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Section 1.12 Spatial Invariants (p. 58) Goal: Search for geometric invariants, such as points of concurrency and collinearity of points. Vocabulary building: 1. angle bisector – a line segment which cuts an angle into two equal parts 2. concur – to come to an agreement; to come together; to meet 3. concurrent lines – 3 or more lines which intersect at the same point 4. collinear points – 3 or more points which are on the same line 5. perpendicular bisector – line or line segment which intersects another line segment to form right angles and to divide the segment into two congruent parts Mathematical Symbols: 1. perpendicular Main ideas: 1. Look at the pictures at the top of p. 58. (We did this experiment in an earlier lesson.) Notice the number of sides on the original shape, and count the number of sides on the polygon created inside the original. If the original polygon had 4 sides, the interior polygon has 8. 2. Look at the picture at the top of p. 59. Read through the material on this page. 3. During class we worked on the “For You to Do” at the bottom of the page. 4. Continue to read p. 60. Put the “Concurrence of Perpendicular Bisectors” theorem and the “Concurrence of Angle Bisectors” theorem into your notes. Make sure you sketch the pictures to go with them. 5. Finally, read p. 61. Make sure you understand what “collinear points” are. In class we worked through the “For You to Do”. We found that when the experiment was finished, the centers of all the circles were collinear. INVESTIGATION 2A – THE CONGRUENCE RELATIONSHIP (p. 72) Many time in life we are asked to compare and contrast things, to look for similarities and differences. This section of chapter 2 will show us how we can do that mathematically. Section 2.1 Getting Started (p. 73) Goal: Define congruence Vocabulary building: 1. congruent figures – two figures are congruent if they have the same size and shape regardless of location or orientation ABC ABC even though they are facing different directions. Mathematical symbols: 1. congruent Main ideas: 1. Read carefully through p. 73. Pay attention to the pictures of the quadrilaterals. Notice how they are all squares but are not all the same size and do not “sit” the same direction. 2. Make sure you understand that “congruence” is made up of two different symbols, and equal sign and the sign for “similar” which means alike but not exactly alike. In the case of congruence, the “not exactly alike” refers to the fact that the figures do not have to face the same direction (orientation). The equal part refers to the fact they must be the same size and shape. Section 2.2 Length, Measure and Congruence (p. 75) Goal: Interpret statements about congruent figures and write statements using the correct notation. Vocabulary building: 1. angle – a figure created when 2 rays intersect; named by a point on the first ray, the vertex, and a point on the second ray; BCD or DCB both names are ok as long as the vertex letter is in the middle (If there is only one angle, it can be named by its vertex C B C D 2. equal – compares numbers 3. line segment – part of a line; is named by its 2 endpoints which are labeled with capital letters; the name of a line segment can be written as AB or AB A B 4. vertex – the point where two rays or segments intersect to form an angle Mathematical Symbols: 1. angle - 2. measure of an angle - m Main ideas: 1. In geometry it is important to know the difference when a shape is being described and when a measurement is being described. “Equal” works with numbers, measurement. “Congruent” refers to shapes. 2. Mathematical symbols help you to know whether you can work with equal (=) or congruence ( ). AB and AB both talk about the line segment between points A and B. AB refers to the length of the segment (ex. It might measure 2 inches), while AB names a line segment, a geometric shape. JK = 1.5 inches RS = 1.5 inches JK = RS because their lengths are the same. JK RS because the bar across the top talks about the shape not the measurement. JK RS because their shapes are identical. 3. Read the Tony and Sasha section very carefully. Pay attention to the statements about the line segments. 4. On p. 76 the book discusses congruence and equality for angles. Pay attention to the paragraph and the picture. You may want to put this in your notes. 5. Pay close attention to the Check Your Understanding questions. These will help you to know if you understand the difference between equal and congruent. Section 2.3 Corresponding Parts (p. 79) Goal: Understand the meaning of corresponding parts. Vocabulary building: 1. corresponding – to be like something else in size and placement 2. quantitative measure – measurements such as area and perimeter which measure quantity 2. similar – alike in shape but not always alike in size; ex. All circles are similar. 4. tick marks – marks (tiny line segments) on a drawing which indicate that parts are congruent. Mathematical Symbols: 1. tick marks Notice the single mark for AB and the triple mark for BC . Since the markings are different, AB is not congruent to BC . Notice also the mark at B . Angles can’t use tick marks but can be marked to show congruence to another angle. Main ideas: 1. It is not always convenient to construct shapes. If we are not going to measure something, we can indicate congruence with symbols on our drawings. (see “tick marks” above) 2. Read through p. 79. Pay attention to the pictures. 