Background information: Labeling angles and lines
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1 Chapter 2 Properties of Angles and Triangles Background information: Labeling angles and lines Lines can be labelled using two letters Angles can be labelled using the symbol accompanied by three letters. The first and third letters indicate points on the two arms. The letter in the middle is the angle. Note that the first and third letters are interchangeable E A G B C H D F Another way to label an angle is by using the symbol accompanied by the angle letter alone. However, this method can only be used when there is only one angle at the vertex point. This method could not be used in the diagram above. 1
2 The sum of the measures of the interior angles of a triangle is
3 Section Exploring Parallel Lines Parallel lines: two lines in the same plane that, no matter how far they extend, do not intersect with each other. Parallel lines are the same distance apart at any given point. 3
4 Transversal a line that intersects two or more lines at distinct points 4
5 Interior Angles any angles formed by a transversal and two parallel lines that lie inside the parallel lines. Exterior Angles any angles formed by a transversal and two parallel lines that lie outside the parallel lines. 5
6 Corresponding Angles one interior angle and one exterior angle that are non adjacent and on the same side of a transversal. 6
7 List all pairs of corresponding angles in the diagram below. 7
8
9 Important Points When a transversal intersects a pair of parallel lines, the corresponding angles are equal e a f b g c h d Conversely, when a transversal intersects two lines and creates equal corresponding angles, the lines are parallel 9
10 NON PARALLEL LINES When a transversal intersects a pair of nonparallel lines the corresponding angles are NOT EQUAL a d b c e f h g 10
11 Vertically Opposite Angles angles that are opposite each other when two lines cross. Vertically opposite angles are equal. Supplementary angles: Angles that add up to 180 o A straight angle measures
12 If A and B are parallel lines, determine the unknown angles. Provide a brief statement showing your reasoning. 12
13 If line AB is parallel to line CD, determine the indicated angles. Provide a brief statement showing your reasoning. E A G B C H 75 0 D F 13
14 Given the following diagram, predict the measures of the angles a through g. Provide a brief statement showing your reasoning. 14
15 4. a) Identify the following A M N Transversal: Corresponding angles: Interior angles: X W Y Z T Q S R Exterior angles: B O b) Are the corresponding angles equal? Explain P M c) Identify the following Transversal: Corresponding angles: Interior angles: A X W Y Z T Q S N R Exterior angles: B d) Are the corresponding angles equal? Explain P O 15
16 5. In the following diagrams is AB parallel to CD? Explain how you know. a) A C b) E R B G 130 o H H 140 o D 130 o A 45 o G Q C B D F corresponding angles are equal therefore the lines are parallel 16
17 Problems Page 72 #, 5 17
18 Section Angles Formed by Parallel Lines Alternate Interior Angles two non adjacent interior angles on opposite sides of a transversal Alternate Exterior Angles two exterior angles formed between two lines and a transversal, on opposite sides of the transversal 18
19 KEY POINTS 60 o 60 o 110 o 110 o 120 o 120 o 120 o 60 o 70 o 110 o 19
20 30 o 20
21 Example 1 pg. 76 Determine the measures of a, b, c, and d. Corresponding angles Vertically opposite Interior angles on same side of transversal are Supplementary alternate interior angles 21
22 Example 2 pg. 77 One side of a cellphone tower will be built as shown. Use the angle measures to prove that braces CG, BF, and AE are parallel. Corresponding angles are equal therefore CG ll AE Alternate interior angles are equal therefor CG II BF 22
23
24 Example 3 Solve for x. Then find angle measure 2x + 8 4x
25 Example 4 Solve for x. Then find angle measure 3x x
26 26
27 27
28 28
29 Practice Problems Page #'s 1 4, 20 and 15 29
30 Find the angle measure of each indicated angle 105 o 75 o 36 o HELP with yesterday's HOMEWORK QUESTIONS Page #'s 1 4, 20 and 15 30
31 Practice Problems Page #'s 1 4, 20 and 15 ANY HOMEWORK QUESTIONS You would like help with 31
32 32
33 15. 33
34 Section Angle Properties in Triangles The exterior angle of a polygon is the angle that is formed by a side of a polygon and the extension of an adjacent side. 34
35 Non adjacent Interior Angles the two angles of a triangle that do not have the same vertex as an exterior angle. In the diagram, angles A & B are non adjacent interior angles to ACD A Exterior Angle B C D 35
36 KEY IDEAS: Angle Properties of Triangles 36
37 Example 1 pg. 87 In the diagram, θ MTH is an exterior angle of ΔMAT. Determine the measures of the unknown angles in ΔMAT. M 40 o A 155 o T H 37
38 Example 2 pg. 88 R N 20 P 38
39 Example 3 Find the measure of each lettered angle. 39
40 TRY Find the measure of each lettered angle. 40
41 Example 4 Find the value of x. Then find the measure of each angle. 41
42 Example #5 Find the value of x and then find the measure of each angle. (x) O (2x) O (2x) O 42
43 Find the value of x and then find the measure of each angle. x o 75 o (4x 54) o 43
44 Problems Pages #'s 2, 3, 11, 14, 15a GOOD AFTERNOON HOME WORK CHECK: PLEASE HAVE READY FOR ME TO SEE ANY HOMEWORK QUESTIONS???? 44
45 45
46 Section Angle Properties in Polygons Convex Polygon a polygon in which each interior angle measures less than Convex non convex (concave) 46
47 Convex Polygons Sides Triangles Angle Sum Formula: The sum of the measures of the interior angles of a convex polygon is Polygon Number of Sides Number of Triangles Sum of Angle Measures triangle 3 quadrilateral 4 pentagon 5 hexagon 6 heptagon 7 octagon 8 47
48 48
49 Names of Different Polygons Number of Sides Name of Polygon 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 11 Hendecagon 12 Dodecagon 49
50 Regular Polygons Polygons in which each side is of equal length. The measure of the interior angles are equal. Example: For regular polygons, the following formula can be used to determine the measure of each individual angle: 50
51 Example 1 Outdoor furniture and structures like gazebos sometimes use a regular hexagon in their building plan. a) What is the sum of the interior angles in the hexagon? b) Determine the measure of each interior angle of a regular hexagon. 51
52 TRY Determine the sum of the interior angles and the measure of each interior angle of a regular 15 sided polygon (pentadecagon). 52
53 Example 2 The sum of the measures of the interior angles of an unknown polygon is What type of polygon is it? HEPTAGON 53
54 Try The sum of the measures of the interior angles of an unknown polygon is What type of polygon is it? NONAGON 54
55 Each Interior angle has an exterior angle that form a straight line making 180 degrees. Clockwise Counter clockwise 55
56 56
57 Example 3 Determine the measure of each exterior angle of a regular convex octagon. 57
58 58
59 Practice Problems Pgs #'s 1-3, 6, 10, 11, 17 ANY QUESTIONS 59
60 Interior Angle Exterior Angle 60
61 61
62 62
63 85 o 50 o 63
64 Know Terminology Completed review sheet is BONUS ON TEST Show all your work, have completed review sheet on your desk at beginning of tomorrows class to get BONUS BRING CALCULATOR, NO SHARING CALCULATORS CLASS AVERAGE >70% 64
65 65
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