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Math 2201 Unit 2: Properties of Angles and Triangles Read Learning Goals, p. 67 text. Ch. 2 Notes §2.1 Exploring the Angles Formed by Intersecting Lines (0.5 class) Read Goal p. 70 text. Outcomes: 1. Define a transversal. pp. 70, 518 2. Define interior angles and identify them in a diagram. pp. 71, 516 3. Define exterior angles and identify them in a diagram. pp. 71, 515 4. Define corresponding angles and identify them in a diagram. pp. 71, 514 5. Define same side interior angles and identify them in a diagram. See notes 6. Define alternate interior angles and identify them in a diagram. pp. 75, 514 7. Define alternate exterior angles and identify them in a diagram. pp. 76, 514 8. Define vertically opposite angles (vertical angles) and identify them in a diagram. See notes This chapter is about the geometry of angles and triangles. You will use inductive reasoning to make conjectures about the properties of angles and triangles and then use deductive reasoning to prove, or counterexamples to disprove, these conjectures. Your proofs will often be in the two-column format. Def n : A transversal is a line that intersects (cuts across) two or more other lines. Transversal A transversal creates two general types of angles, interior and exterior angles. Def n : Interior angles are angles that lie inside the lines that the transversal intersects. In the diagram below, a, b, c, and d are interior angles. Transversal a c 1 d b Def n : Exterior angles are angles that lie outside the lines that the transversal intersects. In the diagram below, e, f , g , and h are exterior angles. Transversal e g f h Def n : Corresponding angles are two angles, one exterior and one interior, that are not adjacent but lie on the same side of the transversal. In the diagram below, a and e, b and f , c and g, and d and h are corresponding angles. Notice that the lines that Transversal form these angles make an “F” shape a E.g.: c a e e g b d f h Def n : Same-side interior angles are two interior angles that lie on the same side of the transversal. In the diagram below, c and e, and d and f are same-side interior angles. Notice that the lines that form these angles make a square-ish “C” shape Transversal E.g.: d c a c e e f g b d f h Def n : Alternate interior angles are two interior angles that lie on opposite sides of the transversal. In the diagram below, c and f , and d and e are alternate interior angles. Notice that the lines that form these angles make a “Z” shape Transversal E.g.: d c a c f e \ e e g 2 f h b d 3 Def n : Alternate exterior angles are two exterior angles that are not adjacent and that lie on opposite sides of the transversal. In the diagram below, a and h, and b and g are alternate interior angles. E.g.: Transversal a c e g b d f h The last pair of angles we need are vertically opposite angles. They are not formed by three lines but by two lines. Def n : Vertically opposite angles (vertical angles) are two angles that are formed when two lines intersect. In the diagram below, a and d , and b and c are vertically opposite angles. a c 4 b d §2.2 Exploring the Angles Formed by Parallel Lines (2 classes) Read Goal p. 73 text. Outcomes: 1. Define parallel lines and use the symbol for parallel lines. p. 75 and notes 2. On a diagram, indicate that lines are parallel using the correct symbol(s). p. 75 3. Generalize, using inductive reasoning, the relationships between pairs of angles (corresponding angles, same-side interior angles, alternate interior angles, and alternate exterior angles ) formed by transversals and parallel lines, with or without technology. pp. 71,78 4. Define the term converse. pp. 71, 514 5. Given a statement, write the converse of the statement. p. 71 6. Determine which angle properties do NOT apply if lines are not parallel. p. 71 7. Define a postulate. See notes 8. State the 4 general types of reasons used in two-column proofs. See notes 9. State the Angle Addition Postulate. See notes Def n : Parallel lines are lines in the same plane that do not intersect. The symbol to”. In the diagram to the right, lines l and m are parallel, so we can write l m . On diagrams that we draw, we place arrow heads (or double or triple arrow heads if necessary) between the ends of the lines to indicate that the lines are parallel. means “is parallel l m l The Relationships between the Angles Formed by Two Parallel Lines and a Transversal Corresponding Angles Draw a transversal that intersects the two parallel lines below. 