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Chapter 11 Notes – Congruence, Similarity, and Transformations – PART 1 – Angles and Polygons Lesson 11-1: Angle and Line Relationships Line and Angle Relationships Names of Special Angles Interior angles lie inside the parallel lines. ∠3, ∠4, ∠5, ∠6 Exterior angles lie outside the parallel lines. ∠1, ∠2, ∠7, ∠8 Alternate interior angles are on opposite sides of the transversal and inside the parallel lines. Alternate exterior angles are on opposite sides of the transversal and outside the parallel lines. Corresponding angles are in the same position on the parallel lines in relation to the transversal. ∠3 and ∠5, ∠4 and ∠6 ∠1 and ∠7, ∠2 and ∠8 ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8 Try It Jun is cutting the tile below as shown. a) Classify the relationship between ∠ a and ∠ b. ∠ 13. b) If m∠ a = 53o, what is the measure of ∠ b? b a **See Example 2 on Page 496 a) Classify the relationship between ∠ 9 and b) If m∠ 13 is 75o, find m∠ 11 and m∠ 15. Angles DEF and WXY are complementary angles, with m∠ DEF = 2xo and m∠ WXY = (3x – 20)o. Find the value of x, and then find m∠ WXY. Lesson 11-2: Triangles **SUM of Interior Angles of ANY Triangle is ___________ Try It Find m∠ F in △DEF The measures of the angles of △XYZ are in the ratio of 2:3:5. What are the measures of D E the angles? F Lesson 11-2: Triangles (Continued) ** Exterior Angles: are made by _________________ one side of the triangle. The exterior angle is _____________________________ to its adjacent interior angle. **To Find Exterior Angles: __________ the two _________________ angles it isn’t touching (REMOTE) Try It K L 5 2 In the figure at left, m∠ 4 = 135o. Find m∠ 2. 1 4 M **Triangles can be classified using their _____________________ and _____________________. By Angles By Sides Acute Scalene – Obtuse - Isosceles – Right - Equilateral – Try It Classify each triangle by its angles and its sides. 106o 45o 45o 38o 36o Lesson 11-3: Polygons **POLYGON - Simple, ____________________ figure 3 or more line ____________________ CANNOT have a ________________________ or ________________________ sides Try It Determine whether the figure is a polygon. If it is, classify the polygon. If it is not a polygon, explain why. **Sum of Interior Angles of a Polygon: _________________, where n is _________________________ - Found by counting the triangles formed by the diagonals from one vertex Example: **To Find Degrees of ONE Interior Angle: Find the Total Degrees and ____________ by sides **REGULAR polygons: All ______________ are congruent and all _________________ are congruent Try It Find the sum of the measures of the interior angles of a heptagon. A stop sign is a regular octagon. What is the measure of one interior angle in a stop sign? **Tessellation: Repetitive pattern of polygons that fit together without ________ or _____________ - Sum of Angles where vertices (corners) meet in tessellation = ______________ Try It: Determine if a tessellation can be made using only regular decagons. Explain. Chapter 11 Notes – Congruence, Similarity, and Transformations – PART 2 – Transformations Lesson 11-4: Translations and Reflections on the Coordinate Plane Original figure is called the _____________________ and the new figure is called the _________ Images will be label with the _________________ symbol (point A becomes A’) **TRANSLATION – is a _______________________ - Image is same ________________ and same ________________ as pre-image (CONGRUENT) - Images face the same ____________________ (ORIENTATION) - To find Coordinates of a Translated Image: 1) Look at the x-axis change: - If it is Left - _________________ the change from the original x-value - If it is Right - _______________ the change to the original x-value 2) Look at the y-axis change: - If it is Down - _______________ the change from the original y-value - If it is Up - ______________ the change from the original y-value Try It The vertices of triangle ABC are A(-3, 7), B(-1, 0), and C(5, 5). Suppose it is translated 4 units right and 5 units down. Find the coordinates of the vertices of its image, and then graph the image and the original on the axes. **REFLECTION – is a __________________ or _______________________ - Image is same ________________ and same ________________ as pre-image (CONGRUENT) - Images face the opposite ____________________ (OPPOSITE ORIENTATION) - To find Coordinates of a Reflected Image: ** Find the Line of Reflection (axis that you reflect over): - If it is the X-AXIS: x-coordinate is the __________________, y-coordinate is the ______________ - If it is the Y-AXIS: x-coordinate is the __________________, y-coordinate