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1.1 Building Blocks of Geometry Geometry Terms Definition: Known words used to describe a new word. Postulate: A statement that is accepted as true without proof sometimes they are called axioms. Theorem: Important statements that are proven. Segment 1. 2. 3. 4. 5. A Begins at one point and ends at another Has points on each end called endpoints Consists of an infinite amount of points Always straight Named by its endpoints, in either order B Can be called: Segment AB Segment BA AB or BA Point 1. Has no dimension (no length, width, thickness) 2. Usually represented by a dot 3. Named using one capital letter • B Line 1. Extends forever in one dimension (length) 2. Has an arrowhead on each end representing the fact that it goes on forever 3. Consists of an infinite amount of points 4. Always straight 5. Named with a lowercase script letter or by two points on the line Can be called: A B l Line l Line AB or AB Line BA or BA Plane 1. 2. 3. 4. Extends forever in 2 dimensions (length & width) A flat surface consisting of infinitely many points Usually represented by a 4-sided figure Named with a capital script letter or 3 noncollinear points on the surface of the plane W A B C Can be named: Plane W Plane ABC, plane BCA, plane CBA, (any three noncollinear points) Different planes in a figure: A D B C E H Plane EFGH F G Plane ABCD Plane BCGF Plane ADHE Plane ABFE Plane CDHG Other planes in the same figure: Any three non collinear points determine a plane! A D B Plane AFGD Plane ACGE C Plane ACH E H F G Plane AGF Plane BDG More Definitions Collinear points – points that lie on the same line More Definitions Coplanar points: points that lie on the same plane Coplanar lines: lines that lie on the same plane A, B, and C are coplanar points Lines l and n are coplanar lines Is Alex between Ty and Josh? Yes! Ty Alex Josh How about now? No, but why not? In order for a point to be between two others, all 3 points MUST BE collinear!! Ray • Piece of a line with only one endpoint (initial point) and continues forever in the other direction A B • Named by the endpoint and a second point named on the ray. (name MUST begin with the endpoint!) AB Opposite Rays • Two rays that share a common initial point and face opposite directions. P Q S • QP and QS are opposite rays. More Definitions • Intersect – two or more figures intersect if they have one or more points in common. • Intersection – all points or sets of points the figures have in common • What is the intersection of: AB & DA BC & AC BC & BC When two lines intersect, their intersection is a point. When two planes intersect, their intersection is a line. B P A R Plane P and Plane R intersect at the line AB Angle symbol: • Two rays that share the same endpoint (or initial point) Sides – the rays XY & XZ Y 5 X Z Named YXZ, ZXY (vertex is always in the middle), or X (if it’s the only X in the diagram). Vertex – the common endpoint; X Angles can also be named by a #. (5) There are 3 different B’s in this diagram; therefore, none of them should be called B. A B? D B C Interior or Exterior? • B is ___________ in the interior • C is ___________ in the exterior on the A • D is ___________ B C D A Assignment Section 9 - 47 1.2 Measuring Length Ruler postulate • The points on a line can be matched with those on the real number line. • The real number that corresponds to a point is the coordinate of the point. • If you find the difference between the coordinates of two points, then take the absolute value, you will have the distance or length between the points. Ruler postulate (continued) A B mAB = AB =a – b or b – a The symbol for the length of AB is AB. Example: Find AB. A Point A is at 1.5 and B is at 5. So, AB = 5 - 1.5 = 3.5 B Determine the length of a given segment. Find the lengths of AB, BC, and CD. AB = – 4 – (–1)= 3 BC = –1 – 4= 5 CD = 4 – 9= 5 Determine whether segments are congruent. In the figure, BC CD, but AB is not congruent to the other segments. Segment Congruence Postulate Segment Congruence Postulate: If two segments have the same length then the segments are congruent. Also if two segments are congruent then they have the same length if measured