A kosarad üres
About the quadrilaterals (reminder)
- Convex quadrilateral: all of its angles are convex. (Figure 37)
- Concave quadrilateral: it has a concave angle. (Figure 38)
- A diagonal divides a quadrilateral into two triangles, therefore the sum of the interior angles of the quadrilateral is 2\latex{\times}180º = 360º. (Figure 39)
Figure 37
Let us find the sum of the exterior angles of a convex quadrilateral. (Figure 40)
\latex{\alpha +\beta +\gamma +\delta =360°}
\latex{\left(\alpha +(\alpha'\right) +\left(\beta +\beta'\right) +\left(\gamma +\gamma '\right) +\left(\delta +\delta '\right) =4\times 180°=720°}
\latex{\left(\alpha +(\alpha'\right) +\left(\beta +\beta'\right) +\left(\gamma +\gamma '\right) +\left(\delta +\delta '\right) =4\times 180°=720°}
These imply: \latex{\alpha '+\beta '+\gamma '+\delta'=360°}.
The sum of the exterior angles of a convex quadrilateral is \latex{ 360º }.
Figure 38
Figure 39
Figure 40
\latex{\delta}
\latex{\alpha}
\latex{\alpha'}
\latex{\beta'}
\latex{\beta}
\latex{\gamma}
\latex{\gamma'}
\latex{\delta'}
\latex{ D }
\latex{ A }
\latex{ B }
\latex{ C }
Special quadrilaterals
- A trapezium is a quadrilateral with a pair of parallel sides. The plural for trapezium is trapezia or trapeziums.
- A parallelogram is a quadrilateral the opposite sides of which are parallel.
- A rhombus is a quadrilateral the sides of which are of equal length. In Figure 42 the relationships of the above quadrilaterals are represented.
- A kite is a quadrilateral two and two adjacent sides of which are of equal length.
Rhombi (or rhombuses) are equilateral kites. (Figure 43)
Figure 42
Figure 43
trapezia
parallelograms
rhombi
rhombi
kites
Figure 41
leg
leg
base
kites
trapezium
parallelogram
rhombus
convex
concave
base
- A rectangle is a quadrilateral all angles of which are right angles. So rectangles are parallelograms with equal angles. (Figure 45)
- A square is a quadrilateral all sides and all angles of which are equal. So a square is an equilateral rectangle and a rhombus with equal angles. (Figure 46)
Figure 45
Figure 46
parallelograms
rectangles
squares
rhombi
rectangles
Figure 44
rectangle
square
Example 1
Decide which of the below statements are true and which are false: a) every parallelogram is a trapezium; b) there is a kite which is not a rhombus; c) there is a rectangle which is a kite; d) there is a rhombus which is not a trapezium; e) every parallelogram is a kite.
Solution
Statements a), b) and c) are true, statements d) and e) are false.
- The definitions imply that it is true. (Two pairs of parallel sides means there is at least one pair of parallel sides.)
- All kites satisfy it which have two sides with different length.
- A square is a rectangle and a kite too.
- Every rhombus is a parallelogram, and every parallelogram is a trapezium, thus every rhombus is a trapezium.
- If a parallelogram is not a rhombus, then its adjacent sides have different length.
Example 2
The two angles on one of the bases of a trapezium are \latex{ 62º } and \latex{ 46º }. Let us calculate the other two angles of the trapezium.
Figure 47
\latex{\delta }
\latex{\delta' }
\latex{\gamma }
\latex{62°}
\latex{46°}
\latex{ D }
\latex{ C }
\latex{ B }
\latex{ A }
Solution
Since AB is parallel with CD, thus the exterior angle \latex{\delta '} of \latex{\delta } and angle DAB are alternate angles, thus \latex{\delta' } = 62º. (Figure 47)
It implies that \latex{\delta } = 180º – \latex{\delta }’ = 180º – 62º = 118º.
Similarly \latex{\gamma }= 180º – 46º = 134º.
It is also true in general that the sum of the angles on a leg of a trapezium is \latex{ 180º }.
Example 3
Two angles of a kite are \latex{ 110º } and \latex{ 44º }. Let us calculate the other two angles of the kite.
Solution
The definition implies that two opposite angles of a kite are equal. Taking this fact into account three cases correspond to the hypotheses, as can be seen in Figure 48.
Figure 48/a
Figure 48/b
Figure 48/c
\latex{110°}
\latex{\gamma }
\latex{\delta}
\latex{44°}
\latex{110°}
\latex{44°}
\latex{\delta}
\latex{\beta}
\latex{110°}
\latex{\gamma }
\latex{\delta}
\latex{44°}
Case I.
Case II.
Case III.
