Properties of Angles and Triangles. Outcomes: G1 Derive proofs that involve the properties of angles and triangles.
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1 Properties of Angles and Triangles Outcomes: G1 Derive proofs that involve the properties of angles and triangles. Achievement Indicators: Generalize, using inductive reasoning, the relationships between pairs of angles formed by transversals and parallel lines, with or without technology. Prove, using deductive reasoning, properties of angles formed by transversals and parallel lines, including the sum of the angles in a triangle. Generalize, using inductive reasoning, a rule for the relationship between the sum of the interior angles and the number of sides (n) in a polygon, with or without technology. Identify and correct errors in a given proof of a property involving angles. Verify, with examples, that if lines are not parallel the angle properties do not apply. Sep 26 9:24 PM 1
2 G2 Solve problems that involve the properties of angles and triangles. Achievement Indicators: Determine the measures of angles in a diagram that involves parallel lines, angles and triangles, and justify the reasoning. Identify and correct errors in a given solution to a problem that involves the measure of angles. Solve a contextual problem that involves angles or triangles. Construct parallel lines, using only a compass or a protractor, and explain the strategy used. Determine if lines are parallel, given the measure of an angle at each intersection formed by the lines and a transversal. Sep 26 9:34 PM 2
3 Define: inductive reasoning (6) conjecture (6) deductive reasoning (28) proof (27) counterexample (19) transitive property (29) two column proof (29) Inductive reasoning drawing a general conclusion by observing patterns and identifying properties in specific examples. The conclusion may or may not be true. specific general Deductive reasoning drawing a specific conclusion through logical reasoning by starting with general assumptions that are known to be true. general specific conjecture A testable expression that is based on available evidence but is not yet proved. counterexample an example that invalidates a conjecture. proof a mathematical argument showing that a statement is true in all cases, or that no counterexample exists. transitive property if two quantities are equal to the same quantity, then they are equal to each other. ex: if a = b and b = c, then a = c. Two colum proofs use deductive reasoning: Steps for deductive reasoning (or two column proofs): Step 1 Begin with an accepted fact. Step 2 Continue with step by step statements with a logical reason using definitions, theorems or postulates. Step 3 Each statement must follow in a sequential order, and lead to a logical conclusion. Sep 26 9:53 PM 3
4 acute angle between 0 and 90 obtuse angle between 90 and 180 straight angle exactly 180 right angle exactly 90 complementary two angles that add up to 90 supplementary two angles that add up to 180 Angles formed by Transversal transversal a line that intersects two or more other lines at distinct points. Sep 26 9:55 PM 4
5 Vocabulary about angles: (previous knowledge) acute obtuse straight right complementary supplementary Vocabulary with parallel lines: transversal interior angles exterior angles corresponding angles alternate interior angles alternate exterior angles vertical angles / opposite angles converse Define each of the terms above, and then draw a diagram to provide an example of each. (Begin p.69 if using text.) Sep 26 9:40 PM 5
6 angles are formed. Interactive activity observe relationships between the different angles created by the transversal. Sep 26 10:01 PM 6
7 Entrance Question E G B D Which pairs of angles are equal in this diagram? A C H F Is there a relationship between the measures of the pairs of angles that are not equal? Oct 2 12:19 AM 7
8 Corresponding Angles "F angles" Any pair of parallel lines makes an F shape with a transversal that crosses them. 4 <4 and <8 are called corresponding angles. 8 Activity (need ruler and protractor): Choose two lines on your notebook paper as parallel lines. With these two lines, draw a transversal. Choose two corresponding angles and measure them with a protractor. Compare the angles. Repeat the process three times using other transversals with different slopes. What do you find? Sep 26 11:00 PM 8
9 Interior Angles when two lines are intersected by a transversal, the four angles between the lines are called interior angles <3, <4, <5 and <6 are interior angles. <3 and <6 are alternate interior angles. Read examples page in text Prove alternate interior angles of parallel lines are equal. Given: l 1 II l 2 Prove: <4 = <5 < 4 and < 5 are alternate interior angles. Two column Proof: Statement Reason 1. l 1 ll l 2 given 2. <1 = <4 vertical angles 3. <1 = <5 corresponding angles 4. <4 = <5 both equal to <1 (substitution) Alternate Interior Angle Theorem If two parallel lines are cut by a transversal, then alternate interior angles are equal. If two lines are cut by a transversal, and the alternate interior angles are equal, then the lines are parallel. Sep 26 10:45 PM 9
