Parallélisme et perpendicularité - 6e

Droites parallèles

Exercice 1 : Droites perpendiculaires, parallèles et nature de quadrilatère

On s'intéresse à la figure ci-dessous :

En plus des marquages, on sait que les droites \( (BA) \) et \( (CD) \) sont parallèles.

En utilisant le codage ou en raisonnant, répondre aux questions suivantes.

Parmi ces droites, lesquelles sont parallèles entre elles ?

A. \( (BA) \) et \( (DC) \)
B. \( (DA) \) et \( (BC) \)
C. \( (EA) \) et \( (BF) \)

Parmi ces angles, le ou lesquels sont des angles droits ?

A. \( \widehat{EFB} \)
B. \( \widehat{BAD} \)
C. \( \widehat{CDA} \)

Quelle est la nature du quadrilatère \( ABCD \) ?

Exercice 2 : Construire une parallèle

Construire la parallèle à \( (d) \) passant par \( A \).

{"init": {"range": [[-9, 9], [-9, 9]], "scale": [35, 35]}, "line": [[[-9.0, -9.0], [9.0, -9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -8.0], [9.0, -8.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -7.0], [9.0, -7.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -6.0], [9.0, -6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -5.0], [9.0, -5.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -4.0], [9.0, -4.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -3.0], [9.0, -3.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -2.0], [9.0, -2.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -1.0], [9.0, -1.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 0.0], [9.0, 0.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 1.0], [9.0, 1.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 2.0], [9.0, 2.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 3.0], [9.0, 3.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 4.0], [9.0, 4.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 5.0], [9.0, 5.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 6.0], [9.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 7.0], [9.0, 7.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 8.0], [9.0, 8.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 9.0], [9.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -9.0], [-9.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-8.0, -9.0], [-8.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-7.0, -9.0], [-7.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, -9.0], [-6.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-5.0, -9.0], [-5.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-4.0, -9.0], [-4.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-3.0, -9.0], [-3.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-2.0, -9.0], [-2.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-1.0, -9.0], [-1.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[0.0, -9.0], [0.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[1.0, -9.0], [1.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[2.0, -9.0], [2.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[3.0, -9.0], [3.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[4.0, -9.0], [4.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[5.0, -9.0], [5.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[6.0, -9.0], [6.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[7.0, -9.0], [7.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[8.0, -9.0], [8.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[9.0, -9.0], [9.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-1.1, 3.0], [-0.9, 3.0], {}], [[-1.0, 2.9], [-1.0, 3.1], {}], [[-6.75, 9.0], [6.75, -9.0], {"subtype": "line"}]], "label": [[[-1.0, 3.0], "A", "above", {}], [[0.0, 0.0], "(d)", "above right", {}]], "drawing_options": {"tools": ["line"]}}

Essais restants : 2

Exercice 3 : identifier des parallélismes et perpendicularités

On considère l'ensemble de droites suivant :

{"init": {"range": [[-8, 8], [-3, 8]], "scale": [30, 30]}, "path": [[[[-3.57187282537192, 3.498817588753323], [-3.221991066496588, 3.856004871290515], [-2.864803783959396, 3.5061231124151826], [-3.214685542834728, 3.1489358298779906], [-3.57187282537192, 3.498817588753323]]], [[[0.0, 0.0], [-0.357187282537192, 0.3498817588753323], [-0.007305523661859714, 0.7070690414125242], [0.3498817588753323, 0.357187282537192], [0.0, 0.0]]], [[[3.498817588753323, 3.57187282537192], [3.1489358298779906, 3.214685542834728], [2.7917485473407986, 3.5645673017100603], [3.141630306216131, 3.9217545842472523], [3.498817588753323, 3.57187282537192]]], [[[-0.0730552366185971, 7.070690414125243], [0.2841320459185949, 6.720808655249911], [-0.06574971295673739, 6.363621372712719], [-0.4229369954939294, 6.713503131588051], [-0.0730552366185971, 7.070690414125243]]], [[[-2.0, 5.0], [-1.5, 5.0], [-1.5, 4.5], [-2.0, 4.5], [-2.0, 5.0]]], [[[-2.0, 0.0], [-2.0, 0.5], [-1.5, 0.5], [-1.5, 0.0], [-2.0, 0.0]]], [[[3.0, 0.0], [2.5, 0.0], [2.5, 0.5], [3.0, 0.5], [3.0, 0.0]]], [[[3.0, 5.0], [3.0, 4.5], [2.5, 4.5], [2.5, 5.0], [3.0, 5.0]]]], "label": [[[-8.0, 7.83638], "(d8)", {"subtype": "line", "stroke": "#2f0d0d", "color": "#2f0d0d"}], [[-8.0, -1.02177], "(d6)", {"subtype": "line", "stroke": "#61175e", "color": "#61175e"}], [[8.0, -0.83725], "(d1)", {"subtype": "line", "stroke": "#6495ed", "color": "#6495ed"}], [[7.83638, 8.0], "(d2)", {"subtype": "line", "stroke": "#a37351", "color": "#a37351"}], [[-2.0, -3.0], "(d4)", {"subtype": "line", "stroke": "#ff6666", "color": "#ff6666"}], [[8.0, 5.0], "(d3)", {"subtype": "line", "stroke": "#ffa400", "color": "#ffa400"}], [[3.0, -3.0], "(d5)", {"subtype": "line", "stroke": "#800000", "color": "#800000"}], [[-8.0, 0.0], "(d7)", {"subtype": "line", "stroke": "#008000", "color": "#008000"}]], "line": [[[-7.6428127709528875, 7.486498186517725], [3.06264, -3.0], {"subtype": "line", "stroke": "#2f0d0d", "color": "#2f0d0d"}], [[0.83725, 8.0], [-7.650118209626507, -0.6645827483167422], {"subtype": "line", "stroke": "#61175e", "color": "#61175e"}], [[7.642812748316742, -0.48736820962650684], [-1.02177, 8.0], {"subtype": "line", "stroke": "#6495ed", "color": "#6495ed"}], [[-2.93864, -3.0], [7.48649820561235, 7.6428127522487985], {"subtype": "line", "stroke": "#a37351", "color": "#a37351"}], [[-2.0, 8.0], [-2.0, -2.5], {"subtype": "line", "stroke": "#ff6666", "color": "#ff6666"}], [[-8.0, 5.0], [7.5, 5.0], {"subtype": "line", "stroke": "#ffa400", "color": "#ffa400"}], [[3.0, 8.0], [3.0, -2.5], {"subtype": "line", "stroke": "#800000", "color": "#800000"}], [[8.0, 0.0], [-7.5, 0.0], {"subtype": "line", "stroke": "#008000", "color": "#008000"}]]}