3. At the bottom of p. 79 the book discusses congruence of triangles. Remember triangles are made up of sides and angles. When we talk about congruent triangles, (or any congruent shapes) we need to be careful of how we look at the figures and how we talk about their parts. ABC FED The way these are named is important. Notice A has the same marking as F . These two angles are said to be “corresponding” since they are marked the same and if we flipped DEF over, it would sit exactly on top of ABC. When we talk about the congruence of two geometric figures, it is very important to name them so their corresponding parts are named at the same time. In the figure of the two triangles, when we talk about ABC FED , we automatically know that AB FE because their letters are named in the same order. Also, AC FD. ABC correspond s to FED which tel ls us those angles are congruent. 4. corresponding parts of congruent figures are congruent (abbreviation: CPCTC) –If two figures are congruent, than all parts (sides or angles) which correspond (match up) are congruent. (This goes with #3 in the notes.) Section 2.4 Triangle Congruence (p. 82) Goal: Understand the meaning of corresponding parts and test for congruence in triangles. Vocabulary building: 1. converse – switching the “if’ and the “then” part of a statement; ex. If today is Wednesday, then we have an early out. – The converse would be “If we have an early out, then today is Wednesday.” 2. postulate – a statement which is accepted as a fact without having to prove it Mathematical Symbols: Main Ideas: 1. Review section 2.3 before starting this section. 2. Carefully read and think about the For Discussion questions on p. 82. These are important to the lesson. 3. When comparing triangles, you can sometimes prove congruence by comparing 3 parts of one triangle to 3 parts of another triangle. For example, if I measure 3 angles of ABC and 3 angles of XYZ, and their measures are the same, are the triangles congruent? If AB XY and A X and AC XZ is that enough informatio n to prove the triangles are congruent? Read and take some notes over “Classifying Your Information” on p. 82. 4. The top of p. 83 lists some possible ways to think about proving triangle congruence. Unfortunately, not all of them work. 5. Triangle Congruence Postulates – These 4 postulates will prove that one triangle is congruent to another: ASA (angle-side-angle), SAS (side-angle-side), SSS (side-sideside), and AAS (angle-angle-side). Angle-side-angle (ASA) – When looking at the triangles, 2 angles and the side in between them should match A A S A A S Side-angle-side (SAS) – When looking at the triangles, 2 sides should match with the angle between them S A S S S A Side-side-side (SSS) – when looking at the triangles, three sides of one triangle are equal in length to three sides of another triangle S S S S S S 6. Understand that there is NOT as SSA postulate. Having two sides next to each other which are congruent and an angle which is not between them will guarantee nothing about the third side of both triangles. 7. Look over Check Your Understanding on p. 84. INVESTIGATION 2B – PROOF AND PARALLEL LINES (p. 88) In this section of chapter 2, we will study parallel lines and the relationships which are formed when a third line is put in the picture. Section 2.5 Getting Started p. 89 Goal: Identify pairs of congruent angles when a transversal cuts parallel lines. Vocabulary building: 1. transversal – a line that intersects 2 or more lines 2. parallel lines – lines in the same plane which never intersect 3. midline – connects the midpoints of 2 sides of a triangle 4. exterior angle – an angle formed outside of a polygon 1 5. supplementary angles – 2 angles whose sum is 180 degrees Notes: The big idea in this lesson is that the measure of an exterior angle is equal to the sum of the two interior angles farthest from it. Section 2.6 Deduction and Proof (p. 91) Goal: Make assumptions and write proofs to understand the need for proofs in mathematics. Vocabulary building: 1. proof – (math def.) a sequence of steps, statements, or demonstrations that lead to a valid conclusion 2. deductive process – use logical thinking to get an answer 3. postulate – a statement which is assumed to be true 4. assumption – something taken for granted 5. median – a segment drawn from a vertex of a triangle to the midpoint of the opposite side 6. vertical angles – formed when 2 lines intersect; (look like the letter X) Notes: 1. In writing a proof it is important to provide evidence, reasons why something is true. 2. Read Minds in Action on p. 92. Sasha is trying to prove to Ivan why his angles must be congruent. This is an example of a proof: We are told that AC = 14 in and AB = 14 in. The measure of angle CAB = 30 degrees. (These are considered “given”s.) Sasha tells Ivan to draw a median from A to M (see vocabulary for this lesson) from C. Now, CM BM because of the definition of a midpoint. AM AM because everything is equal to itself. CAM BAM because of SSS. Finally, Sasha says ACM ABM because corresponding parts of congruent figures are congruent. The items in black are considered statements while the items in red are the reasons the statements are true. This is an example of how a proof works. 3. Try this one: A straight angle is a line. It has 180 degrees. AC is a line (straight angle). If mBDC 72 then mBDA 180 72 108. BQ is also a line (straight angle). Since mBDC 72 , then mCDQ 180 72 108. Finally, by using the same strategy, we would find the measure of angle QDC = 72. This allows us to notice that vertical angles are equal. 