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Date: ______________________ Section 1 – 1: Points, Lines, and Planes Notes A Point: is simply a _______________. Example: Drawn as a ________. Named by a ______________ letter. Words/Symbols: A Line: is made up of ____________ and has no thickness or __________. Drawn with an _________________ at each end. Named by the _____________ representing two points on the line or a lowercase script letter. Points on the same _______ are said to be _____________. Words/Symbols: Example: A Plane: is a _______ surface made up of ____________. Drawn as a ____________ 4-sided figure. Named by a _____________ script letter or by the letters naming three ___________________ points. Points that lie on the same plane are said to be _______________. Words/Symbols: Example: 1 Example #1: Use the figure to name each of the following. a.) Name a line that contains point P. b.) Name the plane that contains lines n and m. c.) Name the intersection of lines n and m. d.) Name a point not on a line. e.) What is another name for line n. f.) Does line l intersect line n or line m? Explain. Example #2: Draw and label a figure for the following relationship. a.) Point T lies on WR. b.) AB intersects CD in plane Q at point P. Example #3: a.) How many planes appear in this figure? b.) Name three points that are collinear. c.) Are points A, B, C, and D coplanar? Explain. suur suur d.) At what point do DB and CA intersect? 2 Date: ______________________ Section 1 – 2: Linear Measure Notes – Part 1 Measure Line Segments A line segment, or ______________, is a measurable part of a line that consists of two points, called _________________, and all of the points between them. A segment with endpoints A and B can be named as _______ or _______. The length or _______________ of AB is written as ________. Example #1: Use a metric ruler to draw each segment. a.) Draw LM that is 42 millimeters long. b.) Draw QR that is 5 centimeters long. Example #2: Use a customary ruler to draw each segment. a.) Draw DE that is 3 inches long. b.) Draw FG that is 2 3 inches long. 4 1 Calculate Measures Betweenness of Points: Point M is between points P and Q if and only if P,Q, and M are ______________ and __________________. Example #4: a.) Find LM. b.) Find XZ. c.) Find DE. d.) Find x and ST if T is between S and U, ST = 7x, SU = 45, and TU = 5x – 3. e.) Find y and PQ if P is between Q and R, PQ = 2y, QR = 3y + 1, and PR = 21. Draw a picture! 2 Date: ______________________ Section 1 – 2: Linear Measure Notes – Part 2 Example: Find the value of x and LM if L is between N and M, NL = 6x – 5, LM = 2x + 3, and NM = 30. Draw a picture! Measure Line Segments Key Concept (Congruent Segments): Two __________________ having the same Ex: measure are __________________. Symbol: Example #1: Name all of the congruent segments found in the kite. 1 Example #2: Find the measurement of RS. Example #3: Use the figures to determine whether each pair of segments is congruent. a.) AB, CD b.) WZ , XY c.) HO, HT d.) MH , TH 2 Date: ______________________ Section 1 – 3: Distance Notes – Part 1 Distance Between Two Points Key Concept (Distance Formulas): Number Line Coordinate Plane The distance d between two points with coordinates (x1, y1) and (x2, y2) is given by d= Pythagorean Theorem: Example #1: Find the distance between E(-4, 1) and F(3, -1). Hint: Draw a triangle! 1 Example #2: Use the number line to find QR. Example #3: Use the number line to find CD. Example #4: Use the number line to find AB and CD. Example #5: Use the Distance Formula to find the distance between the following points. a.) A(10, -2) and B(13, -7) b.) X(-5, -7) and Y(-10, 7) c.) G(-4, 1) and H(3, -1) 2 Date: ______________________ Section 1 – 3: Midpoint Notes – Part 2 Midpoint of a Segment Key Concept (Midpoint): The midpoint M of PQ is the point ___________________ P and Q such that _____________________. Number Line: The coordinate of the midpoint of a __________________ whose endpoints have coordinates a and b is Example #1: The coordinates on a number line of J and K are –12 and 16, respectively. Find the coordinate of the midpoint of JK . Hint: Draw a number line! Example #2: The coordinates on a number line of T and S are 5 and 8, respectively. Find the coordinate of the midpoint of TS . Hint: Draw a number line! Coordinate Plane: The coordinates of the _____________________ of a segment whose endpoints have coordinates (x1, y1) and (x2, y2) are 1 Example #3: Find the coordinates of the midpoint of PQ for P(-1, 2) and Q(6, 1). Example #4: Find the coordinates of the midpoint of GH for G(8, -6) and H(-14, 12). Example #5: Find the coordinates of the midpoint of AB for A(4, 2) and B(8, -6). Example #6: What is the measure of PR if Q is the midpoint of PR ? Segment Bisector: any segment, line, or plane that interests a segment at its _______________ 2 Date: ______________________ Section 1 – 4: Angle Measure Notes – Part 1 Measure Angles Degree: a unit of measure used in measuring ______________ and __________. An arc of a circle with a measure of 1° is ___________ of the entire circle. Ray: is a part of a ___________ It has one ____________________ and extends indefinitely in _________ direction. Symbols: Opposite Rays: two rays _________ and _________ such that B is between A and C Key Concept (Angle): An angle is formed by two ______________________ __________________. The rays are called ____________ of the angle. The common endpoint is the ______________. Symbols: 1 rays that have a common An angle divides a plane into three distinct parts. Points _____, _____, and _____ lie on the angle. Points _____ and _____ lie in the interior of the angle. Points _____ and _____ lie in the exterior of the angle. Example #1: a.) Name all angles that have B as a vertex. b.) Name the sides of ∠5 . c.) Write another name for ∠6 . Example #2: a.) Name all the angles that have W as a vertex. b.) Name the sides of ∠1 . c.) Write another name for ∠WYZ . d.) Name the vertex of ∠4 . 