FIGURE IT OUT
- Find all the other angles inside the following rectangles.
(i) Rectangle ABCD
Given:
ABCD is a rectangle.
The diagonals AC and BD intersect.
∠CAB = 30°.
Solution:
Since ABCD is a rectangle,
∠DAB = 90°
Therefore,
∠DAC + ∠CAB = 90°
∠DAC + 30° = 90°
∴ ∠DAC = 60°
Since opposite sides of a rectangle are parallel,
AB ∥ DC
Therefore,
∠DCA = ∠CAB = 30°
Since ∠DCB = 90°,
∠ACB = 90° − 30°
∴ ∠ACB = 60°
Similarly, for diagonal BD,
∠ABD = 30°
and
∠ADB = 60°
Therefore, the angles at the four corners formed by the diagonals are:
∠CAB = 30°
∠DAC = 60°
∠DCA = 30°
∠ACB = 60°
∠ABD = 30°
∠DBC = 60°
∠ADB = 60°
∠BDC = 30°
Now, let the diagonals AC and BD intersect at O.
The angle between the diagonals is:
∠AOB = 60°
Vertically opposite angles are equal.
Therefore,
∠COD = 60°
The adjacent angles form a straight angle.
Therefore,
∠AOD = 180° − 60°
∠AOD = 120°
Similarly,
∠BOC = 120°
Hence, the four angles at the intersection of the diagonals are:
60°, 120°, 60°, 120°
Answer for (i):
The angles formed by the diagonals at the corners are 30° and 60°.
The angles at the intersection of the diagonals are:
60°, 120°, 60°, 120°
(ii) Rectangle PQRS
Given:
PQRS is a rectangle.
The diagonals PR and QS intersect at O.
∠QOR = 110°.
Solution:
Since ∠QOR = 110°,
and vertically opposite angles are equal,
∠POS = 110°
The adjacent angles form a straight angle.
Therefore,
∠QOR + ∠QOP = 180°
110° + ∠QOP = 180°
∴ ∠QOP = 70°
Similarly,
∠ROS = 70°
Therefore, the four angles at the intersection of the diagonals are:
110°, 70°, 110°, 70°
Now, since the diagonals are symmetrically placed in the rectangle, each diagonal makes half of the 110° angle with the vertical direction.
Therefore,
110° ÷ 2 = 55°
Thus, each diagonal makes an angle of 55° with the vertical side.
Since the angle of a rectangle is 90°,
Angle with the horizontal side = 90° − 55°
= 35°
Therefore, the diagonal divides each corner of the rectangle into:
35° and 55°
Answer for (ii):
The angles formed by the diagonals at each corner are:
35° and 55°
The angles at the intersection of the diagonals are:
110°, 70°, 110°, 70°
FINAL ANSWERS
(i) Rectangle ABCD
Corner angles formed by the diagonals:
30°, 60°, 30°, 60°, 30°, 60°, 60°, 30°
Angles at the intersection:
60°, 120°, 60°, 120°
(ii) Rectangle PQRS
Corner angles formed by the diagonals:
35° and 55°
Angles at the intersection:
110°, 70°, 110°, 70°