4. Write the vertical angle theorem in your notes. (p. 93.) Section 2.7 Parallel Lines (p. 99) Goal: Identify pairs of congruent angles when a transversal cuts parallel lines. Vocabulary building: 1. parallel lines – lines in the same plane which do not intersect 2. alternate interior angles – angles on opposite sides of a transversal between lines (in the diagram see angles 3 and 5; see angles 4 and 6) 3. corresponding angles – found on the same side of the transversal, occupy similar positions (in the diagram see angles 1 and 5; angles 4 and 8; angles 2 and 6; and angles 3 and 7) 4. vertical angles – formed when two lines intersect 5. consecutive angles – found on the same side of the transversal between the lines (in the diagram see angles 4 and 5; angles 3 and 6) 6. alternate exterior angles – found on opposite sides of the transversal outside the lines (in the diagram see angles 1 and 7; angles 2 and 8) 7. corollary – a consequence which comes from a theorem 8. equidistant – equal distance from the same point Notes: 1. It is very important to understand the vocabulary in this section. Make sure you have studied the diagram. 2. AIP theorem – If two lines from congruent alternate interior angles with a transversal, then the two lines are parallel 3. Read through p. 100 which is the proof of the AIP theorem. 4. Make sure you read the rest of the section. We will also find that when parallel lines are cut by a transversal corresponding angles are equal in measure, consecutive angles are supplementary, and alternate exterior angles are equal in measure. Section 2.8 The Parallel Postulate (p. 105) Goal: Identify pairs of congruent angles when a transversal cuts parallel lines. Vocabulary building: Notes: Parallel postulate – if point p is not on line l, exactly one line through p will be parallel to l p l PAI theorem – if two parallel lines are cut by a transversal, alternate interior angles are congruent. line n is parallel to line l; angles 1 and 2 are alternate interior angles and are congruent. Use the picture and what you know about alternate interior, vertical, corresponding, straight, and consecutive angles to find the missing angle measures. m1 70 m2 110 m3 115 (straight angle w/65) m4 65 (vertical angles) m5 115 (vertical to angle 3 or straight angle w/65) m6 70 (straight angle w/angle 2 or triangl e 180) (I’ve included just some of the answers to give you an idea.) Triangle-angle-sum theorem – The interior angles of a triangle equal 180 degrees. Unique perpendicular theorem – If point p is not on line , there is only one line through p which is perpendicular to (See the picture for the parallel postulate, think about a line going straight down to make a right angle with line .) Investigation 2C – Writing Proofs Section 2.9 Getting Started p. 115 Section 2.10 What Does a Proof Look Like? P. 117 Goal: Use a variety of ways to write and present proofs. Vocabulary: Notes: See p. 117 for an example of a 2 column proof. Two column proofs must have a leftcolumn for Statements and a right-column for Reasons. The statements tell what you know while the reasons explain how you know. All proofs begin with the “given” because that is all you know when you start. The very last “statement” is what you are trying to prove, your goal. See p. 118 at the top for an example of a paragraph proof. A paragraph proof is written in logical sentences. You must keep it detailed enough so it makes sense. You still begin with your “given”. In the middle of p. 118 is an example of an outline style proof. This is more informal than the 2 column proof but still used a series of statements and reasons. Notice the symbols for “because” and “therefore”. The reasons are put in ( ). Section 2.11 Analyzing the Statement to Prove p. 123 Goal: Identify the hypothesis and conclusion of a given statement. Vocabulary: 1. hypothesis – the “if” part of an “if . . . then” statement; what you assume to be true. 2. conclusion – the “then” part of an “if . . . then” statement; states what you need to prove 3. equiangular – all angles in the figure have the same measure Notes: An “if . . . then” statement is made of a hypothesis and a conclusion. Ex. If I am hungry, then I will have a snack. The “I am hungry” is the hypothesis while the “I will have a snack” is the conclusion. “if . . .then” statements do not have to be true. The goal of this lesson is to be able to identify the hypothesis and the conclusion and to tell if the statement is true or false. Ex. If a triangle is isosceles, then the two base angles are congruent. The hypothesis is “a triangle is isosceles”, and the conclusion is “the two base angles are congruent.” This is a true statement since an isosceles triangle has 2 sides of equal measure which would make two of the angles be congruent. Ex. Vertical angles are congruent. The words “if” and “then” are not here, however, think of the statement this way: If angles are vertical, then they are congruent. This is a true statement. Section 2.12 Analysis of a Proof p. 125 Goal:Use a variety of ways to write and present proofs. Vocabulary: Notes: 1. “analysis” is creating the proof; deciding what to use; like a rough draft of a paper 2. “visual scan” – This is what we have been doing so far. Draw a picture, mark congruent pieces, see what you have. 3. Look at example 1 p. 126. It explains how to analyze and use a visual scan. 4. Notice that CPCTC is used often as a reason. 5. Flow chart strategy. This is a top-down analysis which starts with what you know and then moves down providing reasons. 6. A flow chart can easily be turned into a 2 column proof. Section 2.13 The Reverse List p. 131 Section 2.14 Practicing Your Proof Writing Skills p. 135 Goal: Use the perpendicular bisector theorem and the isosceles triangle theorem to prove that parts of a figure are congruent. Vocabulary: 1. perpendicular bisector – a segment which divides another segment into two equal parts while making a right angle at the point of intersection 2. Isosceles triangle – a triangle which has at least 2 congruent sides and 2 congruent angles Notes: