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Review of Geometry Prepared by Title V Staff: Daniel Judge, Instructor Ken Saita, Program Specialist East Los Angeles College EXIT TOPICSBACKNEXT © 2002 East Los Angeles College. All rights reserved. Click one of the buttons below or press the enter key
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Topics Lines Angles Triangles Click on the topic that you wish to view... EXIT TOPICSBACKNEXT
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Lines EXIT TOPICSBACKNEXT
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When a pair of lines are drawn, the portion of the plane where the lines do not intersect is divided into three distinct regions. Region 1 Region 3 Region 2 EXIT TOPICSBACKNEXT
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These regions are referred to as: Interior Region – Region bounded by both lines. Exterior Region – The remaining outside regions. exterior interior EXIT TOPICSBACKNEXT
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Parallel Lines – Lines that never intersect. l1l1 l2l2 Notation l 1 l 2 EXIT TOPICSBACKNEXT
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Transversal – A line that intersects two or more lines in different points. l1l1 l2l2 Note: l 1 is not parallel to l 2 ( l 1 l 2 ) EXIT TOPICSBACKNEXT
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Transversal l1l1 l2l2 Note: l 1 is parallel to l 2 ( l 1 l 2 ) EXIT TOPICSBACKNEXT
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Angles EXIT TOPICSBACKNEXT
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Angles are formed when lines intersect. l1l1 l2l2 Note: ( l 1 l 2 ) A B C D EXIT TOPICSBACKNEXT
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A and B are said to be adjacent. (neighbors) l1l1 l2l2 A B C D EXIT TOPICSBACKNEXT
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l1l1 l2l2 A B C D Adjacent Angles – Angles that share a common vertex and a common side between them. EXIT TOPICSBACKNEXT
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l1l1 l2l2 A B C D Note: B and C are adjacent (neighbors) C and D are adjacent (neighbors) D and A are adjacent (neighbors) EXIT TOPICSBACKNEXT
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l1l1 l2l2 A B C D Vertical Angles – The pairs of non-adjacent angles formed by the intersection of two lines. EXIT TOPICSBACKNEXT
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l1l1 l2l2 A B C D Note: A and C are vertical angles B and D are vertical angles EXIT TOPICSBACKNEXT
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Q: What’s special about vertical angles? Answer – They have the same measure. (they are congruent) l1l1 l2l2 110° 70° EXIT TOPICSBACKNEXT
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Fact – When you intersect two lines at a point l1l1 l2l2 A C BD A C (congruent) B D (congruent) EXIT TOPICSBACKNEXT
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Two angles are said to be supplementary if their sum measures 180°. Adjacent angles formed by two intersecting lines are supplementary. l1l1 l2l2 A C BD A and B are supplementary angles. EXIT TOPICSBACKNEXT
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Can you find any other supplementary angles in the figure below? l1l1 l2l2 A C BD EXIT TOPICSBACKNEXT
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Note: Angles whose sum measures 90° are said to be complementary. EXIT TOPICSBACKNEXT
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Revisiting the transversal, copy this picture in your notebook. l1l1 l2l2 Note: ( l 1 l 2 ) AB C D H G E F EXIT TOPICSBACKNEXT
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Angles in the interior region between the two lines are called interior angles. Angles in the exterior region are called exterior angles. l1l1 l2l2 AB C D H G E F Interior Exterior EXIT TOPICSBACKNEXT
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Q: Which are the interior angles and exterior angles? l1l1 l2l2 AB C D H G E F EXIT TOPICSBACKNEXT
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l1l1 l2l2 AB C D H G E F Answer— InteriorExterior C A D B E G F H EXIT TOPICSBACKNEXT
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Q: Which angles are adjacent? Q: Which angles are vertical? Q: Which angles are supplementary? l1l1 l2l2 AB C D H G E F EXIT TOPICSBACKNEXT
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Consider a transversal consisting of the two parallel lines. l1l1 l2l2 A C B D FE GH EXIT TOPICSBACKNEXT
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l1l1 l2l2 A C B D FE GH We know, A D B C E H G F since they are all vertical angles. EXIT TOPICSBACKNEXT
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Q: Are any other angles congruent? EXIT TOPICSBACKNEXT
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Yes! If we could slide l 2 up to l 1, we would be looking at the following picture. EXIT TOPICSBACKNEXT
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l1l1 l2l2 A C B D FE GH This means the following is true: A and E have the same measure (congruent) B and F have the same measure (congruent) C and G have the same measure (congruent) D and H have the same measure (congruent) EXIT TOPICSBACKNEXT
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Having knowledge of one angle in the special transversal below, allows us to deduce the rest of the angles. l1l1 l2l2 120° C B D FE GH l 1 l 2 What are the measures of the other angles? EXIT TOPICSBACKNEXT
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Answer: l1l1 l2l2 120°60° l 1 l 2 60° 120° 60° 120° Why? EXIT TOPICSBACKNEXT
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Triangles EXIT TOPICSBACKNEXT
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One of the most familiar geometric objects is the triangle. In fact, trigonometry is the study of triangles EXIT TOPICSBACKNEXT
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Triangles have two important properties 1. 3 sides 2. 3 interior angles A BC EXIT TOPICSBACKNEXT
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We also have some special triangles. EXIT TOPICSBACKNEXT
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Right Triangle — One interior angle of the triangle measures 90° (has a right angle) EXIT TOPICSBACKNEXT
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Equilateral Triangle — 1. All of the sides are congruent (have the same measure). EXIT TOPICSBACKNEXT
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Equiangular Triangle — 1. All of the interior angles are congruent (have the same measure). EXIT TOPICSBACKNEXT
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Note – Equiangular triangles are also equilateral triangles. Equilateral triangles are also equiangular triangles. EXIT TOPICSBACKNEXT
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Isosceles Triangle — 1. Two of the interior angles of the triangle are congruent (have the same measure). 2. Two of the sides are congruent. EXIT TOPICSBACKNEXT
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The sum of the interior angles of any triangle measures 180° A BC That is, A + B + C = 180° EXIT TOPICSBACKNEXT
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Why? EXIT TOPICSBACKNEXT
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Form a transversal with two parallel lines. A BC EXIT TOPICSBACKNEXT
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Fill in the missing vertical angles. A BC EXIT TOPICSBACKNEXT
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Solution-- A BC A BC EXIT TOPICSBACKNEXT
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Fill in the remaining angles. A BC A BC EXIT TOPICSBACKNEXT
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Solution-- A BC A BC Do you notice anything? BC EXIT TOPICSBACKNEXT
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That is, B + A + C = 180° A BC A BC Note – The order in which we add doesn’t matter. BC EXIT TOPICSBACKNEXT
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A BC A + B + C = 180° (This is true for any triangle) EXIT TOPICSBACKNEXT
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End of Review of Geometry Title V East Los Angeles College 1301 Avenida Cesar Chavez Monterey Park, CA 91754 Phone: (323) 265-8784 Email Us At: menteprog@hotmail.com Our Website: http://www.matematicamente.org EXIT TOPICSBACKNEXT
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