2 Date: ______________________ Section 1 – 4: Angle Measure Notes – Part 2 Measure Angles Key Concept (Classify Angles): RIGHT ANGLE: Model: Measure: ACUTE ANGLE: OBTUSE ANGLE: Model: Model: Measure: Measure: Example #1: Measure each angle, then classify as right, acute, or obtuse. a.) b.) c.) d.) 1 e.) f.) Example #2: Measure each angle named and classify it as right, acute, or obtuse. a.) ∠TYV b.) ∠WYT c.) ∠TYU d.) ∠VYX e.) ∠SYV 2 Date: ______________________ Section 1 – 4: Angle Measure Notes – Part 3 Congruent Angles Key Concept (Congruent Angles): Angles that have the same _____________________ are congruent angles. Arcs on the figure also indicate which angles are ___________________. Example #1: State whether each pair of angles is congruent, and if so write a congruence statement. a.) b.) Example #2: Find the value of x and the measure of one angle. 1 Angle Bisector: a _________ that divides an angle into _________ congruent angles. Ex: uuur If PQ is the angle bisector of ___________, then _____________________. Example #3: In the figure, QP and QR are opposite rays, and QT bisects ∠RQS . a.) If m∠RQT = 6 x + 5 and m∠SQT = 7 x − 2 , find m∠RQT . b.) Find m∠TQS if m∠RQS = 22a − 11 and m∠RQT = 12a − 8 . Example #4: In the figure, YU bisects ∠ZYW and YT bisects ∠XYW . a.) If m∠1 = 5 x + 10 and m∠2 = 8 x − 23 , find m∠2 . b.) If m∠WYZ =82 and m∠ZYU = 4r + 25 , find r. 2 Date: ______________________ Section 1 – 5: Angle Relationships Notes – Part 1 Pairs of Angles Key Concept (Angle Pairs): Adjacent Angles: are two angles that lie in the same ____________, have a common _____________, and a common ___________, but no common interior ____________ Examples: Vertical Angles : are two non-adjacent angles formed by two __________________ lines Examples: Linear Pair : Non-example: a pair of ________________ angles whose non-common sides are opposite __________. Example: Non-example: 1 Example #1 : Name an angle pair that satisfies each condition. a.) two angles that form a linear pair b.) two acute vertical angles c.) an angle supplementary to ∠VZX d.) two acute adjacent angles Key Concept (Angle Relationships): Complementary Angles: two angles whose measures have a sum of ________ Examples: Supplementary Angles: two angles whose measures have a sum of ________. Examples: Example #2: Find the measures of two supplementary angles if the measure of one angle is 6 less than 5 times the measure of the other angle. Example #3: Find the measures of two complementary angles if the difference in the measures of the two angles is 12. Example #4: The measure of an angle’s supplement is 33 less than the measure of the angle. Find the measure of the angle and its supplement. 2 Date: ______________________ Section 1 – 5: Angle Relationships Notes – Part 2 Perpendicular Lines Lines that form right angles are _____________________. Key Concept (Perpendicular Lines): Perpendicular lines intersect to form _________ right angles. Perpendicular lines intersect to form _________________ _______________ angles. ________________ and _________ can be perpendicular to lines or to other line segments and rays. The right angle symbol in the figure indicates that the lines are ___________________. Symbol: _______ is read is perpendicular to. suur suuur Example #1: Find x so that KO ⊥ HM . Example #2: Find x and y so that BE and AD are perpendicular. 1 Assumptions: Example #3: Determine whether or not each of the following statements can be assumed or not. All points shown are coplanar. P is between L and Q. PN ≅ PL ∠QPO and ∠OPL are supplementary. PN ⊥ PM L, P, and Q are collinear. ∠QPO ≅ ∠LPM PQ ≅ PO LP ≅ PQ ∠LMP and ∠MNP are adjacent angles. ∠LPN and ∠NPQ are a linear pair. ∠OPN ≅ ∠LPM PM , PN , PO, and LQ intersect at P. Example #4: Determine whether each statement can be assumed from the figure below. Explain. a.) m∠VYT = 90 b.) ∠TYW and ∠TYU are supplementary c.) ∠VYW and ∠TYS are complementary 2 Date: ______________________ Section 1 – 5: Angle Relationships Extra Examples Example #1: Two angles are complementary. One angle measures 24° more than the other. Find the measures of the angles. Example #2: Find the measures of two supplementary angles if the measure of one angle is 4 less than 3 times the measure of the other angle. Example #3: The measure of an angle’s supplement is 22 less than the measure of the angle. Find the measure of the angle and its supplement. suur suur Example #4: Find the value of x so that AC and BD are perpendicular. 1 Date: ______________________ Section 1 – 6: Polygons Notes Polygons A polygon is a ______________ figure whose sides are all segments. The sides of each angle in a polygon are called ___________ of the polygon, and the vertex of each angle is a _____________ of the polygon. Examples: Polygons can be ________________ or ________________. Examples: _____________________ ________________________ 1 Number of Sides 3 Regular Polygon polygon are quadrilateral Polygon: in congruent which and 5 ___________________. 6 Ex: all all a convex the ________ the angles are heptagon octagon 9 decagon 12 n Example #1: Name each polygon by the number of sides. Then classify it as convex or concave, regular or irregular. a.) b.) Perimeter The perimeter of a polygon is the sum of the _______________ of its sides, which are _________________. Example #2: Find the perimeter of each polygon. a.) b.) c.) 2