Parallélogrammes & Obtention

Le plan - Mathématiques 5e

Exercice 1 : Démontrer qu'un quadrilatère est un parallélogramme

Voici le quadrilatère \(EFGH\) :

On sait que \[(EF) // (GH)\] \[EF = GH = 5,9 cm\]
Compléter la démonstration pour montrer que le quadrilatère \(EFGH\) est un parallélogramme.

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"left"}, {"type": "statement_input", "fields": [], "align": "left", "name": "concl", "check": null}]}, "SimpleComparison": {"previous_statement": true, "next_statement": true, "colour": 210, "inputs": [{"type": "value_input", "fields": [], "align": "left", "name": "value_1", "check": null}, {"type": "dummy_input", "fields": [{"name": "op", "values": [["=", "EQ"], ["\u2260", "NE"], ["<", "LT"], ["\u2264", "LE"], [">", "GT"], ["\u2265", "GE"], ["appartient \u00e0", "BELONGS"]], "type": "dropdown"}], "align": "left"}, {"type": "value_input", "fields": [], "align": "left", "name": "value_2", "check": null}]}, "theorem_Parallelogram2": {"previous_statement": true, "next_statement": true, "colour": 23, "inputs": [{"type": "dummy_input", "fields": [{"text": "Si un quadrilat\u00e8re a ses", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "diagonales qui se coupent en", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "leur milieu alors c'est un", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "parall\u00e9logramme", "type": "text"}], "align": "left"}]}, "Parallel": {"previous_statement": true, "next_statement": true, "colour": 210, "inputs": [{"type": "value_input", "fields": [], "align": "left", "name": "value_1", "check": "Line"}, {"type": "dummy_input", "fields": [{"text": "//", "type": "text"}], "align": "left"}, {"type": "value_input", "fields": [], "align": "left", "name": "value_2", "check": "Line"}]}, "VarValueLineCD": {"compound": {"blocks": {"VarValueLine": {"output": "Line", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValueLine", "value": "(GH)"}}, "output": "Line", "colour": 330}, "theorem_Parallelogram3": {"previous_statement": true, "next_statement": true, "colour": 23, "inputs": [{"type": "dummy_input", "fields": [{"text": "Si un quadrilat\u00e8re a deux", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "c\u00f4t\u00e9s oppos\u00e9s parall\u00e8les et de", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "m\u00eame longueur alors c'est un", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "parall\u00e9logramme", "type": "text"}], "align": "left"}]}, "VarValueLengthAB": {"compound": {"blocks": {"VarValueLength": {"output": "Length", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValueLength", "value": "EF"}}, "output": "Length", "colour": 330}, "ParallelogramABCD": {"compound": {"blocks": {"Parallelogram": {"compound": {"blocks": {"QuadrilateralProperty": {"previous_statement": true, "next_statement": true, "colour": 210, "inputs": [{"type": "dummy_input", "fields": [{"name": "element", "default": "", "type": "text_input"}, {"text": "est un", "type": "text"}, {"name": "property_type", "values": [["parall\u00e9logramme", "parallelogram"], ["losange", "losange"], ["carr\u00e9", "square"], ["rectangle", "rectangle"]], "type": "dropdown"}], "align": "left"}]}}, "ast": {"type": "QuadrilateralProperty", "property_type": "parallelogram"}}, "previous_statement": true, "next_statement": true, "colour": 210}}, "ast": {"type": "Parallelogram", "element": "EFGH"}}, "previous_statement": true, "next_statement": true, "colour": 210}, "VarValueLineAB": {"compound": {"blocks": {"VarValueLine": {"output": "Line", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValueLine", "value": "(EF)"}}, "output": "Line", "colour": 330}, "Start": {"next_statement": true, "colour": 65, "tooltip": "D\u00e9marrage de la d\u00e9monstration", "inputs": [{"type": "dummy_input", "fields": [{"text": "D\u00e9monstration", "type": "text"}], "align": "left"}], "creatable": false, "on_event": "start"}, "theorem_Parallelogram1": {"previous_statement": true, "next_statement": true, "colour": 23, "inputs": [{"type": "dummy_input", "fields": [{"text": "Si un quadrilat\u00e8re a ses c\u00f4t\u00e9s", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "oppos\u00e9s parall\u00e8les deux \u00e0 deux", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "alors c'est un parall\u00e9logramme", "type": "text"}], "align": "left"}]}}, "toolbox": [{"type": "SimpleComparison", "value_1": {"type": "VarValueLengthAB"}, "value_2": {"type": "VarValueLengthCD"}, "op": "EQ"}, {"type": "Parallel", "value_1": {"type": "VarValueLineAB"}, "value_2": {"type": "VarValueLineCD"}}, {"type": "theorem_Parallelogram1"}, {"type": "theorem_Parallelogram2"}, {"type": "theorem_Parallelogram3"}, {"type": "theorem_Parallelogram4"}]}, "ast": [[{"type": "Start", "deletable": false, "movable": 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Exercice 2 : Reconnaître un carré par ses côtés et ses angles

Parmi les quadrilatères suivants, lesquels sont des carrés ?

1.

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2.

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3.

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4.

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Exercice 3 : Problème - Reconnaître un quadrilatère particulier en utilisant le codage et la définition d'un cercle

On considère deux cercles de centre O suivant la figure ci-dessous.

Donnez la nature la plus précise possible du quadrilatère HLFJ.

Exercice 4 : Reconnaître un parallélogramme par ses côtés

Parmi les quadrilatères suivants, lesquels sont des parallélogrammes ?

1.

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Sachant que \((FG) // (HI)\) et \((FI) // (GH)\)
2.

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Sachant que \((OP) // (QS)\)
3.

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Sachant que \((GJ) // (HI)\) et \(GJ = HI\)
4.

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Sachant que \((LM) // (NO)\) et \(LM = NO\)

Exercice 5 : Démontrer qu'un quadrilatère est un parallélogramme

Démontrer que le quadrilatère ABCD est un parallélogramme.
Si plusieurs blocs "On sait que, or, donc" sont nécessaires, il faut les écrire à la suite les uns des autres et non imbriqués les uns dans les autres.

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"ast": {"type": "QuadrilateralProperty", "property_type": "parallelogram"}}, "previous_statement": true, "next_statement": true, "colour": 210}}, "ast": {"type": "Parallelogram", "element": "ABCD"}}, "previous_statement": true, "next_statement": true, "colour": 210}, "Perpendicular": {"previous_statement": true, "next_statement": true, "colour": 210, "inputs": [{"type": "value_input", "fields": [], "align": "left", "name": "value_1", "check": "Line"}, {"type": "dummy_input", "fields": [{"text": "\u22a5", "type": "text"}], "align": "left"}, {"type": "value_input", "fields": [], "align": "left", "name": "value_2", "check": "Line"}]}, "VarValueLineBC": {"compound": {"blocks": {"VarValueLine": {"output": "Line", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValueLine", "value": "(BC)"}}, "output": "Line", "colour": 330}, "Start": {"next_statement": true, "colour": 65, "tooltip": "D\u00e9marrage de la d\u00e9monstration", "inputs": [{"type": "dummy_input", "fields": [{"text": "D\u00e9monstration", "type": "text"}], "align": "left"}], "creatable": false, "on_event": "start"}, "Implication": {"previous_statement": true, "next_statement": true, "colour": 120, "tooltip": "\u00c0 partir d'hypoth\u00e8ses et d'une propri\u00e9t\u00e9 d\u00e9montre un r\u00e9sultat", "inputs": [{"type": "dummy_input", "fields": [{"text": "On sait que", "type": "text"}], "align": "left"}, {"type": "statement_input", "fields": [], "align": "left", "name": "hypo", "check": null}, {"type": "dummy_input", "fields": [{"text": "or,", "type": "text"}], "align": "left"}, {"type": "statement_input", "fields": [], "align": "left", "name": "theo", "check": null}, {"type": "dummy_input", "fields": [{"text": "donc", "type": "text"}], "align": "left"}, {"type": "statement_input", "fields": [], "align": "left", "name": "concl", "check": null}]}, "theorem_PerpendicularParallelTrans": 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