether it is convex or concave
22 LESSON 1.6 EXAMPLE 2 Classify polygons (DO NOT DO THIS PROBLEM, JUST READ THROUGH IT, THE SOLUTIONS ARE GIVEN BELOW) Classify the polygon by the number of sides. Tell whether the polygon is equilateral, equiangular, or regular. Explain your reasoning. a. b. c. Solution a. The polygon has 4 sides, so it is a quadrilateral. The angles in the interior of the polygon are congruent, so it is equiangular. Not all of the sides are congruent, so it is not equilateral. So, the polygon is not regular. b. The polygon has 8 sides. It is equilateral and equiangular, so it is a regular octagon. c. The polygon has 3 sides, so it is a triangle. It is not equilateral or equiangular, so it is not regular.
23 EXAMPLE 3 Find side lengths (DO NOT DO THIS PROBLEM, JUST READ THROUGH IT, THE SOLUTIONS ARE GIVEN BELOW) The figure shown at the right is a regular y pentagon. Find the length of a side. First, write and solve an equation to find the value of x. Use the fact that the sides of a regular pentagon are congruent. 2x 1 = x + 3 Write equation. x 1 = 3 Subtract x from each side. x = 4 Add 1 to each side. Then find a side length. Evaluate one of the expressions when x = 4. 2x 1 = 2(4) 1 = 7 The length of a side of the pentagon is 7 centimeters. Exercises for Examples 2 and 3 (complete these problems below) 5. Classify the polygon by the number of sides. Tell whether the polygon is equilateral, equiangular, or regular. Explain your reasoning. 6. The figure shown at the right is a regular hexagon. Find the length of a side.
24 CHAPTER 1 Standardized Practice Test Please complete this Practice Test, show work for each problem. Multiple Choice 1. Which statement about the figure is true? A Lines x and y intersect at point A. B Points A, B, and C are collinear. C EC and ED are opposite rays. D Another name for AE is AB. 2. Name three points that are collinear. 6. The endpoints of CD are C(6, 1) and D( 4, 1). Find the midpoint M of CD. A M(10, 2) B M( 10, 2) C M(2, 0) D M(1, 0) 7. JK has a length of 4.5 units. If LM has endpoints L(3, 1) and M( 1, 4), how much longer than JK is LM? A 0.5 unit C 2.5 units B 2 units D JK is longer. 8. Name the acute angles in the given figure. A G, H, and I C F, G, and I B H, G, and J D G, J, and I 3. What is the intersection of plane HGY and plane HFX? A B HZ HZ C Point H D Plane EFH 4. What is the length of ST? A 2 B 4 C 2 D 6 5. If WX XY, what is the length of WZ? A CAD and DAE B BAC and FAE C BAF and CAE D BAD and FAD 9. If the measure of RST is 134º, find the measure of QST. A 67º B 33º C 34º D 98º 10. m A is 42º greater than m B. If A and B are supplementary, find m A and m B. A m A = 111º, m B = 69º B m A = 42º m B = 48º C m A = 42º m B = 138º D m A = 66º m B = 24º A 7 B 10 C 3 D 4
25 Standardized Practice Test continued 11. Name a pair of vertical angles in the figure shown. Gridded Answer 17. Find the area, in square units, of a triangle with vertices X( 7, 2), Y(8, 2), and Z(6, 7). A 2 and 4 B 1 and 4 C 3 and 5 D There are none. 12. Which describes the following polygon? A equilateral B equiangular C regular D none of these 13. Which of the following is a convex polygon? A C B D 14. Point M is the midpoint of AB. If AM = 12x + 8 and MB = 10x + 15, find the length of AB. A 3.5 B 4 C 50 D Find CD. A 5 B 28 C 56 D Find the length of diagonal BC of ABCD to the nearest hundredth. A 2.83 units B 5.66 units C 6.32 units D 7.21 units Short Response 18. A swimmer stands somewhere in a circular pool. The distance to the farthest side (through the center of the pool) is 3 times the distance to the nearest side. The circumference of the pool is 100 feet. a. How close is the swimmer to the nearest side? b.how far must the swimmer swim to get to the center? Extended Response 19. You are a surveyor. You take your first measurement facing due north. You turn to the right to take your second measurement and then right again, 4 times as far, to take your third measurement. You are now facing due west. a. How many degrees did you turn to take your second measurement? b.how many degrees should you have turned after your second measurement if you wanted to take your third measurement facing south? c. How many degrees must you turn to the left in order to take a fourth measurement in the opposite direction of your second measurement?
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28 Example: Simplify the radicals (Do not complete this problem, just read through it. The solutions are below). a. 98b = 49 2 b = 49 2 b = 7 2b b. 121x! = 121 x! = 11x! Complete the problems below. Show your work.
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