\latex{\alpha =44°, \beta =\delta =110°,}
\latex{\alpha +2\beta +\gamma =360°,}
\latex{264° +\gamma =360°,}
\latex{\gamma =96°.}
\latex{\alpha +2\beta +\gamma =360°,}
\latex{264° +\gamma =360°,}
\latex{\gamma =96°.}
\latex{\alpha =110°, \beta =\delta =44°,}
\latex{\alpha +2\beta +\gamma =360°,}
\latex{198° +\gamma =360°,}
\latex{\gamma =162°.}
\latex{\alpha +2\beta +\gamma =360°,}
\latex{198° +\gamma =360°,}
\latex{\gamma =162°.}
\latex{\alpha =110°, \gamma =44°,}
\latex{\alpha +2\beta +\gamma =360°,}
\latex{154° +2\beta =360°,}
\latex{2\beta =206°,}
\latex{\beta =103°,}
\latex{\beta =\delta=103°.}
\latex{\alpha +2\beta +\gamma =360°,}
\latex{154° +2\beta =360°,}
\latex{2\beta =206°,}
\latex{\beta =103°,}
\latex{\beta =\delta=103°.}
\latex{ A }
\latex{ C }
\latex{ B }
\latex{ D }
\latex{ B }
\latex{ D }
\latex{ C }
\latex{ A }
\latex{ B }
\latex{ D }
\latex{ A }
\latex{ C }
Exercises
{{exercise_number}}. The interior angles of a convex quadrilateral are \latex{\alpha ,\,\beta ,\,\gamma ,\,\delta } the corresponding exterior angles are \latex{\alpha',\,\beta',\,\gamma',\,\delta' } respectively. Calculate the corresponding interior and exterior angles, if
- \latex{\alpha =100°,\;\beta =72°,\;\gamma '=84°;}
- \latex{\alpha =70°,\;\beta =135°,\;\gamma '=108°;}
- \latex{\alpha =82°,\;\alpha' =98°,\;\delta=121°;}
- \latex{\alpha =138°,\;\beta' =88°,\;\delta '=170°}.
{{exercise_number}}. Give the interior and exterior angles of the trapezium if two opposite angles of it are
- \latex{60°}and \latex{120°;}
- \latex{45°}and \latex{100°;}
- \latex{162.5°}and \latex{13°;}
- \latex{81°}and \latex{81°}.
{{exercise_number}}. Give the interior and exterior angles of the trapezium if the ratio of its interior angles is
- \latex{2:3:4:5;}
- \latex{4:7:10:13;}
- \latex{3:5:7:9;}
- \latex{5:5:7:8}.
{{exercise_number}}. Calculate the interior and exterior angles of the parallelogram if one of its interior angles is
- \latex{30°;}
- \latex{53°;}
- \latex{129°;}
- \latex{143.2°}.
{{exercise_number}}. Give the interior and exterior angles of the parallelogram if the ratio of two of its interior
angles is
- \latex{1:3;}
- \latex{4:5;}
- \latex{5:7;}
- \latex{a:b}.
{{exercise_number}}. Calculate the interior and exterior angles of the rhombus if one of its interior angles is
- \latex{30°;}
- \latex{57°;}
- \latex{83°;}
- \latex{174°}.
{{exercise_number}}. Calculate the interior angles of the kite if two opposite interior angles are
- \latex{32°}and \latex{131°;}
- \latex{68°}and \latex{111°;}
- \latex{23.4°}and \latex{68.6°;}
- \latex{282°}and \latex{16°}.
{{exercise_number}}. Calculate the interior angles of the kite if two adjacent interior angles are
- \latex{122°}and \latex{38°};
- \latex{45°}and \latex{131°};
- \latex{220°}and \latex{18°};
- \latex{72°}and \latex{83°}.
{{exercise_number}}. The angle included by the diagonal and one of the sides of a rectangle is
- \latex{30°;}
- \latex{11°;}
- \latex{38.6°;}
- \latex{\alpha}.
Calculate the angle included by the diagonals.
{{exercise_number}}. Prove that the sum of the two interior angles on a side of a convex quadrilateral is equal to the sum of the exterior angles belonging to the other two interior angles.
{{exercise_number}}. Decide which of the below statements are true and which are false.
- There is a non-convex trapezium.
- There is trapezium which has exactly one right angle.
- There is trapezium the legs of which are parallel.
- There is parallelogram which has a right angle.
- Every parallelogram has a right angle.
- There is parallelogram which is not a trapezium.
- If a parallelogram has a right angle, then it is a rectangle.
- There is a rectangle which is a rhombus.
- Every rhombus is a trapezium.
- If a kite is a rectangle, then it is a square.
- If all four sides of a parallelogram are equal, then it is a square.
