# Unit 1: Similarity, Congruence, and Proofs

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1 Unit 1: Similarity, Congruence, and Proofs This unit introduces the concepts of similarity and congruence. The definition of similarity is explored through dilation transformations. The concept of scale factor with respect to dilations allows figures to be enlarged or reduced. Rigid motions lead to the definition of congruence. Once congruence is established, various congruence criteria (e.g., ASA, SSS, and SAS) can be explored. Once similarity is established, various similarity criteria (e.g., AA) can be explored. These criteria, along with other postulates and definitions, provide a framework to be able to prove various geometric proofs. In this unit, various geometric figures are constructed. These topics allow students a deeper understanding of formal reasoning, which will be beneficial throughout the remainder of Analytic Geometry. KEY STANDARDS Understand similarity in terms of similarity transformations MCC9-12.G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. MCC9-12.G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. MCC9-12.G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems involving similarity MCC9-12.G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. MCC9-12.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 20

2 Understand congruence in terms of rigid motions MCC9-12.G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. MCC9-12.G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. MCC9-12.G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Prove geometric theorems MCC9-12.G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. MCC9-12.G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. MCC9-12.G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Make geometric constructions MCC9-12.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. MCC9-12.G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. 21

3 UNDERSTAND SIMILARITY IN TERMS OF SIMILARITY TRANSFORMATIONS KEY IDEAS 1. A dilation is a transformation that makes a figure larger or smaller than the original figure based on a ratio given by a scale factor. When the scale factor is greater than 1, the figure is made larger. When the scale factor is between 0 and 1, the figure is made smaller. When the scale factor is 1, the figure does not change. When the center of dilation is the origin, you can multiply each coordinate of the original figure, or pre-image, by the scale factor to find the coordinates of the dilated figure, or image. Example: The diagram below shows ABC dilated about the origin with a scale factor of 2 to create ABC ' ' '. 22

4 2. When the center of dilation is not the origin, you can use a rule that is derived from shifting the center of dilation, multiplying the shifted coordinates by the scale factor, and then shifting the center of dilation back to its original location. For a point (x, y) and a center of dilation (x c, y c ), the rule for finding the coordinates of the dilated point with a scale factor of k is (k(x x c ) + x c, k(y y c ) + y c ). When a figure is transformed under a dilation, the corresponding angles of the pre-image and the image have equal measures. For ABC and ABC ' ' ' below, A A', B B', and C C'. When a figure is transformed under a dilation, the corresponding sides of the pre-image and the image are proportional. For ABC and ABC ' ' ' below, AB BC AC. AB ' ' BC ' ' AC ' ' So, when a figure is under a dilation transformation, the pre-image and the image are similar. For ABC and ABC ' ' ' below, ABC A' B ' C '. 23

5 3. When a figure is dilated, a segment of the pre-image that does not pass through the center of dilation is parallel to its image. In the figure below, AC A' C ' since neither segment passes through the center of dilation. The same is true about AB and A' B ' as well as BC and B ' C '. When the segment of a figure does pass through the center of dilation, the segment of the pre-image and image are on the same line. In the figure below, the center of dilation is on AC, so AC and AC ' ' are on the same line. 24

6 PROVE THEOREMS INVOLVING SIMILARITY KEY IDEAS 1. When proving that two triangles are similar, it is sufficient to show that two pairs of corresponding angles of the triangles are congruent. This is called Angle-Angle (AA) Similarity. Example: The triangles below are similar by AA Similarity because each triangle has a 60 angle and a 90 angle. The similarity statement is written as ABC DEF, and the order in which the vertices are written indicates which angles/sides correspond to each other When a triangle is dilated, the pre-image and the image are similar triangles. There are three cases of triangles being dilated: The image is congruent to the pre-image (scale factor of 1). The image is smaller than the pre-image (scale factor between 0 and 1). The image is larger than the pre-image (scale factor greater than 1). When two triangles are similar, all corresponding pairs of angles are congruent. When two triangles are similar, all corresponding pairs of sides are proportional. When two triangles are congruent, the triangles are also similar. A two-column proof is a series of statements and reasons often displayed in a chart that works from given information to the statement that needs to be proven. Reasons can be given information, based on definitions, or based on postulates or theorems. A paragraph proof also uses a series of statements and reasons that works from given information to the statement that needs to be proven, but the information is presented as running text in paragraph form. 31

