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Transcript
Part 1: Transformations/Rigid Motions Rigid Motions and transformations -Rigid Motions produce congruent figures -Translation, Rotation, Reflections are all rigid motions -Rigid Motions preserve size, shape and angle measure, they only change the position of a figure Translations Ta,b a →how to move your pre-image left/right b →how to move your pre-image up/down Vectors are drawn from pre-image to image and show distance and direction of the slide Reflections Notation: rline 1. Graph line of reflection 2. Count how far away each point is on the line and count the opposite going the other way Special reflections Point Reflections YOU MUST MEMORIZE Special compositions -Perform Compositions from right to left! -Glide Reflections: SHOW MATHEMATICALLY the slopes are the same for the vector and line of reflection. This justifies that they are parallel. - Composition of reflections over parallel lines are the same as one translation Rotations and more special compositions Notation: R Either know your rules, or Rotate paper! Rotate counterclockwise for positive angles, and clockwise for negative angles! - Composition of reflections over perpendicular lines are the same as one rotation Additional Notes Rotational Symmetry Order 360 𝑛 Regents-Style Questions 1. In the diagram below, congruent figures 1, 2, and 3 are drawn. Which sequence of transformations maps figure 1 onto figure 2 and then figure 2 onto figure 3? 1) a reflection followed by a translation 2) a rotation followed by a translation 3) a translation followed by a reflection 4) a translation followed by a rotation 2. In the diagram below, and are graphed. Use the properties of rigid motions to explain why . More Transformations Practice Problems Continued in the back Part 2: Triangle Theorems/Properties Name of Theorem or In words/ Symbols relationship 1. Side angle relationship The longest side is across from the largest angle. The medium length side is across from the medium-sized angle. The shortest side is across from the smallest angle 2. Triangle inequality Theorem The sum of the lengths of the two smaller sides of a triangle is greater than the length of the largest side. 3. Pythagorean Theorem a) c2 = a2 + b2 a) To find a missing side 2 b) to Classify triangles 4.Isosceles triangle Theorem 2 b) If c < a + b C is longest side ( hypotenuse) 2 If c2 > a2 + b2 it is acute it is obtuse If c2 = a2 + b2 it is right The base angles of an isosceles triangle are equal in measure. The sides opposite the base angles in an isosceles triangle (called legs) are equal in length. Diagrams/ Hints/ Techniques Draw arrows! Add up the two smaller sides and compare to the largest side. If the sum is greater, it’s a triangle! 2,3,4 (2+3) > 4 ? yes! If you see a right angle, it’s a right triangle! use P.T to solve for a missing side. WATCH OUT! If asked “does this make a triangle” you must use Theorem # 2- NOT PYTHAGOREAN. Angles opposite are congruent! If you see expressions, make them equal to each other! 5. Exterior angle theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles of the triangle. IN + IN =OUT 6. Exterior angles in a polygon. a). ONE exterior a) 360/n Exterior angles and formed by extending a side of the triangle. b) Sum is AlWAYS 360 degrees b) Sum of Exterior 7. Interior angles in a polygon. a) The supplement of on Exterior angle-they are linear pairs! Exterior + Interior = 180 Remember! One exterior and one interior angle add up to 180 degrees! a) ONE interior b) Sum of Interior. 8. Segments in a triangle: 9. Points of concurrence. 10.Centroid and Ratios b) Number of △ ′𝑠 times 180. ( n-2)180 Medians- Goes to the midpoint of the opposite side creating two equal segments Altitudes-Are perpendicular to the opposite side creating right angles Perpendicular bisectors- Goes to the midpoint of opposite side and is perpendicular to it. Angle bisectors- bisects the angle at the vertex it goes through making 2 congruent angles. ** In Isosceles and Equilateral triangles these segments coincide! 2 or more medians Centroid : Always inside the triangle. Cuts each median into a 2:1 ratio 2 or more Altitudes Orthocenter: Inside for acute triangles, on the triangle for right triangles and outside for obtuse triangles. 2 or more angle bisectors Incenter: Always inside the triangle. 2 or more perpendicular bisectors Circumcenter: Inside for acute triangles, on the triangle for right triangles and outside for obtuse triangles. Centroid cuts every median into a 2:1 ratio. Use this ratio to set up equation 2X + 1x= whole length of median. ALL OF MY CHILDREN ARE BRING IN PEANUT BUTTER COOKIES. Read carefully- What is the segment they want? Sometimes you need to substitute back in! Regents-Style Questions *Note: The Regents will occasionally address triangle questions by combining them with other core units of study. The following are a few problems that exemplify this. We will cover these additional topics in future Regents Review Sessions. 