Understanding Cyclic Quadrilaterals and Equilateral Triangles
The problem involves a cyclic quadrilateral ABCD and an equilateral triangle ABC. We need to find the measure of angle CDA.
Properties Utilized
- Equilateral Triangle: All angles in an equilateral triangle measure $60^{\circ}$.
- Cyclic Quadrilateral: The sum of opposite angles in a cyclic quadrilateral is $180^{\circ}$.
Step-by-Step Solution
-
Since triangle ABC is equilateral, all its angles are $60^{\circ}$. Therefore, $\angle ABC = 60^{\circ}$.
-
ABCD is a cyclic quadrilateral. A property of cyclic quadrilaterals is that opposite angles sum to $180^{\circ}$. This means $\angle ABC + \angle CDA = 180^{\circ}$.
-
Substitute the known value of $\angle ABC$ into the equation:
$ 60^{\circ} + \angle CDA = 180^{\circ} $
-
Solve for $\angle CDA$:
$ \angle CDA = 180^{\circ} - 60^{\circ} $
$ \angle CDA = 120^{\circ} $
Thus, the angle CDA is $120^{\circ}$.
Similar Questions
- The angles of a quadrilateral are in the ratio of 1 : 3 : 4 : 7. What is the difference between the largest and the smallest angle?
- If one angle of a parallelogram is $39^\circ$ less than twice the smallest angle, then the smallest angle of the parallelogram is:
- Find the area of a rhombus whose perimeter is 164 cm and one diagonal is of length 80 cm.
- Find the area of a trapezium whose parallel sides are 10 cm and 20 cm and non-parallel sides are equal to 10 cm.
- If one angle of a parallelogram is $18^\circ$ less than twice the smallest angle of the same parallelogram, then the largest angle of the parallelogram is:
- If one angle of a parallelogram is $48^\circ$ less than twice the smallest angle, then the measure of the largest angle of the parallelogram will be:
- If one angle of a parallelogram is $30^\circ$ less than twice the measure of the smallest angle, then the measure of the largest angle of the parallelogram is:
- In a rhombus ABCD, if $\angle ACB = 40^\circ, \text{then } \angle ADB = ?$
- P is the mid-point of side BC of a parallelogram ABCD such that $\angle BAP = \angle DAP$.
If AD = 10 cm, then CD = ? - The angles of a quadrilateral are in the ratio of 1 : 2 : 3 : 4. What is the measure of the greatest of these angles?
Important Questions from Quadrilaterals
The ratio between the length and breadth of a rectangular park is 3 : 2. If a man cycling along the boundary at the speed of 12 km per hour completes one round in 8 minutes, then the area of the park in square meter will be
The side of a rhombus is 26 cm. The length of one of its diagonals is 20 cm. The sum of the lengths of the diagonals of this rhombus is equal to the perimeter of a rectangle. If the difference between the length and breadth of the rectangle is 6 cm, then what is the area of the rectangle?
PQRS is a cyclic quadrilateral. If ∠P is 4 times ∠R, and ∠S is 3 times ∠Q, then the average of ∠Q and ∠R is:
ABCD is a trapezium in which AB || DC and DC is perpendicular to BC. If ∠DAB = 110°, then ∠ABC - ∠ADC =_____.
If the quadrilateral has an inscribed circle, then the sum of a pair of opposite sides equals:
