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4-1 Congruence and Transformations Name: ______________________________ Period:______________ Objective: Draw, identify, and describe transformations in the coordinate plane. Use properties of rigid motions to determine whether figures are congruent and to prove figures are congruent. Identify each transformation. Write the coordinates of the pre-image and image. Consider the triangle whose vertices are A(-3, 2), B(-1, -4), and C(3, 0). Use a piece of graph paper to perform the following transformations. Make sure to draw an x and y-axis below. 1. Plot the original triangle in pencil. Use the color blue to perform a translation 5 units to the left and 2 units up. 2. Go back to the original triangle in pencil. Use the color red to draw its reflection about the y-axis. 3. Use the color black to dilate the original triangle by a scale factor of 2. 4. Use the color ____________________ to rotate the image about the origin 180. An isometry is a transformation that preserves length, angle measure, and area. A rigid transformation is another name for an isometry. Which of the transformations above represent an isometry? Determine whether the polygons with the given vertices are congruent. A. A(-3, 1), B(2, 3), C(1, 1) P(-4, -2), Q(1, 0), R(0, -2) Homework for section 4-1. 7. The polygon Q(3, 2), R(6, 5), S(6, 2) undergoes the transformation: (x, y) → (4x, 4y). Name the coordinates of the image points. 8. Apply the transformation M to the polygon with the given vertices. Name the coordinates of the image points. Identify and describe the transformation. M(x, y) (x A(3, 10), B y C 4.2 Classifying Triangles Objective: Classify triangles by their angle measures and side lengths. Use triangle classification to find angle measures and side lengths. Consider the triangle whose vertices are J(4, 8), K(10, 14), and L(16, 8). Find the length of each side of this triangle using the Distance Formula and classify the triangle by the lengths of its sides. Determine if the following triangles classifications are possible. An answer of “yes” is to be accompanied by a drawing of that triangle. a. Right, Isosceles b. Scalene, Obtuse c. Equilateral, Right Homework: Textbook pages 227-228 problems 1-11, 18-19, 35-37 4-3 Angle Relationships in Triangles Objective: Find the measures of interior and exterior angles of a triangle. The measure of one of the acute angles in a right triangle is given. Find the measure of the other acute angle. 3. 44.9 _____________ 4. (90 z) _____________ 5. 0.3 _____________ 6. The measures of the angles of a triangle are in the ration 1: 4: 7. What are the measures of the angles? (Hint: Let x, 4x, and 7x represent the angle measures. 12. Solve for x and then find the measure of . Let’s Prove the Triangle Sum Theorem!!! Yay! (I’m sure you’re thinking let’s not in your head.) Given: Prove: , line l is parallel to through B. Statements Reasons 1. 1. Given 2. 2. Given 3. 3. 4. 4. 5. 5. 6. 6. Prove the Exterior Angle Theorem Statement 1 ∆PQR is a triangle Reason Given 2 Triangle Sum Theorem 3 Definition of linear pair. 4 If two angles form a linear pair, they are supplementary. 5 Definition of supplementary angles. 6 Statements 2, 5 and Substitution Property. 7 Subtraction Property. Homework: Textbook page 236 problems 16-22, 24. For problem #24 write out the whole proof not just the blanks. 4-4 Congruent Triangles Objective: Use properties of congruent triangles. Prove triangles congruent by using the definition of congruence.