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7 th grade Geometry Discipline: Orange Belt Training Order of Mastery: Constructions/Angles 1. Investigating triangles (7G2) 4. Drawing shapes with given conditions (7G2) 2. Complementary Angles (7G5) 3. Supplementary Angles (7G5) 5. Vertical/adjacent angles (7G5) 6. Solving for angles (7G5) Welcome to the Orange Belt Constructions/Angles Triangles, triangles, triangles oh my! What do you know about triangles?! Write everything you can possibly think of (including pictures!) that has to do with triangles the box below Things I know about triangles Hopefully you made a comment in the box above about how many sides a triangle has and possibly you made a comment about the number of angles. Speaking of angles, there are LOTS of different types of angles in the world including complementary, supplementary, vertical, and adjacent and you can use them to solve geometry problems every day. Get ready to do some serious drawing and problem solving with triangles and angles (and maybe even some quadrilaterals!) in this discipline. It s going to be lots of fun! Good Luck Grasshopper. Standards Covered: 7.G.2 - Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. 7.G.5 - Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. 2
1. Investigating triangles (7G2) A triangle is a closed figure with three sides and three angles. EQUILATERAL triangle ISOSCELES triangle SCALENE triangle All three sides the same length Two sides with the same length All three sides different lengths Acute Angled triangle Right Angled triangle Obtuse Angled triangle Has three acute angles A right triangle has one 90 Has one obstuse angle Above there are six different classifications of triangles. They are called classifications because sometimes a triangle can have two of the classifications from above. For each pair of classifications below, try to draw a triangle that could be made with them. If so, draw it. If not, write Not possible and explain why you think that. (your work for this will most likely need to be done on a separate paper) PAIR OF CLASSIFICATIONS Possible/not possible Drawing/explanation Acute equilateral triangle Isosceles right triangle Obstuse scalene triangle Equilateral right triangle Scalene right triangle Scalene acute triangle Is it possible for a triangle to have three classifications from above? Why or why not? 3
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The interior angles of triangles (that is, the three angles inside a triangle) are very unique. In fact they always add up to the same thing 180. Check it out: Now try cutting this triangle out and cutting it on the dotted lines. Put the interior angles (the ones with stars) together and it should form a straight line! 5
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Stained glass commissioning: You have been asked to design a stained glass piece featuring the triangle(similar to the one to the right). The requirements are as follows: *At least 2 isosceles right triangles *At least 2 scalene and acute triangles *At least 1 equilateral triangle *Any 2 other triangles of your choosing Sketch out your stained glass triangle design below and label it just like the image above. (you may need to use graph paper) color it if you have time! Ratios and Triangles: The angles of a triangle are in the ratio of 1:2:3. Find the measure of the smallest angle of the triangle. The Stumper: What would be the largest possible angle that a triangle could have? What would be the smallest? Could these two angles be in the same triangle and what would that triangle look like? 7
2. Drawing shapes with given conditions When given measurements for a shape, sometimes there could be more than one shape drawn from those measurements. Or sometimes you could be given measurements and NOT be able to draw the given shape. Check out the triangle inequality theorem below that describes this Triangle Inequality Theorem Any side of a triangle is less than the sum of the other two sides. Here is an example of a failed triangle because a+b is smaller than c. 1. Which of the following could represent the lengths of the sides of a triangle? Justify your answer using the triangle inequality theorem. a. 1, 2, 3 b. 6, 8, 15 c. 5, 7, 9 2. Using a ruler, draw two triangles (showing all side lengths) and show that they both satisfy the requirements of the triangle inequality theorem. 3. Given the measurements of 5 inches, 4 inches, and 7 inches, can you draw a triangle? 8
4. Using the graph paper to the right, draw a triangle that has one side that is 7 units long and one triangle that is 9 units long. Is there more than one triangle you could draw? How many can you draw? What do you think the length of the hypotenuse is? (you may need to estimate) 4. Draw a shape that has 3 total sides and one pair of congruent sides (sides with the same length) and one right angle. What would you call this shape? 5. (algebra connection) A triangle has angles of X, X+30, and X+60 degrees. Write an equation to determine the value of X. 9
A quick look at quadrilaterals For each quadrilateral in the diagram below, write two observations of each (what is unique about that quadrilateral). Based on your work above, write a definition for quadrilateral in this box. 10
6. Use a ruler to draw one (or more?) quadrilaterals with two sides of 4 inches and two sides of 5 inches. What are the different types of quadrilaterals you could draw? 7. I am a quadrilateral with two right angles, one set of parallel lines and two sets of perpendicular lines. What is the name of me? 8. I am a quadrilateral with two sets of congruent sides (Sides with the same length) and two sets of parallel sides. What is the name of me? 11
9. I have 4 sides. Exactly one of my angles is a right angle. Exactly two of my sides are parallel. My non-parallel sides have lengths of 2cm and 4 cm. Can you draw me? 10. I am a shape with 4 congruent sides and 4 congruent angles? What is the name of me? 11. Partner game: Guess the Shape! Step 1: Draw a figure below (could be any type of triangle or quadrilateral) Step 2: Make a list of clues that would enable a partner to draw your figure. Extra challenge what is the smallest number of clues that works?! 12
3. Complementary/Supplementary Angles (7G5) Complementary Angles If two angles add to 90, we say they complement each other. Complementary comes from Latin completum meaning "completed" because the right angle is thought of as being a complete angle. Supplementary Angles If two angles add up to 180 we say they are supplementary. Notice that they make a straight line. Spelling: be careful, it is not "Complimentary Angle with an I... that would be an angle that says something nice about you! Notice that the angles don t have to be right next to each other to be supplementary. This is also true for complementary angles. In the space below, use a protractor to draw at least one set of complementary angles and one set of supplementary angles (with all angles labeled!) 13
Use your knowledge of complementary and supplementary angles to solve for the missing angle(s). 14
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17. Create one of your own solve for the angle problems modeled after the problems on the previous page. Attempt to make this problem very challenging and then give it to your favorite friend to solve. 18. If the ratio of two complementary angles is 2:1, what is the measure of the larger angle? 19. If the ratio of two complementary angles is 7:2, what is the measure of the smaller angle? 16
4. Vertical/adjacent angles (7G5) Vertical Angles Two angles are vertical if they share a vertex and are opposite each other where two lines intersect. Adjacent Angles Two angles are adjacent if they have a common side and a common vertex and don t overlap. Interestingly enough, vertical angles are EQUAL! This will come in very handy! In this figure, angle ABC is adjacent to angle CBD. Which of these pictures depicts adjacent angles. Explain why or why not for each: 17
1. Solve for a, b, and c in the image here: Which angles are adjacent and which are vertical (There are multiple answer for this so be careful!) 18
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11. Solve for x in this figure: 12. Measure and label all angles in this picture: 20
13. Using a protractor, draw a figure with at least: 2 complementary angles 2 supplementary angles 5 vertical angles 5 adjacent angles Measure and label each angle and have a friend check over you figure for everything above before completing your packet! 21