Unit 2 Triangles. Unit 2.1 Notes: Basic Triangle Properties

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1 Unit 2 Triangles Unit 2.1 Notes: Basic Triangle Properties Theorem: The sum of the angles in a triangle is ALWAYS mm AA + mm BB + mm CC = Find the measure of each missing angle. a) b) 2. Find mm AA 3. Triangle Inequality Conjecture: The SUM of the lengths of any two sides of a triangle is 4. State whether the given three numbers could be the measures of the sides of a triangle a) 4, 10, 12 b) 12, 22, 8 c) 10, 6, 17 d) 11, 7, 19 e) 9, 12, 3 f) 4, 9, 6

2 5. Side-Angle inequality Conjecture: *The SHORTEST side of a triangle is always opposite the: *The LONGEST side of a triangle is always opposite the: 6. Order the sides of each triangle from shortest to longest: a) b) c) d) 7. Order the angles in each triangle from smallest to largest.

3 8. Theorem: Base angles of isosceles triangles are congruent Equilateral Triangle 9. Use the properties of Equilateral and Isosceles Triangles to find the value of xx in each triangle below: d) e) mm 2 = xx + 69 f)

4 Unit 2.2 Notes: Parts of a Triangle 1. Median: 2. Altitude: 3. Angle Bisector: 4. Perpendicular Bisector:

5 5. Each triangle below shows a triangle with one or more of its medians: d) e) 6. Each triangle below shows a triangle with one or more of its angle bisectors: d) e)

6 7. Midsegment: H G I J F 8. In each triangle M, N, and P are the midpoints of the sides. Name a segment parallel to the one given. a) b) 9. Find the missing length indicated: d) e) f)

7 Unit 2.3 Notes: Similarity and Dilations Solving Proportions: CROSS MULTIPLY! 1. Solve each proportion: a) 8 3 = xx 8 b) 2 5 = nn 6 c) 6 5 = xx 7 d) 8 4 = 6 3xx e) xx + 1 = f) xx 3 = 2xx 2 5 Similar Polygons: Scale Factor: Enlargement vs. Reduction

8 2. The polygons in each pair are similar. Find the scale factor of the smaller polygon to the larger. **You can use proportions to find the missing lengths of similar polygon** 3. Find the missing length. The triangles in each pair are similar. (HINT: redraw all sets of triangles that overlap!) a) DEF DKL b) MLK MFG c) VUP RQP c) FGJ LKJ d) MLN SLT e) BCD BML

9 A dilation is a transformation in which the image (the final polygon-the one you create) and the preimage (the initial polygon) are similar to each other. The Center of Dilation is a point through which all dilation takes place. All the points are either stretched or compressed through this point. A dilation with center C and scale factor k is a transformation that maps every point P in the plane to a point P' so the following properties are true: a) Shape, orientation, and angles are preserved. (meaning they don't change) b) All sides change by a single scale factor, k. (The scale factor k is a positive number such that k=cp'/cp and k is not 1.) c) The corresponding preimage and image sides are d) The corresponding points of the figure are collinear with the center of dilation. (If P is not the center point P, then the image point P' lies on CP.) 4. Use the origin as the center of dilation. Plot the preimage using the points given, then use the scale factor to find the coordinates of the vertices of the image, and plot them. a) MM(0, 4), NN(3, 4), LL(3, 0); kk = 2 bb) G(2, 8), H(6, 6), I(4, 2), k(-2, 2); k= 1 2

10 Unit 2.4 Notes: Similar Triangle Proofs Similar triangles are two triangles that three congruent ANGLES and three proportional SIDES. There are THREE ways to show that two triangles are similar: SSS Similarity Theorem: Q S P T R U 1. Example: Determine if the two triangles are similar using the SSS Similarity Theorem. SAS Similarity Theorem: Q S P R T U 2. Example: Determine if the two triangles are similar using the SAS Similarity Theorem.

11 AA Similarity Theorem: Q S P R T U 3. Example: Determine if the two triangles are similar using the AA Similarity Theorem. 4. Example: Determine if the triangles are similar. If so, state how you know they are similar and write a similarity statement.

12 5. Example: Prove the triangles are similar. Statements Reasons Statements Reasons 6. Example: Find xx.

13 Unit 2.5 Notes: Right Triangle Similarity An altitude of any right triangle splits the triangle into similar triangles. We end up with a small right triangle, a medium right triangle, and the original right triangle. To make it easier to see the corresponding angles and sides if we pull the three triangles apart and draw them side by side. 1. Example: Find the missing length indicated. a) b) c) d) e) f) Parallel Line Similarity: A line parallel to one side of a triangle splits the other two sides of the triangle into

14 2. Example: Find the missing length indicated. 3. Example: Find the value of xx. Angle Bisector Similarity If one angle of a triangle is bisected, ( ), then the angle bisector of the triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle. 3. Example: Find the value of xx.