10 Vertical angles: Prove vertical angles are equal. Given: <1 and <2 are vertical angles Prove: <1 = <2 Two column Proof: Statement Reason 1. <1 + <3 = 180 supplementary angles / straight line 2. <2 + <3 = 180 supplementary angles / straight line 3. <1 + <3 = <2 + <3 both equal 180 (substitution) 4. <1 = <2 subtraction of equal parts Sep 26 10:20 PM 10
11 Co interior Angles When two lines l1 and l2 are intersected by a transversal, then the interior angles on the same side of the transversal are called cointerior angles <3 and <5 are co-interior angles. 8 <4 and <6 are co-interior angles. Prove co interior angles of parallel lines are supplementary. Given: l 1 // l 2 Prove: <3 + <5 = 180 Proof: Statement Reason 1. l1//l2 given 2. <3 = < 6 alternate interior angles 3. <5 + <6 = 180 angles on a line (supplementary) 4. <5 + <3 = 180 substitute < 3 for <6 (equal angles) 5. <3 + <5 = 180 rewrite of step 4 Co interior Angles Theorem If two parallel lines are cut by a transversal, then co interior angles are supplementary. If two lines are cut by a transversal, then, if co interior angles are supplementary, the lines are parallel. Sep 29 5:07 PM 11
12 Find the missing angles, and state reasons for each answer. Example 1: <1 = <2 = Example 2: < 1 = < 2 = < 3 = Example 3: 53 Example 4: Sep 29 5:38 PM 12
13 Summary Parallel Lines and a Transversal Vertical Angles < 1 = < 4 < 2 = < 3 < 5 = < 8 < 6 = < 7 Alternate interior Angles < 3 = < 6 < 4 = < 5 Corresponding Angles < 1 = < 5 < 2 = < 6 < 3 = < 7 < 4 = < 8 Co interior Angles < 3 + < 5 = 180 < 4 + < 6 = 180 Sep 29 5:41 PM 13
14 Prove the following: 1. Given c//d <1 = <3 Prove a//b c d 2 b a Statement Reason 1. c//d 2. given 3. <3 = <4 4. <1 = <4 5. a//b alternate interior angles Prove the following: Given: BE bisects <ABC CE bisects <BCD <2 + <3 = 90 Prove: AB//CD Statement Reason 1. BE bisects <ABC 2. <1 = <2 definition of bisect 3. CE bisects <BCD <2 + <3= <2 + <2 + <3 +<3 = 180 addition 7. <1 + <2 + <3 + <4 = AB//CD Oct 1 11:37 PM 14
15 Given: <1 = <5 Prove: <2 = <4 Statement Reason Given: <1 is supplementary to <4 Prove: <2 = <3 Statement Reason Oct 1 11:58 PM 15
16 Text page 72, selected questions Text page 78 85, selected questions Sep 30 8:58 PM 16
17 Angle Properties in Triangles The Sum of Angles in a Triangle Use what we have learned about parallel lines to prove that the angles in a triangle add to 180. D C 4 3 A 1 2 B Given ABC Prove: <1 + <2 + <3 = 180 Proof: Statement Reason 1. Draw line DC parallel to AB construction 2. <3 + <4 = <DCB angle addition 3. <DCB + <2 = 180 co interior angles 4. <3 + <4 + <2 = 180 substitution from step 2 5. <1 = <4 alternate interior angles 6. <1 + <2 + <3 = 180 substitution Sum of the Angles in a Triangle Theorem (SATT) The sum of angles in a triangle is 180. Sep 29 5:24 PM 17
18 Use angle sums to determine angle measures. In the diagram, <MTH is an exterior angle of MAT. Determine the measures of the unknown angles in MAT. M 40 A 155 T H Textbook Examples pages Application Questions pages Oct 2 12:09 AM 18
19 Journal If you are given one interior angle and one exterior angle of a triangle, can you always determine the other interior angles of the triangle? Explain, using diagrams. Oct 2 12:14 AM 19
20 Angle Properties in Polygons How is the number of sides in a polygon related to the sum of its interior angles and the sum of its exterior angles? Polygon # of sides # of triangles Sum of Angle Measures triangle quadrilateral 4 pentagon 5 hexagon 6 heptagon 7 octagon 8 Draw the polygons listed in the table above. Create triangles to help you determine the sum of the measures of their interior angles. Record your results in a table like the one above. Make a conjecture about the relationship between the sum of the measures of the interior angles of a polygon, S, and the number of sides of the polygon, n. Use your conjecture to predict the sum of the measures of the interior angles of a dodecagon (12 sides). Verify your prediction using triangles. Oct 2 12:26 AM 20
21 Sum of Exterior Angles of Polygons Draw a rectangle. Extend each side of the rectangle so that the rectangle has one exterior angle for each interior angle. Determine the sum of the measures of the exterior angles. What do you notice about the sum of the measures of each exterior angle of your rectangle and its adjacent interior angle? Would this relationship also hold true for the exterior and interior angles of the irregular quadrilateral shown? Explain. Make a conjecture about the sum of the measures of the exterior angles of any quadrilateral. Test your conjecture. Draw a pentagon. Extend each side of the pentagon so that the pentagon has one exterior angle for each interior angle. Based on your diagram, revise your conjecture to include pentagons. Test your revised conjecture. Do you think your revised conjecture will hold for polygons that have more than five sides? Explain and verify by testing. Oct 2 1:00 AM 21
22 Oct 2 10:36 AM 22
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