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Proofs Chloe Clegg, Georgina Platt, William Snapp, Komal Parikh, David Korsunskiy, Joseph Higginbotham Dilations Dilation : a transformation that makes a figure larger or smaller than the original figure based on a ratio given by a scale factor. Scale Factor Rules : x>1= figure is made larger 0<x<1 = figure is made smaller Center of Dilation = Origin You have to multiply each coordinate of the pre- image by the scale factor to get the dilated figure. Center of Dilations If the center of Dilation is not on the origin then you use the formula : k(x-xc) + xc k(y-yc) + yc k= scale factor of dilation (x,y) = point of dilation (xc ,yc) = center point of dilation Center of Dilations cont’d. When a figure is transformed under a dilation , the CORRESPONDING ANGLES of the pre- image and the image have equal measurement. A’ Example : A triangle ABC and triangle A’B’C’ <A=<A’ <B=<B’ <C=<C’ B C B’ C’ Dilations cont’d When a figure is dilated, a segment of the pre-image that does not pass through the center of the 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’ A’ B’ A B C C’ Dilations cont’d 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 A’C’ are on the same line. A’ B’ A B C C’ Dilations Examples 1. Dilate the following triangles by 3 A B A C B C Examples cont’d What are the ordered pairs of the new larger triangles? What are the angles of the new larger triangles? Rigid Motions 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 ) Congruent figures always have the same side lengths and angle measures as each other. Rule: Two triangles are congruent if and only if their corresponding sides and corresponding angles are congruent = CPCTC CPCTC= Corresponding Parts of Congruent Triangles are Congruent Proof Theorems If two triangles are similar, two corresponding angles are congruent. This is the Angle Angle(AA) Similarity. Rules : 1) If two triangles are similar ,all corresponding pairs of angles are congruent. 2) If two triangles are similar all corresponding pairs of sides are proportional. 3) Congruent = similarity Sides and Angles There are four ways you can show two triangles are congruent to each other: ASA : Angle Side Angle Two angles and the included side of a triangle are congruent to the two angles and the included side of the other triangle SSS: Side-Side-Side Three sides of a triangle have to be congruent to the other three sides of another triangle Sides and Angles Cont’d SAS : Side-Angle-Side Two sides and the included angle of a triangle have to be congruent to two sides and the included angle of another triangle. AAS: Angle- Angle- Side Two angles and a non included side of a triangle are congruent to the angles and the corresponding nonincluded side of the other triangle Sides and Angles SSA : Side -Side Angle ● There is no way to show triangle congruence through SSA! ● The two sides and a non included angle one triangle are congruent to two sides of a non - included angle of a second triangle does not make them congruent. ALSO.. ● Two triangles having three angles congruent to each other does not make them congruent it only makes them similar! Geometric Theorems Auxiliary line : a line drawn in a diagram that makes other figures such as congruent triangles or angles formed by a transversal. Key Ideas on lines and angles: Vertical Angle Theorem : Vertical angles are congruent Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then alternate interior angles formed by the transversal are congruent Corresponding Angles Postulate :If two parallel lines are cut by a transversal then corresponding angles formed by the transversal are congruent. Geometric Theorems Key Points about Triangles: Triangle Angle Sum Theorem : The sum of the measure of the angles of a triangle is 180 degrees. Isosceles Triangle Theorem: If two sides of a triangle are congruent , then the angles opposite those sides are also congruent . Triangle Midsegment Theorem: If a segment joins the midpoints of two sides of a triangle , then the segment is parallel to the third side and half its length Geometric Theorems ● ● ● ● Key Points on Parallelograms : Opposite sides are congruent and opposite angles are congruent The diagonals of a parallelogram bisect each other If the diagonals of a quadrilateral bisect each other , then the quadrilateral is a parallelogram A rectangle is a parallelogram with congruent diagonals. Examples Lines AB and CD are parallel and line EF is a transversal,m∠BGH=2x and m∠DHF=5x−51. Find the measurement, in degrees, of ∠CHF. E G B A 2x C D H F More examples Find the degree of each angle in the Isosceles triangle below Find the value of X in the Right Triangle below. 60° 68° 3X 5X-40 2x Two Column Proof 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. The reasons can be given information. Paragraph proof : also uses a series of statements and reasons that works from a given information to the statement that need to be proven , but the information is presented as running text in paragraph form. Two Column Proof Example In the triangle below, line AC is parallel to line DE: Prove that line DE divides line AB and CB proportionally. B D A E C Steps Statement Justification Step 1 DE given Step 2 < BDE ≅<BAC two lines cut by transversal corresponding angles are congruent Step 3 <DBE ≅<ABC reflexive property of congruence b/c same angle Step 4 DBG ≅ AC ABC AA Step 5 BA/ BD = BC/BE corresponding sides of similar triangles are proportional Step 6 BD + DA= BA BE+ BC= BC segment addition Step 7 BD + DA / BD = BE + EC/BE substitution Step 8 1+DA/BD = 1+ EC/BE simplify Steps Cont’d Statement Justification Step 9 DA/BD = EC/ BE subtraction Step 10 DE divides AB and CB proportionally definition of proportionality Lets Try Another One.. Geometric Constructions Geometric constructions uses only a straightedge and a compass to make shapes with specific angles or side lengths https://www.khanacademy.org/math/geometry/geometric-constructions/geo-bisectors/v/constructing-a-perpendicular-bisectorusing-a-compass-and-straightedge