Pour les deux questions suivantes, on donnera la réponse sous forme de liste séparée par des points-virgules.
S'il n'y en a pas, écrire "Aucune".

Donner la ou les droites parallèles à la droite \((d1)\).

Donner la ou les droites perpendiculaires à la droite \((d1)\).

Exercice 4 : Identifier si des droites sont parallèles, sécantes, ou perpendiculaires

Voici une figure représentant deux droites \( (d_1) \) et \( (d_2) \) :

En faisant attention aux marquages de la figure, compléter les phrases suivantes à propos de la figure avec "sont", "ne sont pas", "semblent" ou "ne semblent pas" :

Les droites \( (d_1) \) et \( (d_2) \) perpendiculaires.

Les droites \( (d_1) \) et \( (d_2) \) sécantes.

Les droites \( (d_1) \) et \( (d_2) \) parallèles.

Exercice 5 : Tracer une parallèle et une perpendiculaire dans une figure avec plusieurs droites

Construire la parallèle et la perpendiculaire à \((k)\) passant par \(V\).

{"init": {"range": [[-9, 9], [-9, 9]], "scale": [35, 35]}, "line": [[[-9.0, -9.0], [9.0, -9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -8.0], [9.0, -8.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -7.0], [9.0, -7.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -6.0], [9.0, -6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -5.0], [9.0, -5.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -4.0], [9.0, -4.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -3.0], [9.0, -3.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -2.0], [9.0, -2.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -1.0], [9.0, -1.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 0.0], [9.0, 0.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 1.0], [9.0, 1.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 2.0], [9.0, 2.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 3.0], [9.0, 3.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 4.0], [9.0, 4.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 5.0], [9.0, 5.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 6.0], [9.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 7.0], [9.0, 7.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 8.0], [9.0, 8.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 9.0], [9.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -9.0], [-9.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-8.0, -9.0], [-8.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-7.0, -9.0], [-7.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, -9.0], [-6.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-5.0, -9.0], [-5.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-4.0, -9.0], [-4.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-3.0, -9.0], [-3.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-2.0, -9.0], [-2.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-1.0, -9.0], [-1.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[0.0, -9.0], [0.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[1.0, -9.0], [1.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[2.0, -9.0], [2.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[3.0, -9.0], [3.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[4.0, -9.0], [4.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[5.0, -9.0], [5.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[6.0, -9.0], [6.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[7.0, -9.0], [7.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[8.0, -9.0], [8.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[9.0, -9.0], [9.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[3.9, -2.0], [4.1, -2.0], {}], [[4.0, -2.1], [4.0, -1.9], {}], [[-9.0, -3.66667], [9.0, 2.33333], {"subtype": "line"}], [[-4.0, -9.0], [9.0, 4.0], {"subtype": "line"}]], "label": [[[4.0, -2.0], "V", "above", {}], [[-7.0, -3.0], "(d)", "above left", {"color": "blue"}], [[-3.0, -8.0], "(k)", "above left", {"color": "blue"}]], "drawing_options": {"tools": ["line"]}}

Essais restants : 2

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