Properties of Angles and Triangles
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1 1 Properties of Angles and Triangles Day 1: Exploring Parallel Lines NOTES Where are parallel lines found in day- to- day life? Types of lines! Parallel Lines: Intersecting Lines: Perpendicular Lines: Angles formed by intersecting lines: Perpendicular: Other:
2 2 EXERCISES 1. What are the measures of the following angles? Use the protractor to find out 2. Using only a protractor and a pencil, draw 3 different sets of parallel lines. How many different sets of parallel lines do you think exist? 3. Try to figure out if the lines A and B parallel in each diagram. Make and explain your educated guess? (Hint: try comparing the diagrams to each other)
3 3 Day 2: Angles Formed By Parallel Lines NOTES Who is Euclid? What did he do for mathematics? Transversal Lines: Angles, Angles, and more Angles! Alternate Interior Angles: Alternate Exterior Angles: Corresponding Angles: Interior Angles: Supplementary Angles: Complementary Angles: Vertically Opposite Angles: Right Angle: Angles around a point: Congruent angles: With these definitions we can inductively and deductively prove many things about angles formed between parallel lines and transversals. Note: if you don t have a parallel line, create one so you can use the definitions!
4 4 Example! Find all angles in the picture Statements Reasons EXERCISES 1. Find the labeled unknown angle! 2. Label and find all of the angles in the image.
5 5 3. You want to build a wire fence with 2 sets of parallel lines crossing each other (think chain link fence). You want one of the vertically opposite angles that form at the junctions to be a number of your choice between (pick a number and stick with it! write it down). Find all other angles you can for the fence (won t need to do it for the whole fence because there will be a pattern!) 4. Prove, showing all steps, that the angles given are in fact the same. Statements Reasons 5. Find all the angles corresponding to the lines this artwork.
6 6 Day 3 Angle Properties of Triangles NOTES Conjecture: Example What angles are equal in this image? What would the angle a be? This property is how Euclid proved this conjecture! let s check it out Review: Types of Triangles and their angles: Equilateral Triangle: Isosceles Triangle: Scalene Triangle: Right Triangle: Acute Triangle: Obtuse Triangle:
7 7 Examples! 1. If we have a right triangle with an angle of 34 what would the other angle be? 2. Find the unknown lettered angles! Note: We have to find the unknown angles in a specific order, as you ll see that finding a helps us to find b, which helps us to find c! 3. Say you are building a birdhouse, you don t have a ruler, but you do have a protractor! You have heard that birds really like obtuse scalene triangles, so you want to incorporate them. Instead of measuring all the angles you know that you can just measure a few and calculate the rest! Find all the angles for birdhouse. EXERCISES 1. Is it possible to have a triangle with 2 right angles? Why or why not? 2. Determine the unknown angles m & n
8 8 3. Prove that angle A is Prove that side r is parallel to side s 5. A manufacturer is designing a lawn chair, as shown. Determine the missing angles Day 4 Angle Properties of Polygons NOTES Polygon: (regular and irregular) Convex Polygons: Concave Polygons: Polygons can be divided into! good thing cause we know a lot about those already
9 9 The sides of polygons! Quadrilateral: Pentagon: Hexagon: Heptagon: Octagon: Nonagon: Decagon: Hendecagon: Can you see the pattern!! Conjecture: Reasoning: If we have a regular polygon, we can make a conjecture about what an individual angle inside it will be. Now wait, what about the exterior angles of polygons? Lets look at it Conjectures:
10 10 Examples! 1. What is the missing angle in this heptagon? 2. Can a convex polygon have summed interior angles of 250? Why or why not? EXERCISES 1. Determine the measure of each interior angle of a loonie 2. The sum of the measures of the interior angles of a polygon is How many sides does this polygon have? 3. What is the total measure of all interior angles of this regular hexagon? (try using the angle given to figure this one out)
11 11 4. Determine angle a. What type of polygons can you see? What type of triangle is the one in the middle? 5. Three exterior angles of a convex pentagon measure 70, 60, and 90. The other exterior angles are congruent. Figure out the measures of the interior angles of the pentagon. (hint: draw a picture!) 6. Determine the values of a, b, c and d.
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