7 UNDERSTAND CONGRUENCE IN TERMS OF RIGID MOTIONS KEY IDEAS A rigid motion is a transformation of points in space consisting of a sequence of one or more translations, reflections, and/or rotations (in any order). This transformation leaves the size and shape of the original figure unchanged. Two plane or solid figures are congruent if one can be obtained from the other by rigid motion (a sequence of rotations, reflections, and translations). Congruent figures have the same side lengths and same angle measures as each other. Two triangles are congruent if and only if their corresponding sides and corresponding angles are congruent. This is sometimes referred to as CPCTC, which means Corresponding Parts of Congruent Triangles are Congruent. When given two congruent triangles, you can use a series of transformations, reflections, and rotations to show the triangles are congruent. You can use ASA (Angle-Side-Angle) to show two triangles are congruent. If two angles and the included side of a triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. ABC DEF by ASA. 6. You can use SSS (Side-Side-Side) to show two triangles are congruent. If three sides of a triangle are congruent to three sides of another triangle, then the triangles are congruent. GIH JLK by SSS. 36

8 7. You can use SAS (Side-Angle-Side) to show two triangles are congruent. If two sides and the included angle of a triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. MPN QSR by SAS. 8. You can use AAS (Angle-Angle-Side) to show two triangles are congruent. If two angles and a non-included side of a triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent. VTU YWX by AAS. Important Tips If two sides and a non-included angle of one triangle are congruent to two sides and a non-included angle of a second triangle, the triangles are not necessarily congruent. Therefore, there is no way to show triangle congruency by Side-Side- Angle (SSA). If two triangles have all three angles congruent to each other, the triangles are similar, but not necessarily congruent. Thus, you can show similarity by Angle- Angle-Angle (AAA), but you cannot show congruence by AAA. 37

9 EOCT Practice Items 1) In this diagram, CD is the perpendicular bisector of AB. The two-column proof shows that AC is congruent to BC. Step Statement Justification 1 CD is the perpendicular bisector of AB Given 2 AD BD Definition of bisector 3 CD CD Reflexive Property of Congruence 4 ADC and BDC are right angles Definition of perpendicular lines 5 ADC BDC All right angles are congruent 6 ADC BDC? 7 AC BC CPCTC Which theorem would justify Step 6? A. AAS B. ASA C. SAS D. SSS [Key: C] 45

10 2) In this diagram, STU is an isosceles triangle where ST is congruent to UT. The paragraph proof shows that S is congruent to U. It is given that ST is congruent to UT. Draw TV that bisects T. By the definition of an angle bisector, STV is congruent to UTV. By the Reflexive Property, TV is congruent to TV. Triangle STV is congruent to triangle UTV by SAS. S is congruent to U by?. Which step is missing in the proof? A. CPCTC B. Reflexive Property of Congruence C. Definition of right angles D. Angle Congruence Postulate [Key: A] 46

11 MAKE GEOMETRIC CONSTRUCTIONS KEY IDEAS 1. To copy a segment, follow the steps given: Given: AB Construct: PQ congruent to AB Procedure: 1. Use a straightedge to draw a line, l. 2. Choose a point on line l and label it point P. 3. Place the compass point on point A. 4. Adjust the compass width to the length of AB. 5. Without changing the compass, place the compass point on point P and draw an arc intersecting line l. Label the point of intersection as point Q. 6. PQ AB. 2. To copy an angle, follow the steps given: Given: Construct: Procedure: ABC QRY congruent to ABC 1. Draw a point R that will be the vertex of the new angle. 2. From point R, use a straightedge to draw RY, which will become one side of the new angle. 47

12 3. Place the compass point on vertex B and draw an arc through point A. 4. Without changing the compass, place the compass point on point R, draw an arc intersecting RY, and label the point of intersection point S. 5. Place the compass point on point A and adjust its width to where the arc intersects BC. 6. Without changing the compass width, place the compass point on point S and draw another arc across the first arc. Label the point where both arcs intersect as point Q. 7. Use a straightedge to draw RQ. 8. QRY ABC 3. To bisect an angle, follow the steps given: Given: ABC Construct: BY, the bisector of Procedure: ABC 1. Place the compass point on vertex B. 2. Open the compass and draw an arc that crosses both sides of the angle. 3. Set the compass width to more than half the distance from point B to where the arc crosses BA. Place the compass point where the arc crosses BA and draw an arc in the angle s interior. 4. Without changing the compass width, place the compass point where the arc crosses BC and draw an arc so that it crosses the previous arc. Label the intersection point Y. 5. Using a straightedge, draw a ray from vertex B through point Y. 48

