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1 MEDIANS ARE PROPORTIONAL ALTITUDES ARE PROPORTIONAL ANGLE BISECTORS ARE PROPORTIONAL PERIMETERS ARE PROPORTIONAL ANGLE BISECTORS FORM PROPOR. SEG. PROBLEM 6 PROBLEM 7 PROBLEM 8 PROBLEM 9 STANDARDS 4 and 5 END SHOW SPECIAL SEGMENTS IN A TRIANGLE PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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2 Standard 4: Students prove basic theorems involving congruence and similarity. Los estudiantes prueban teoremas básicos que involucran congruencia y semejanza. Standard 5: Students prove triangles are congruent or similar and are able to use the concept of corresponding parts of congruent triangles. Los estudiantes prueban que triángulos son congruentes o semejantes y son capaces de usar el concepto de partes correspondientes de triángulos congruentes. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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3 STANDARDS 4 and 5 ALTITUDE: A segment from a vertex of a triangle perpendicular to the line containing the opposite side. ALTURA: Un segmento desde el vértice de un triángulo perpendicular a la línea conteniendo el lado opuesto. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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4 STANDARDS 4 and 5 PERPENDICULAR BISECTOR: A line or segment that passes through the midpoint of a side of a triangle and is perpendicular to that side.. BISECTRIZ PERPENDICULAR: Una línea o segmento que pasa a través del punto medio de un lado de un triángulo y es perpendicular a ese lado. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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5 STANDARDS 4 and 5 MEDIAN: A segment that connects a vertex of a triangle to the midpoint of the opposite side. MEDIANA: Un segmento que conecta un vertice de un triángulo a el punto medio del lado opuesto. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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6 STANDARDS 4 and 5 ANGLE BISECTOR: A segment from a vertex to the opposite side that bisects (divides in two equal parts) the angle of the triangle. BISECTRIZ ANGULAR: Un segmento desde un vértice a el lado opuesto que biseca (divide en dos partes iguales) el ángulo de el triángulo. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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7 STANDARDS 4 and 5 Draw an isosceles triangle and draw an altitude, a perpendicular bisector, a median and an angle bisector from the vertex angle. What do you observe? Altitude PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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8 STANDARDS 4 and 5 Draw an isosceles triangle and draw an altitude, a perpendicular bisector, a median and an angle bisector from the vertex angle. What do you observe? Perpendicualr Bisector PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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9 STANDARDS 4 and 5 Draw an isosceles triangle and draw an altitude, a perpendicular bisector, a median and an angle bisector from the vertex angle. What do you observe? Median PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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10 STANDARDS 4 and 5 Draw an isosceles triangle and draw an altitude, a perpendicular bisector, a median and an angle bisector from the vertex angle. What do you observe? Angle Bisector PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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11 STANDARDS 4 and 5 Draw an isosceles triangle and draw an altitude, a perpendicular bisector, a median and an angle bisector from the vertex angle. What do you observe? Altitude Perpendicualr Bisector Median Angle Bisector Lets put them all together So, all of them occupy the same Geometric Space in the triangle! Could you do something similar with an equilateral triangle in all the vertices? PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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12 A B C K L M KL AB MK CA LM BC STANDARDS 4 and 5 CABMKL KL AB MK CA LM BC OR REVIEW: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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13 A B C K L M AB KL CA MK BC LM N KN D AD In simlar triangles MEDIANS are proportional to sides: STANDARDS 4 and 5 KL AB MK CA LM BC KN AD OR PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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14 J I K S J U T R TU IK STANDARDS 4 and 5 = JS = IJK TJU, R is the midpoint of TU and S is the midpoint of IK. TU= 10, JR= 6, IK = 30. Find JS. S R J U T I K JS JR (6) = (30)(6) 10 JS = JS JS=18 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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15 A B C K L M In similar triangles ALTITUDES are proportional to sides: AB KL CA MK BC LM O KO E AE STANDARDS 4 and 5 KL AB MK CA LM BC KO AE OR PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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16 F AF P KP A B C K L M In similar triangles ANGLE BISECTORS are proportional to sides: AB KL CA MK BC LM STANDARDS 4 and 5 KL AB MK CA LM BC KP AF OR PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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17 JR JS JT JT+TI STANDARDS 4 and 5 S R J U T I K J I K S J U T R JS is an angle bisector. If TJ = 14, IT=22, and JS= 30, what is the value for JR? Suppose IJK TJU = JR = (30) = (14)(30) 36 JR = JR 30 JR = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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18 A B C K L M AB KL CA MK BC LM In similar triangles PERIMETERS proportional to sides: KL MK LM++ ABCA BC ++ STANDARDS 4 and 5 KL AB MK CA LM BC KLMKLM++ AB CA BC++ OR PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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19 JK JU JT+JU+TU PERIMETER IJK STANDARDS 4 and 5 S R J U T I K Given: JT=20, JU=30, TU=70, AND JK=90, what is the perimeter of IJK? Suppose TJU IJK. J I K S J U T R = PERIMETER IJK = 120 PERIMETER IJK = 3 (120) PERIMETER IJK = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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20 A B C D AB AC DC BD STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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21 IS SK JI JK STANDARDS 4 and 5 S R J U T I K JS is an angle bisector. JI= X+5, JK=X+3, IS=3, and SK=2. Find the value for X. Suppose IJK TJU. = = 3(X+3) = 2(X+5) 3X + 9 = 2X X = 2X X X = 1 X+5 X X PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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