is the ______________ Try It The vertices of figure MNOP are M(-8, 6), N(5, 9), O(2, 1), and P(-10, 3). Graph the figure and the image of the figure after a reflection over the y-axis. List the new coordinates of the image. Lesson 11-5: Rotations on the Coordinate Plane **ROTATION – is a _______________________ - Image is same ________________ and same ________________ as pre-image (CONGRUENT) - Figures move around a FIXED point called the ______________________ - Can be a __________________ of the figure (then that point doesn’t move) - Can be the ______________________ (0,0) - Figures can Rotate _____________________________ or __________________________________ **Rotational Symmetry: If the figure can be rotated less than ___________ around its center and it looks like the __________________________ figure **To figure out degree : Divide ____________ by number of times the pattern _____________ Try It Draw the figure shown after a 90o clockwise rotation about point A. A Triangle EFG has vertices E(2, 1), F(1, -1), and G(4, -1). Graph the figure and its image after a clockwise rotation of 90o about vertex F. Then give the coordinates of the vertices for triangle E’F’G’. Parallelogram ABCD has vertices A(-3, -1), B(1, -2), C(-1, -4), and D(-5, -3). Graph the parallelogram and its image after a rotation of 180o about the origin. Give the coordinates of the vertices for parallelogram A’B’C’D’. Determine whether or not the star shown has rotational symmetry. If it does, what is the angle of rotation? Lesson 11-6: Congruence and Transformations **Two figures are congruent if you can make the _______________ after a series of transformations (Translations, Reflections, Rotations) of the ______________ **Helpful hints: USE _________________________ Translation – Orientation is the ___________________ Reflection – Orientation is _____________________ Rotation – Orientation is _______________________ Try It Determine if the two figures are congruent by using transformations. Explain. Chapter 11 Notes – Congruence, Similarity, and Transformations – PART 3 – Similarity Lesson 11-7: Dilations on the Coordinate Plane **DILATION - ____________________ or ____________________ a figure by a __________________________ - Think about __________________ and _______________________ figures - Similar figures are the same ____________________ but different ___________________ - To find the Coordinates after a Dilation 1) Multiply each x-coordinate by the ______________________________ 2) Multiply each y-coordinate by the ______________________________ - To find the Scale Factor 1) Find the lengths of 2 ____________________________ sides - Subtract the ______________________ 2) Make a __________________ comparing the length of the _________________ to the length of the __________________ 3) Simplify the ratio Try It A figure has vertices D(-4, 4), E(4, 8), F(8, 8), and G(4,0). Graph the figure and the image of the polygon after a dilation with a scale factor of ¼ . A triangle has vertices H(-1, -2), I(1,4), and J(2,1). Find the coordinates of the triangle after a dilation with a scale factor of 4. Peri is enlarging a page from a book on the copy machine. The original page measured 14 centimeters by 21 centimeters. If the enlarged image measures 17.5 centimeters by 26.25 centimeters, what is the scale factor of the dilation? Lesson 11-8: Similarity and Transformations **SIMILAR FIGURES – Same __________________ but different __________________ - Two figures are similar if the _____________________ can be obtained by a sequence of transformations and dilations from the ___________________ - Compare _________________________ to decide if figures are similar - Multiply Lengths of the original figure By _____________________________ to find the lengths of the image ** SCALE FACTOR – If it is less than one: Dilation is a ___________________________ - If it is greater than one: Dilation is an ________________________________ Try It Determine if the two triangles are Determine if the two rectangles are similar by using transformations. similar by using transformations. Explain your reasoning. Explain your reasoning. H K H I J K J E F G L M O N Triangle RST has sides that are 20 mm, 10 mm, and 25 mm in length. It is related to △FGH by a scale factor of 0.8. What are the lengths of the sides of △FGH? A bakery transfers a photo image onto a sheet cake. The original photo, which is 8 inches by 10 inches, is reduced by a scale factor of 0.8. The baker decides the image is still too large, and reduces the image by a scale factor of 0.9. What are the dimensions of the final image that will appear on the cake? Is the reduced image similar to the original?