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Transcript
Unit 1 Describe and Identify the three undefined terms, Understand Segment Relationships and Angle Relationships Part 1 Definitions: Points, Lines, Planes and Segments Undefined Terms Points, Line and Plane are all considered to be undefined terms. – This is because they can only be explained using examples and descriptions. – They can however be used to define other geometric terms and properties A < B > Point – A location, has no shape or size – Label: Line – A line is made up of infinite points and has no thickness or width, it will continue infinitely.There is exactly one line through two points. – Label: Line Segment – Part of a line – Label: Ray – A one sided line that starts at a specific point and will continue on forever in one direction. – Label: F < E A B > Collinear – Points that lie on the same line are said to be collinear – Example: Non-collinear – Points that are not on the same line are said to be non-collinear (must be three points … why?) – Example: Plane – A flat surface made up of points, it has no depth and extends infinitely in all directions. There is exactly one plane through any three non-collinear points Coplanar – Points that lie on the same plane are said to be coplanar Non-Coplanar – Points that do not lie on the same plane are said to be non-coplanar Intersect The intersection of two things is the place they overlap when they cross. – When two lines intersect they create a point. – When two planes intersect they create a line. Space Space is boundless, three-dimensional set of all points. Space can contain lines and planes. Practice Use the figure to give examples of the following: 1. 2. 3. 4. Name two points. Name two lines. Name two segments. Name two rays. 5. 6. 7. 8. 9. Name a line that does not contain point T. Name a ray with point R as the endpoint. Name a segment with points T and Q as its endpoints. Name three collinear points. Name three non-collinear points. Congruent When two segments have the same measure they are said to be congruent Symbol: Example: A B > < < > D C AB CD Midpoint / Segment Bisector The midpoint of a segment is the point that divides the segment into two congruent segments The Segment Bisector is a segment, line or ray that intersects another segment at its midpoint. Example Q is the Midpoint of PR, if PQ=6x-7 and QR=5x+1, find x, PQ, QR, and PR. Between Point B is between point A and C if and only if A, B and C are collinear and AB BC AC < A B C > 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 Example Find the length XY in the figure shown. Example If S is between R and T and RS = 8y+4, ST = 4y+8, and RT = 15y – 9. Find y. Part 3 Angles Angle An angle is formed by two non-collinear rays that have a common endpoint. The rays are called sides of the angle, the common endpoint is the vertex. Kinds of angles Right Angle Acute Angle Obtuse Angle Straight Angle / Opposite Rays Congruent Angles Just like segments that have the same measure are congruent, so are angles that have the same measure. Angle Addition Postulate – If R is in the interior of <PQS, then m<PQR + m<RQS = m<PQS P R S Q – If m<PQR + m<RQS = m<PQS, then R is in the interior of <PQS Example If m<BAC = 155, find m<CAT and m<BAT T C (3x+14) ° B (4x-20) ° A Example <ABC is a straight angle, find x. D A (11x-12) ° (2x+10) ° B C Angle Bisector A ray that divides an angle into two congruent angles is called an angle bisector. Example Ray KM bisects <JKL, if m<JKL=72 what is the m<JKM? Adjacent Angles are two angles that lie in the same plane, have a common vertex, and a common side, but no common interior points C B ADB is adjacent to D A BDC Vertical Angles Two non-adjacent angles formed by two intersecting lines 1 1 is vertical to 2 2 Vertical Angles have the same measure and are congruent Linear Pair A pair of adjacent angles who are also supplementary 1 2 1 and 2 are a liner pair and m 1 + m 2 =180 Angle Relationships Complementary Angles - Two angles whose measures have a sum of 90 Supplementary Angles - are two angles whose measures have a sum of 180 Examples Part 3 Polygons Polygon Closed figure whose sides are all segments. – To be a Polygon 2 things must be true • Sides have common endpoints and are not collinear • Sides intersect exactly two other sides Examples Non-Examples Naming a Polygon The sides of each angle in a polygon are the sides of the polygon The vertex of each angle is a vertex of the polygon They are named using all the vertices in consecutive order Example A B D C The number of sides determines the name of the polygon 3 - Triangle 4 - Quadrilateral 5 - Pentagon 6 - Hexagon 7 - Heptagon 8 - Octagon 9 - Nonagon 10 - Decagon 12 - Dodecagon Anything else …. N - gon (where n represents the number of sides) Concave VS Convex Concave Convex Regular Polygon A regular polygon is a convex polygon whose sides are all congruent and whose angles are all congruent Perimeter The perimeter of a polygon is the sum of the lengths of its sides. Example l s a b s s w w s c p =a + b + c l p =s + s +s + s p = 4s p =l + w + l +w p = 2l + 2w Perimeter of the Coordinate Plane Find the perimeter of the triangle ABC with A(-5,1), B(-1,4), C(-6,-8) Area Area of a polygon is the number of square units it encloses h h b b 1 A= 2 bh A = bh Circle C = 2 šr A = š r2 r Unit 1 The End!