1. The coordinates of the vertices of 1) right 2) acute 3) obtuse 4) equiangular are , , and . Which type of triangle is 2. The following is a 4-point question. This student only earned two points. Reason Why Student Only Earned 2/4 Points: More Triangle Practice Problems Continued in the back ? Part 3: Congruency Concept Key Ideas/Tips Beginning a Proof *mark your picture *annotate question (givens) *use the tools from your tool box *1st write your givens *last step = what you're trying to prove *make a plan *use a checklist for your shortcuts! *# your steps! *All of your givens and markings should be a step in your proof! Triangle Congruency Shortcuts *WE CANNOT USE AAA or SSA!!! * Proving Parts *To prove triangles are congruent, you need to use a shortcut. are Congruent *To prove parts (sides and angles) of triangles are congruent, we: 1st: prove triangles are congruent 2nd: use CPCTC (Corresponding Parts of Congruent Triangles are Congruent) Proving Definitions Addition Postulate With IF AB = EF CD = CD Then, AB + CD = EF + CD AD = ED Substitution (Reflexive Property) (Addition Postulate) (Substitution Postulate) = Subtraction Postulate With Substitution Transitive Property IF AC = BD, BC = BC Then, AC- BC = BD - BC AB = CD (Reflexive Property) (Subtraction Postulate) (Substitution Postulate) ̅̅̅̅̅ ; ̅̅̅̅ AB = 5; ̅̅̅̅ 𝐴𝐵 ≅ 𝐷𝐸 𝐷𝐸 ≅ ̅̅̅̅ 𝐺𝐻 So, AB = GH (by the TRANSITIVE PROPERTY) <1 ≅ < 4 Using Supplements So, <2 ≅ <3 (since 2 is the supplement of <1 and <3 is the supplement of < 4) Indirect Proof *Use when proving something is not true or ≇ or to show something is not true. Steps: 1. Assume the opposite of the prove statement 2. Prove normally 3. Contradict something in the given. ***Proof Pieces are Available on the website. You may use them to help remember key details. Access at any time Example Regents Problem: Given: Quadrilateral ABCD is a parallelogram with diagonals and intersecting at E Prove: Describe a single rigid motion that maps onto . More Proof/Congruency Practice Problems Continued in the back Review Session 1 Practice Problems 1. The vertices of have coordinates , , and ). Under which transformation is the image not congruent to ? 1) a translation of two units to the right and two units down 2) a counterclockwise rotation of 180 degrees around the origin 3) a reflection over the x-axis 4) a dilation with a scale factor of 2 and centered at the origin 2. Triangle ABC and triangle DEF are graphed on the set of axes below. Which sequence of transformations maps triangle ABC onto triangle DEF? 1) a reflection over the x-axis followed by a reflection over the y-axis 2) a 180° rotation about the origin followed by a reflection over the line 3) a 90° clockwise rotation about the origin followed by a reflection over the y-axis 4) a translation 8 units to the right and 1 unit up followed by a 90° counterclockwise rotation about the origin 3. Quadrilateral ABCD is graphed on the set of axes below. When ABCD is rotated 90° in a counterclockwise direction about the origin, its image is quadrilateral A'B'C'D'. Is distance preserved under this rotation, and which coordinates are correct for the given vertex? 1) no and 2) no and 3) yes and 4) yes and 4. If is the image of , under which transformation will the triangles not be congruent? 1) reflection over the x-axis 2) translation to the left 5 and down 4 3) dilation centered at the origin with scale factor 2 4) rotation of 270° counterclockwise about the origin 5. A sequence of transformations maps rectangle ABCD onto rectangle A"B"C"D", as shown in the diagram below. Which sequence of transformations maps ABCD onto A'B'C'D' and then maps A'B'C'D' onto A"B"C"D"? 1) a reflection followed by a rotation 2) a reflection followed by a translation 3) a translation followed by a rotation 4) a translation followed by a reflection 6. In the diagram of parallelogram FRED shown below, . If 1) 2) 3) 4) , what is is extended to A, and is drawn such that ? 124° 112° 68° 56° 7. In the diagram below, which single transformation was used to map triangle A onto triangle B? 1) 2) 3) 4) line reflection rotation dilation translation 8. Which statement is sufficient evidence that 1) and 2) , , 3) There is a sequence of rigid motions that maps onto , onto , and onto . 4) There is a sequence of rigid motions that maps point A onto point D, onto , and onto . is congruent to ? 9. Triangle ABC is graphed on the set of axes below. Graph and label reflection over the line . 10. As graphed on the set of axes below, Is congruent to is the image of , the image of after a sequence of transformations. ? Use the properties of rigid motion to explain your answer. 11. In parallelogram ABCD shown below, diagonals Prove: Statements 1. In parallelogram ABCD shown below, diagonals and intersect at E. and intersect at E. Reasons 1. Given after a Guided Practice Right Triangles: 12. To find the sides of a right triangle use the Pythagorean Theorem. a2 + b2 = c2 {Remember that the a and b make up right angle} 13) Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles of the triangle. 14) Isosceles Triangles: The base angles of an isosceles triangle are equal in measure. The sides opposite the base angles in an isosceles triangle (called legs) are equal in length. 15) Centroid: 2:1 Ratio for segments ̅̅̅̅̅ and ̅̅̅̅ In triangle ABC, ̅̅̅̅̅ 𝐴𝐷, 𝐶𝐹, 𝐵𝐸 are medians. If ̅̅̅̅ 𝐶𝐹 = 33, find CG and FG. 16) Side-Angle Relationship The longest side is across from the largest angle. The medium length side is across from the medium-sized angle. The shortest side is across from the smallest angle In triangle DOG, m<D = 40, m<O 60, and m<G = 80. State the longest side of the triangle __________________ State the shortest side of the triangle _________________ 17. In the accompanying diagram, bisects and . Prove: 𝐻𝐼 ≅ 𝐾𝐿 Statements 1. bisects and . Reasons 1. Given 18) Given: Prove: 19) Graph and state the coordinates of ∆𝐴′𝐵′𝐶′, the image of ∆𝐴𝐵𝐶 with A(1,2), B(3,0) and C(6,8)after the composition 𝑇2,0o 𝑅180° . 20. Graph triangle ABC. A(1, 1), B(4, 5), C(3, 2) and reflect it through point (-2, 1). Label your image. Use rigid motions to explain why the pre-image and image are congruent. 21. In the diagram of trapezoid ABCD below, diagonals and Which statement is true based on the given information? 1) 3) 2) 4) intersect at E and .