Reteach. Understanding Points, Lines, and Planes. P point P

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1 Name Date Class 1-1 Understanding Points, Lines, and Planes A point has no size. It is named using a capital letter. All the figures below contain points. line Figure Characteristics Diagram Words and Symbols 0 endpoints extends forever in two directions line AB or AB P point P line segment or segment ray plane endpoints has a finite length 1 endpoint extends forever in one direction extends forever in all directions segment XY or XY ray RQ or RQ A ray is named starting with its endpoint. plane FGH or plane V Draw and label a diagram for each figure. 1. point W. line MN 3. JK 4. EF Name each figure using words and symbols Name the plane in two different ways

2 Name Date Class 1-x 1-1 Understanding Points, Lines, and Planes continued Term Meaning Model collinear points that lie on the same line noncollinear points that do not lie on the same line F and G are collinear. F, G, and H are noncollinear. coplanar points or lines that lie in the same plane noncoplanar points or lines that do not lie in the same plane W, X, and Y are coplanar. W, X, Y, and Z are noncoplanar. Figures that intersect share a common set of points. In the first model above, FH intersects FG at point F. In the second model, XZ intersects plane WXY at point X. Use the figure for Exercises Name each of the following. 9. three collinear points 10. three noncollinear points 11. four coplanar points 1. four noncoplanar points 13. two lines that intersect CD 14. the intersection of JK and plane R 1-7

3 Name Date Class x-x 1- Measuring and Constructing Segments The distance between any two points is the length of the segment that connects them. The distance between E and J is EJ, the length of EJ. To find the distance, subtract the numbers corresponding to the points and then take the absolute value. Use the figure above to find each length. EJ cm 1. EG. EF 3. FH On PR, Q is between P and R. If PR 16, we can find QR. PQ QR PR 9 x 16 x 7 QR Find JK. Find BC Find SV. Find XY Find DF. Find ST. 1-14

4 Name Date Class x-x 1- Measuring and Constructing Segments continued Segments are congruent if their lengths are equal. AB BC The length of AB equals the length of BC. AB BC AB is congruent to BC. Copying a Segment Method sketch using estimation draw with a ruler construct with a compass and straightedge Steps Estimate the length of the segment. Sketch a segment that is about the same length. Use a ruler to measure the length of the segment. Use the ruler to draw a segment having the same length. Draw a line and mark a point on it. Open the compass to the length of the original segment. Mark off a segment on your line at the same length. Refer to triangle ABC above for Exercises 10 and Sketch LM that is congruent to AC. 11. Use a ruler to draw XY that is congruent to BC. 1. Use a compass to construct ST that is congruent to JK. The midpoint of a segment separates the segment into two congruent segments. In the figure, P is the midpoint of NQ. 13. PQ is congruent to. 14. What is the value of x? 15. Find NP, PQ, and NQ. 1-15

5 Name Date Class x-x 1-3 Measuring and Constructing Angles An angle is a figure made up of two rays, or sides, that have a common endpoint, called the vertex of the angle. There are four ways to name this angle. Y Use the vertex. XYZ or ZYX Use the vertex and a point on each side. Use the number. Name each angle in three ways Name three different angles in the figure. Angle acute right obtuse straight Model Possible Measures 0 a 90 a a 180 a 180 Classify each angle as acute, right, obtuse, or straight. 4. NMP 5. QMN 6. PMQ 1-

6 Name Date Class x-x 1-3 Measuring and Constructing Angles continued You can use a protractor to find the measure of an angle. GEF is obtuse. DEG is acute. Use the figure above to find the measure of each angle. 7. DEG 8. GEF The measure of XVU can be found by adding. mxvu mxvw mwvu Angles are congruent if their measures are equal. In the figure, XVW WVU because the angles have equal measures. VW is an angle bisector of XVU because it divides XVU into two congruent angles. Find each angle measure. 9. mcfb if AFC is a straight angle. 10. mefa if the angle is congruent to DFE. 11. mefc if DFC AFB. 1. mcfg if FG is an angle bisector of CFB. 1-3

7 Name Date Class x-x 1-4 Pairs of Angles Angle Pairs Adjacent Angles Linear Pairs Vertical Angles have the same vertex and share a common side adjacent angles whose noncommon sides are opposite rays nonadjacent angles formed by two intersecting lines 1 and are adjacent. 3 and 4 are adjacent and form a linear pair. 5 and 6 are vertical angles. Tell whether 7 and 8 in each figure are only adjacent, are adjacent and form a linear pair, or are not adjacent Tell whether the indicated angles are only adjacent, are adjacent and form a linear pair, or are not adjacent and and 4 6. and 3 Name each of the following. 7. a pair of vertical angles 8. a linear pair 9. an angle adjacent to

8 Name Date Class x-x 1-4 Pairs of Angles continued Angle Pairs Complementary Angles sum of angle measures is 90 Supplementary Angles sum of angle measures is 180 m1 m 90 In each pair, 1 and are complementary. m3 m4 180 In each pair, 3 and 4 are supplementary. Tell whether each pair of labeled angles is complementary, supplementary, or neither Find the measure of each of the following angles. 1. complement of S 13. supplement of S 14. complement of R 15. supplement of R 16. LMN and UVW are complementary. Find the measure of each angle if mlmn (3x 5) and muvw x. 1-31

