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If one side of a parallelogram is $24 °$ less than twice the smallest angle, then the measure of the largest angle of the parallelogram from the option given below is:

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176 °

68 °

112 °

102 °

answer is C.

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Detailed Solution

Construct a parallelogram

A B C D

, as shown in the figure given below.
The opposite angles of a parallelogram are equal.
So,

∠ A = ∠ C, ∠ B = ∠ D

.
So, the measures of the angles of parallelogram

A B C D

are

x, 2 x − 24, x, a n d 2 x − 24

Calculate the value of

x

.
By angle sum property of the quadrilateral,

∠ A + ∠ B + ∠ C + ∠ D = 360 ° ⇒ x + 2 x − 24 + x + 2 x − 24 = 360 ° ⇒ 6 x − 48 = 360 ° ⇒ 6 x = 408 °

⇒ x = 68 °

Substitute the value of

x

.
The measure of the other angle is given by:

(2 x − 24) ° = (2 × 68 − 24) ° (2 x − 24) ° = 112 °

Hence, the measure of the largest angle is

112 °

.
Hence, option 3) is correct.

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