5 Measure any two corresponding angles. What do you notice about the measures? Measure a different pair of corresponding angles. What do you notice about those measures? Use inductive reasoning to make a conjecture about the measures of corresponding angles when two lines are cut by a transversal. Conjecture: When two parallel lines are cut by a transversal, the measures of corresponding angles are ___________________. Should you make this conjecture if the lines are not parallel? (Y or N) _____________ Alternate Interior Angles Draw a transversal that intersects the two parallel lines below. Measure any two alternate interior angles. What do you notice about the measures? Measure a different pair of alternate interior angles. What do you notice about those measures? Use inductive reasoning to make a conjecture about the measures of alternate interior angles when two lines are cut by a transversal. Conjecture: When two parallel lines are cut by a transversal, the measures of alternate interior angles are ___________________. Should you make this conjecture if the lines are not parallel? (Y or N) _____________ Alternate Exterior Angles Draw a transversal that intersects the two parallel lines below. 6 Measure any two alternate exterior angles. What do you notice about the measures? Measure a different pair of alternate exterior angles. What do you notice about those measures? Use inductive reasoning to make a conjecture about the measures of alternate exterior angles when two lines are cut by a transversal. Conjecture: When two parallel lines are cut by a transversal, the measures of alternate exterior angles are ___________________. Should you make this conjecture if the lines are not parallel? (Y or N) _____________ Def n : If the sum of the measures of two angles equals 180 then the angles are supplementary. 130 50 Since 130 50 180 , the two angles above are supplementary. Same-Side Interior Angles Draw a transversal that intersects the two parallel lines below. Measure any two same-side interior angles. What do you notice about the measures? Measure a different pair of same-side interior angles. What do you notice about those measures? Use inductive reasoning to make a conjecture about the measures of same-side interior angles when two parallel lines are cut by a transversal. Conjecture: When two parallel lines are cut by a transversal, the same-side interior angles are ___________________. Should you make this conjecture if the lines are not parallel? (Y or N) _____________ Def n : A theorem is a formula or statement that can be proved from other formulas or statements. The last four conjectures can be grouped into one statement called the Parallel Lines Theorem (PLT). 7 Parallel Lines Theorem (PLT) (See In Summary, p. 78 text) If two parallel lines are cut by a transversal then, the measures of corresponding angles are equal. the measures of alternate interior angles are equal. the measures of alternate exterior angles are equal. same-side interior angles are supplementary. Note that the PLT consists of a premise (If two parallel lines are cut by a transversal) and four conclusions (each of the bulleted statements). Also note that there are simple two-column proofs for some of the statements that make up the PLT on pp. 75. However, we will be focusing on using the PLT to complete other proofs and to find the missing angles in diagrams involving parallel lines. Def n : The converse of a given statement is a statement formed by switching the premise and the conclusion of the given statement. Let’s form the four converses that arise from the PLT. They can be used to prove that two lines are parallel. (See In Summary, p. 78 text) Converse of PLT #1: If two lines are cut by a transversal and the measures of corresponding angles are equal then the lines are parallel. Converse of PLT #2: If two lines are cut by a transversal and the measures of alternate interior angles are equal then the lines are parallel. Converse of PLT #3: If two lines are cut by a transversal and the measures of alternate exterior angles are equal then the lines are parallel. Converse of PLT #4: If two lines are cut by a transversal and the same-side interior angles are supplementary then the lines are parallel. Do # 5, p. 72 text in your homework booklet. The Relationships between the Angles Formed by Two Intersecting Lines Vertically Opposite (Vertical) Angles Draw two intersecting lines below. 