13 6. ABY YBC, so BY is the bisector of ABC. 4. To construct a perpendicular bisector of a line segment, follow the steps given: Given: AB Construct: The perpendicular bisector of AB Procedure: 1. Adjust the compass to a width greater than half the length of AB. 2. Place the compass on point A and draw an arc passing above AB and an arc passing below AB. 3. Without changing the compass width, place the compass on point B and draw an arc passing above and below AB. 4. Use a straightedge to draw a line through the points of intersection of these arcs. 5. The segment is the perpendicular bisector of AB. Note: To bisect AB, follow the same steps listed above to construct the perpendicular bisector. The point where the perpendicular bisector intersects AB is the midpoint of AB. 49

14 5. To construct a line perpendicular to a given line through a point not on the line, follow the steps given: Given: Line l and point P that is not on line l Construct: The line perpendicular to line l through point P Procedure: 1. Place the compass point on point P. 2. Open the compass to a distance that is wide enough to draw two arcs across line l, one on each side of point P. Label these points Q and R. 3. From points Q and R, draw arcs on the opposite side of line l from point P so that the arcs intersect. Label the intersection point S. 4. Using a straightedge, draw PS. 5. PS QR. 6. To construct a line parallel to a given line through a point not on the line, follow the steps given: Given: Line l and point P that is not on line l Construct: The line parallel to line l through point P 50

15 Procedure: 1. Draw a transversal line through point P crossing line l at a point. Label the point of intersection Q. 2. Open the compass to a width about half the distance from points P to Q. Place the compass point on point Q and draw an arc that intersects both lines. Label the intersection of the arc and PQ as point M and the intersection of the arc and l as point N. 3. Without changing the compass width, place the compass point on point P and draw an arc that crosses PQ above point P. Note that this arc must have the same orientation as the arc drawn from points M to N. Label the point R. 51

16 4. Set the compass width to the distance from points M to N. 5. Place the compass point on point R and draw an arc that crosses the upper arc. Label the point of intersection S. 6. Using a straightedge, draw a line through points P and S. 7. PS l 52

17 7. To construct an equilateral triangle inscribed in a circle, follow the steps given: Given: Circle O Construct: Equilateral Procedure: ABC inscribed in circle O 1. Mark a point anywhere on the circle and label it point P. 2. Open the compass to the radius of circle O. 3. Place the compass point on point P and draw an arc that intersects the circle at two points. Label the points A and B. 4. Using a straightedge, draw AB. 5. Open the compass to the length of AB. 6. Place the compass point on A. Draw an arc from point A that intersects the circle. Label this point C. 7. Using a straightedge, draw AC and BC. 8. Equilateral ABC is inscribed in circle O. 53

18 8. To construct a square inscribed in a circle, follow the steps given: Given: Circle O Construct: Square ABCD inscribed in circle O Procedure: 1. Mark a point anywhere on the circle and label it point A. 2. Using a straightedge, draw a diameter from point A. Label the other endpoint of the diameter as point C. This is diameter AC. 3. Construct a perpendicular bisector to AC through the center of circle O. Label the points where it intersects the circle as point B and point D. 54

19 4. Using a straightedge, draw AB, BC, CD, and AD. 5. Square ABCD is inscribed in circle O. 9. To construct a regular hexagon inscribed in a circle, follow the steps given: Given: Circle O Construct: Regular hexagon ABCDEF inscribed in circle O Procedure: 1. Mark a point anywhere on the circle and label it point A. 2. Open the compass to the radius of circle O. 3. Place the compass point on point A and draw an arc across the circle. Label this point B. 4. Without changing the width of the compass, place the compass point on B and draw another arc across the circle. Label this point C. 5. Repeat this process from point C to a point D, from point D to a point E, and from point E to a point F. 6. Use a straightedge to draw AB, BC, CD, DE, EF, and AF. 7. Regular hexagon ABCDEF is inscribed in circle O. 55

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