by a fair ruler. If AB & XY have the same length, Then AB = XY, and AB XY Symbol for congruent Segment Addition Postulate If B is between A & C, then AB + BC = AC. If AB + BC = AC, then B is between A & C. A B C Example: If DE = 2, EF = 5, and DE = FG, find FG, DF, DG, & EG. D E F FG = 2 DF = 7 DG = 9 EG = 7 G Questions Assignment Practice A, B and Section 11 - 27 1.3 Measuring Angles A protractor is a device used for measuring angles. As on a ruler the intervals on a protractor are equal. Angle Measurement • mA means the “measure of A” • Measure angles with a protractor. • Units of angle measurement are degrees (o). • Angles with the same measure are congruent angles. • If mA = mB, then A B. Measure of an Angle • The rays of an angle can be matched up with real numbers • (from 0 to 180) on a protractor so that the measure of the equals the absolute value of the difference of the numbers. 55o 20o mA = 55 - 20 = 35o Determine the measure of a given angle. Find the measures of angle BVC. m BVC =125 – 50= 75 Add measures of angles. m AVC = m AVB + m BVC = 25 + 75 = 100 Angle Addition Postulate If P is in the interior of RST, then mQRP + mPRS = mQRS. If mQRP = 5xo, S P mPRS = 2xo, & mQRS = 84o, find x. Q 5x + 2x = 84 7x = 84 x = 12 R mQRP = 60o mPRS=24o Angle Congruence Postulate If two angles have the same measure, then they are congruent. If two angles are congruent, then they have the same measure. Types of Angles • Acute angle – Measures between 0o & 90o • Right angle – Measures exactly 90o • Obtuse angle – Measures between 90o & 180o • Straight angle – Measures exactly 180o Adjacent Angles Two angles that share a common vertex & side, but have no common interior parts. (they have the same vertex, but don’t overlap) such as 1 & 2 2 1 Example: • Name an acute angle 3, 2, SBT, or TBC • Name an obtuse angle ABT • Name a right angle 1, ABS, or SBC • Name a straight angle ABC S T 3 1 2 A B C Midpoint • The point that bisects a segment. • Bisects? splits into 2 equal pieces 12x + 3 A 10x + 5 M 12x + 3 = 10x + 5 2x = 2 x=1 B Segment Bisector A segment, ray, line, or plane that intersects a segment at its midpoint. k A M B Angle Bisector A ray that divides an angle into two congruent adjacent angles. A D B C BD is an angle bisector of ABC. Example: If FH bisects EFG and mEFG = 120o, then what is mEFH? E 120 o 60 2 H F mEFH 60 o G Example: Solve for x. * If they are congruent, set them equal to each other, then solve! x + 40 = 3x - 20 40 = 2x - 20 60 = 2x 30 = x Which angles are adjacent? 1 & 2, 2 & 3, 3 & 4, 4 & 1 Then what do we call 1 & 3? 2 1 3 4 Vertical Angles – two angles that share a common vertex & whose sides form 2 pairs of opposite rays. 1 & 3, 2 & 4 Linear Pair A linear pair is two adjacent angles whose non-common sides are opposite rays. These angles form a straight line and their sum is 180°. Example • Vertical angles? 1 & 4 • Adjacent angles? 1 & 2, 2 & 3, 1 5 3 & 4, 4 & 5, 5 & 1 • Linear pair? 5 & 4, 1 & 5 • Adjacent angles not a linear pair? 1 & 2, 2 & 3, 3 & 4 2 3 4 Important Facts • Vertical Angles are congruent. • The sum of the measures of the angles in a linear pair is 180o. Example: If m5 = 130o, find o =130 m3 m6 =50o m4 =50o 4 5 3 6 A Example: Find x and y mABE mABD mDBC mEBC E B D C x = 40 y = 35 mABE = 125o mABD = 55o mDBC = 125o mEBC = 55o Complementary Angles • Two angles whose sum is 90o 35o 1 2 1 & 2 are complementary A & B are complementary 55o A B Supplementary Angles Two angles whose sum is 180o 1 & 2 are supplementary. X & Y are supplementary. 130o X 50o Y Example: A & B are supplementary. mA is 5 times mB. Find mA & mB. mA + mB = 180o mA = 5(mB) Now substitute! 5(mB) + mB = 180o 6(mB)=180o mB=30o mA=150o Perpendicular Bisector A perpendicular bisector intersects at the midpoint AND is perpendicular to the segment. Parallel Lines Parallel Lines Two lines are parallel lines if they lie in the same plane and do not intersect. Perpendicular Lines Two lines are perpendicular lines if they intersect to form right angles. Skew Lines Skew are lines that do NOT lie in the same plane and do NOT intersect. Questions Find the measure of each of the angles. Questions Questions Questions Assignment Practice B and Section 14 - 44