Answer:
The measures of the angles of the rhombus are 72° , 108° , 72° , 108°
Step-by-step explanation:
* Lets revise the properties of the rhombus
- It has 4 equal sides
- Each two opposite angles are equal
- Each two adjacent angles are supplementary (their sum = 180°)
- The diagonals are perpendicular to each other
- The diagonals bisect the vertices angles (divide each vertex into two
equal parts)
* Now lets solve the problem
- Let the name of the rhombus is ABCD
- The two diagonals are AC and BD
- Let the diagonals AC and BD make the angles BAC and ABD with
side AB
∵ The diagonals of the rhombus bisect the vertex angles
∴ m∠BAC is half m∠A
∴ m∠ABD is half m∠B
- There is a ratio between the measures of the angles BAC and ABD
∵ m∠BAC : m∠ABD = 6 : 9
∵ m∠A + m∠B = 180° ⇒ consecutive angles
∴ m∠BAC + m∠ABD = 1/2 × 180° = 90°
∵ m∠BAC : m∠ABD : their sum
6 : 9 : 15⇒(6 + 9)
? : ? : 90° ⇒(the sum of the angles)
∴ m∠BAC = 6/15 × 90° = 36°
∴ m∠ABD = 9/15 × 90° = 54°
∵ m∠BAC is half m∠A
∴ m∠A = 36° × 2 = 72°
∵ m∠ABD is half m∠B
∴ m∠B = 54° × 2 = 108°
* The measures of the angles of the rhombus are 72° , 108° , 72° , 108°