9 Name Date Class x-x 1-5 Using Formulas in Geometry The perimeter of a figure is the sum of the lengths of the sides. The area is the number of square units enclosed by the figure. Model Figure Rectangle Square Perimeter P w or ( w) P 4s Area A w A s Find the perimeter and area of each figure. 1. rectangle with 4 ft, w 1 ft. square with s 8 mm The perimeter of a triangle is the sum of its side lengths. The base and height are used to find the area. Perimeter Area P ab c A 1 bh or bh Find the perimeter and area of each triangle

10 Name Date Class x-x 1-5 Using Formulas in Geometry continued Circles Circumference Area Models distance around the circle space inside the circle Words pi times the diameter or times pi times the radius pi times the square of the radius Formulas C d or C r A r C r A r C (4) A (4) C 8 A 16 C 5.1 m A 50.3 m Find the circumference and area of each circle. Use the key on your calculator. Round to the nearest tenth. 7. circle with a radius of 11 inches 8. circle with a diameter of 15 millimeters

11 Name Date Class x-x 1-6 Midpoint and Distance in the Coordinate Plane The midpoint of a line segment separates the segment into two halves. You can use the Midpoint Formula to find the midpoint of the segment with endpoints G(1, ) and H(7, 6). x 1 x , y y M M, M 8 8, M(4, 4) M is the midpoint of HG. Find the coordinates of the midpoint of each segment QR with endpoints Q(0, 5) and R(6, 7) 4. JK with endpoints J(1, 4) and K(9, 3) Suppose M(3, 1) is the midpoint of CD and C has coordinates (1, 4). You can use the Midpoint Formula to find the coordinates of D. x x y y M(3, 1) M, 1 1 x-coordinate of D 3 3 x x y-coordinate of D y 1 1 Set the coordinates equal. 1 1 x Replace (x 1, y 1 ) with (1, 4). 1 y 4 y 6 1 x Multiply both sides by. 4 y 5 x Subtract to solve for x and y. 6 y The coordinates of D are (5, 6). 5. M(3, ) is the midpoint of RS, and R has coordinates (6, 0). What are the coordinates of S? 6. M(7, 1) is the midpoint of WX, and X has coordinates (1, 5). What are the coordinates of W? 1-46

12 Name Date Class x-x 1-6 Midpoint and Distance in the Coordinate Plane continued The Distance Formula can be used to find the distance d between points A and B in the coordinate plane. d 1 1 ( x x ) ( y y ) (7 1) (6 ) (x 1, y 1 ) (1, ); (x, y ) (7, 6) 6 4 Subtract Square 6 and 4. 5 Add. < 7. Use a calculator. The distance d between points A and B is the length of AB. Use the Distance Formula to find the length of each segment or the distance between each pair of points. Round to the nearest tenth. 7. QR with endpoints Q(, 4) and R(3, 9) 8. EF with endpoints E(8, 1) and F(1, 1) 9. T(8, 3) and U(5, 5) 10. N(4, ) and P(7, 1) You can also use the Pythagorean Theorem to find distances in the coordinate plane. Find the distance between J and K. c a b 5 6 Pythagorean Theorem a 5 units and b 6 units 5 36 Square 5 and Add. c 61 or about 7.8 Take the square root. Use the Pythagorean Theorem to find the distance, to the nearest tenth, between each pair of points Side b is 6 units. Side a is 5 units. 1-47

13 Name Date Class x-x 1-7 Transformations in the Coordinate Plane In a transformation, each point of a figure is moved to a new position. Reflection Rotation Translation ABC ABC JKL JKL RST RST A figure is flipped over a line. A figure is turned around a fixed point. A figure is slid to a new position without turning. Identify each transformation. Then use arrow notation to describe the transformation

14 Name Date Class x-x 1-7 Transformations in the Coordinate Plane continued Triangle QRS has vertices at Q(4, 1), R(3, 4), and S(0, 0). After a transformation, the image of the figure has vertices at Q(1, 4), R(4, 3), and S(0, 0). The transformation is a rotation. A translation can be described using a rule such as (x, y) (x 4, y 1). Preimage Apply Rule Image R(3, 5) R(3 4, 5 1) R(7, 4) S(0, 1) S(0 4, 1 1) S(4, 0) T(, 1) T( 4, 1 1) T (6, ) Draw each figure and its image. Then identify the transformation. 5. Triangle HJK has vertices at H(3, 1), J(3, 4), and K(0, 0). After a transformation, the image of the figure has vertices at H(1, 3), J(1, ), and K(4, ). 6. Triangle CDE has vertices at C(4, 6), D(1, 6), and E(, 1). After a transformation, the image of the figure has vertices at C(4, 6), D(1, 6), and E(, 1). Find the coordinates for each image after the given translation. 7. preimage: XYZ at X(6, 1), Y(4, 0), Z(1, 3) rule: (x, y) (x, y 5) 8. preimage: FGH at F(9, 8), G(6, 1), H(, 4) rule: (x, y) (x 3, y 1) 9. preimage: BCD at B(0, ), C(7, 1), D(1, 5) rule: (x, y) (x 7, y 1) 1-55

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