8 Measure one pair of vertical angles. What do you notice about the measures? Measure a different pair of vertical angles. What do you notice about those measures? Use inductive reasoning to make a conjecture about the measures of vertical angles when two lines intersect. Conjecture: When two lines intersect, the measures of the vertical angles are ___________________. E.g.: Find the value of alpha α , beta β , and gamma γ in the diagram below. Since AC BM , we can use the PLT to find two of the missing angles. Since CAB and MBE are corresponding angles, they have the same measures so m MBE 65 . Since ACB and MBC are alternate interior angles, they have the same measures so m MBC 80 . Since ABE is a straight angle, we know that 180 . Substituting values for and gives 65 80 180 145 180 145 145 180 145 35 E.g.: Use the PLT to find the values of x and y and the measures of the marked angles. 6 x 65 5 y 15 3y 5 2x 5 Since the angles marked 3 y 5 and 5 y 15 are same-side interior angles, they are supplementary. Therefore, 9 3 y 5 5 y 15 180 8 y 20 180 8 y 20 20 180 20 8 y 160 8 y 160 8 8 y 20 So these two angles have measures 3 20 5 65 and 5 20 15 115 . Since the angles marked 6 x 65 and 5 y 15 are vertically opposite angles, we know they have the same measure. Therefore 6 x 65 115 6 x 65 65 115 65 6 x 180 6 x 180 6 6 x 30 Finally, the angle marked 2 x 5 must be 65 since it is vertically opposite to the angle marked 3 y 5 . Do #’s 4 a, b, 15, 20 pp. 79-82 text in your homework booklet. Def n : A postulate is a statement that is accepted without proof. E.g.: Angle Addition Postulate (AAP): If point B lies in the interior of m AOB m BOC m AOC . A AOC , then B O C So now you have four different types of reasons that you can use in a two-column proof. Given information Definitions Postulates (including properties of algebra) Theorems that you have already proved. Now let’s use the PLT and its converses to complete some two-column proofs. 10 E.g.: #16, p. 81 text. Statements AB DE; DE FG m BAC m ACF m CDE m DCF m ACD m ACF m DCF m ACD m BAC m CDE Reasons Given PLT (alternate interior angles) PLT (alternate interior angles) Angle Addition Postulate Substitution E.g.: #12, p. 80 text. Statements m FOX m FRS SR XO m FXO m FPQ Reasons Given Converse of PLT (corresponding angles) PQ XO Given Converse of PLT (corresponding angles) PQ SR Transitive Property 11 §2.3 Angle Properties in Triangles (1 class) Read Goal p. 86 text. Outcomes: 1. Define an auxiliary line and use it to complete proofs. See notes 2. Prove the angle sum of a triangle theorem (ASTT). pp. 86-87 3. Define and identify an exterior angle of a polygon. pp. 87, 95, 515 4. Define and identify non-adjacent interior angles (remote interior angles). pp. 88, 516 5. Determine and prove the relationship between an exterior angle of a triangle and its nonadjacent interior angles (remote interior angles). p. 88 Def n : An auxiliary line is a line (ray, segment) added to a diagram to help complete a proof. We are going to use an auxiliary line to prove that the sum of the measures of the angles of any triangle is 180 . This statement is known as the angle sum of a triangle theorem (ASTT). E B D A Statements Draw EF parallel to AD m EBA m ABD m DBF 180 m EBA m BAD; m DBF m BDA m BAD m ABD m BDA 180 F Reasons Auxiliary Line Definition of straight angle. PLT (Alternate interior angles). Substitution Def n : An exterior angle of a polygon is the angle formed by a side of a polygon and the extension of an adjacent side. In the triangle below, ACD is an exterior angle. 12 Carefully extend any side of the triangle drawn below to form an exterior angle. Find the measure of this exterior angle. Measure the two angles that are inside the triangle and as far away from the exterior angle as possible. These are called the remote interior angles. B A C Make a conjecture about the measure of the exterior angle and the measures of the two remote interior angles. Conjecture: The measure of the exterior angle is equal to the __________ of the two remote interior angles. E.g.: Find m S in the diagram to the right. S x According to the last conjecture, 3 x x 80 3 x x x x 80 2 x 80 2 x 80 2 2 x 40 R 80 3x T So m S 40 . Let’s prove our last conjecture and make it into a theorem. 2 1 m m m m 1 m 3 m 1 m 1 m Statements 2 m 3 180 4 180 2 m 3 m 3 m 4 2m 4 3 4 Reasons ASTT Definition of a straight angle. Substitution Subtraction Property Do #’s 3, 5, 7, 10, 14, 15, pp.90-92 text in your homework booklet. 