Answer :
To solve this problem, we need to find the sizes of the angles of the parallelogram given the specific relationship between them. Here's a detailed step-by-step solution:
1. Define the Angles:
- Let the smallest angle be [tex]\( x \)[/tex].
- According to the problem, one angle is 40° less than three times the smallest angle. Therefore, the other angle can be represented as [tex]\( 3x - 40 \)[/tex].
2. Property of Parallelograms:
- In a parallelogram, opposite angles are equal.
- Consecutive angles in a parallelogram are supplementary, which means their measures add up to 180°.
3. Set Up the Equation:
- Since the consecutive angles are supplementary, we set up the following equation:
[tex]\[ x + (3x - 40) = 180 \][/tex]
4. Solve the Equation:
- Combine like terms:
[tex]\[ x + 3x - 40 = 180 \][/tex]
[tex]\[ 4x - 40 = 180 \][/tex]
- Add 40 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[ 4x = 220 \][/tex]
- Divide by 4 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = 55 \][/tex]
5. Determine the Other Angle:
- Substitute [tex]\( x = 55 \)[/tex] back into the expression for the other angle:
[tex]\[ 3x - 40 = 3(55) - 40 = 165 - 40 = 125 \][/tex]
6. Verify the Angles:
- The two angles we found are 55° and 125°.
- Check that they are supplementary:
[tex]\[ 55° + 125° = 180° \][/tex]
This confirms that our solution is correct.
Therefore, the sizes of any two angles of the parallelogram that satisfy the given relationship are:
[tex]\[ \boxed{55°, 125°} \][/tex]
Thus, the correct answer is:
d) 55°, 125°
1. Define the Angles:
- Let the smallest angle be [tex]\( x \)[/tex].
- According to the problem, one angle is 40° less than three times the smallest angle. Therefore, the other angle can be represented as [tex]\( 3x - 40 \)[/tex].
2. Property of Parallelograms:
- In a parallelogram, opposite angles are equal.
- Consecutive angles in a parallelogram are supplementary, which means their measures add up to 180°.
3. Set Up the Equation:
- Since the consecutive angles are supplementary, we set up the following equation:
[tex]\[ x + (3x - 40) = 180 \][/tex]
4. Solve the Equation:
- Combine like terms:
[tex]\[ x + 3x - 40 = 180 \][/tex]
[tex]\[ 4x - 40 = 180 \][/tex]
- Add 40 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[ 4x = 220 \][/tex]
- Divide by 4 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = 55 \][/tex]
5. Determine the Other Angle:
- Substitute [tex]\( x = 55 \)[/tex] back into the expression for the other angle:
[tex]\[ 3x - 40 = 3(55) - 40 = 165 - 40 = 125 \][/tex]
6. Verify the Angles:
- The two angles we found are 55° and 125°.
- Check that they are supplementary:
[tex]\[ 55° + 125° = 180° \][/tex]
This confirms that our solution is correct.
Therefore, the sizes of any two angles of the parallelogram that satisfy the given relationship are:
[tex]\[ \boxed{55°, 125°} \][/tex]
Thus, the correct answer is:
d) 55°, 125°