13 §2.4 Angle Properties in Polygons (1 class) Read Goal p. 94 text. Outcomes: 1. Define and give an example of a convex polygon. pp. 96, 514 2. Determine how the number of sides of a convex polygon is related to the sum of its interior angles. p. 94 3. Determine the sum of the exterior angles of a convex polygon. p. 95 4. Define and give an example of a regular polygon. p. 97 5. Determine how the number of sides of a regular polygon is related to the measure of each interior angle. p. 96 Def n : A convex polygon is a polygon in which each interior angle measures less than 180 . In the figure below, some of the polygons are convex and some are non-convex (concave). Label the figure correctly. ________________________ ______________________________ Complete the table below for the sum of the measures of the interior angles of a convex polygon. (See pp. 94, 96 text) Polygon Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon n-gon # Sides (n) 3 4 5 6 7 8 # Triangles 1 2 3 4 5 6 n n2 Sum of Measures of Interior Angles (S) 180 360 Make a conjecture about the relationship between the number of sides of a convex polygon, n, and the sum of the measures of the interior angles, S. Conjecture: The sum of the measures of the interior angles of a convex polygon is ________________ ___________________________________________________________________________________. 14 Below, write the expression for the sum of the measures of the interior angle of a regular polygon with n sides. (See p. 99) E.g.: The sum of the measures of the interior angles of a convex polygon is known to be between 2500 and 2600 . How many sides does the polygon have? The sum of the measures of the interior angles of a convex polygon is n 2 180 . Let n 2 180 2600 180n 360 2600 180n 360 360 2600 360 180n 2960 180 n 2960 180 180 n 16.4 Since the largest the sum of the interior angles can be is 2600 , the number of sides must be less than 16.4 . So the polygon has 16 sides. Complete the table below for the sum of the measures of the exterior angles of a convex polygon. (See p. 95 text) Polygon Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon n-gon # Sides 3 4 5 6 7 8 # Exterior Angles 3 4 5 6 7 8 n n Sum of Measures of Exterior Angles 360 360 Make a conjecture about the sum of the measures of the exterior angles of a convex polygon. Conjecture: The sum of the measures of the exterior angles of a convex polygon is _______________. E.g.: The sum of the measures of the interior angles of a polygon is four times the sum of the measures of its exterior angles, one at each vertex. How many sides does the polygon have? 15 Let n be the number of sides of the polygon, then ***The sum of the measures of the interior angles of a polygon is n 2 180 . The sum of the measures of its exterior angles is 360 . ***Four times the sum of the measures of its exterior angles is 4 360 1440 . Therefore, n 2 180 1440 n 2 180 180 n28 n2 2 8 2 1440 180 n 10 So the polygon has 10 sides. Def n : A regular polygon is a polygon in which side is the same length and each interior angle has the same measure. E.g.: 16 Complete the table below for the measure of each interior angle of a regular polygon. (See p. 97 text) Polygon # Sides (n) Sum of Measures of Interior Angles (S) 3 180 4 360 Regular (Equilateral) Triangle Regular Quadrilateral (Square) Regular Pentagon 5 Regular Hexagon 6 Regular Heptagon 7 Regular Octagon 8 n-gon n Measure of Each Interior Angle 180 60 3 360 90 4 n 2180 Make a conjecture that relates the measure of each interior angle to the number of sides of a regular polygon. Conjecture: The measure of each interior angle of a regular polygon is _________________________ ___________________________________________________________________________________. Below write the expression for the measure of each interior angle of a regular polygon with n sides. (See p. 99) E.g.: The measure of each interior angle of a regular polygon is eight times that of an exterior angle. How many sides does the polygon have? Let n be the number of sides of the polygon, then The sum of the measures of the interior angles of a polygon is n 2 180 . ***The measure of each interior angle is n 2 180 n The sum of the measures of its exterior angles is 360 . The measure of an exterior angle is 360 n 360 ***Eight times the measure of an exterior angle is 8 . n 17 Therefore, n 2 180 8 360 n n 180n 360 2880 n n Since the two fractions above are equal and their denominators are equal, then their numerators must also be equal. Therefore, 180n 360 2880 180n 360 360 2880 360 180n 3240 180 n 3240 180 180 n 18 So the regular polygon has 18 sides. Do #’s 1, 3, 6-8, 11, pp. 99-101 text in your homework booklet. 18 §2.5 Exploring Congruent Triangles (One-half class) Read Goal p. 104 text. Outcomes: 1. Explain what is meant by congruent triangles. See notes. 2. Define and give an example of an included (contained) angle. See notes 3. Define and give an example of a non-included (non-contained) angle. See notes 4. Define and give an example of an included (contained) side. See notes 5. Determine five ways to show that two triangles are congruent. p. 105 Def n : Congruent triangles ' s are triangles in which corresponding sides have the same length and corresponding angles have the same measure. In the diagram to the right, JMK PRQ . This means that: Sides Angles JM PR m J m P JK PQ m K m Q m M m R MK QR If we want to work backwards and show that two triangles are congruent, do we have to show that three sets of corresponding sides are equal in length and that three sets of corresponding angles have the same measure? Luckily, the answer is NO! Proving (showing) that Two Triangles are Congruent Method 1: SSS (side-side-side) If three pairs of corresponding sides are equal, then the two triangles are congruent. Using the diagram below, if you can show that AB DE , BC EF , and AC DF then you have proved that ABC DEF . 19 Def n : An included (contained) angle is an angle that is formed by two sides of the triangle. Method 2: SAS (side-angle-side) If two pairs of corresponding sides and the included (contained) angles are equal, then the two triangles are congruent. Using the diagram below, if you can show that AB DE , BC EF , and m B m E then you have proved that ABC DEF . Def n : An included (contained) side is a side that is formed by two angles of the triangle. Method 3: ASA (angle-side-angle) If two pairs of corresponding angles and the included (contained) sides are equal, then the two triangles are congruent. Using the diagram below, if you can show that m B m E , BC EF , and m C m F then you have proved that ABC DEF . 20 Method 4: AAS (angle-angle-side) If two pairs of corresponding angles and the non-included (non-contained) sides are equal, then the two triangles are congruent. Using the diagram below, if you can show that m B m E , m C m F , and AC DF then you have proved that ABC DEF . Method 5: HL (hypotenuse-leg) If the hypotenuse and a leg of one right triangle are equal to the hypotenuse and a leg of another right triangle, then the two triangles are congruent. Using the diagram below, if you can show that, AB DE and AC DF then you have proved that ABC DEF . Do #’s 1-3, p. 106 text in your homework booklet. 21 §2.6 Proving Congruent Triangles (2 classes) Read Goal p. 107 text. Outcomes: 1. Prove that triangles, or that corresponding parts of triangles, are congruent. To show that parts of triangles are congruent, it is often necessary to first prove that the triangles themselves are congruent using deductive reasoning (often SSS, SAS, ASA, AAS, or HL). These proofs will often be in the two-column format. Proof using SSS Prove that m NQY m PQY Statements NQ PQ, NY PY YQ YQ NQY PQY m NQY m PQY Reasons Given Common side SSS Corresponding parts of congruent triangles are equal. (CPCTE) Proof using SAS Prove that AB ED Statements AC EC, BC DC m ACB m ECD ABC EDC AB ED Reasons Given Vertically opposite angles are equal. SAS Corresponding parts of congruent triangles are equal. (CPCTE) 22 Proof using ASA Prove that BP CP Statements AP 5, DP 5 AP DP m BPA m CPD BAP is a right angle CDP is a right angle m BAP 90 m CDP 90 m BAP m CDP BPA CPD BP CP Reasons Given Substitution Vertically opposite angles are equal. Given Definition of a right angle. Substitution ASA CPCTE Proof using AAS Prove that PS QS m SPR m SQR m SRP m SRQ RS RS SPR SQR PS QS Statements Reasons Given Common side AAS CPCTE 23 Proof using HL Prove that m QPS m RPS Statements QP RP SP SP QSP is a right angle RSP is a right angle QSP is a right triangle RSP is a right triangle QSP RSP m QPS m RPS Reasons Given Common side Given Definition of a right triangle. HL CPCTE Do #’s 1-2, 4-6, 8, 9, 11, pp. 112-114 text in your homework booklet. Do #’s 2-5, 7, 8, 10, 12, 13, 15-17, pp. 119-120 text in your homework booklet. 24