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Zorluk: OrtaQuadrilaterals and Polygons

In a parallelogram, the measure of one interior angle is 30∘30^\circ less than twice the measure of the consecutive interior angle. What is the measure of the larger angle?

  1. 110∘110^\circCevap
  2. B
    50∘50^\circ
  3. C
    130∘130^\circ
  4. D
    230∘230^\circ

Cevap

The measure of the larger angle is 110∘110^\circ.
The correct answer correctly identifies that consecutive angles in a parallelogram are supplementary. Setting up the equation x+(2x−30)=180∘x + (2x - 30) = 180^\circ and solving for xx yields 70∘70^\circ. Substituting this back into the expression for the larger angle gives 2(70)−30=110∘2(70) - 30 = 110^\circ.

Adım Adım Çözüm

1
Identify the geometric property relating the angles.
Consecutive interior angles of a parallelogram are supplementary, meaning they add up to 180∘180^\circ.
This is a fundamental property of parallelograms formed by parallel lines cut by a transversal.
2
Translate the word problem into an algebraic equation.
Let the smaller angle be xx. The larger angle is 2x−302x - 30. Therefore, x+(2x−30)=180x + (2x - 30) = 180.
This captures both the relationship described ('30∘30^\circ less than twice') and the supplementary geometric property.
3
Solve the algebraic equation for xx.
Combine terms: 3x−30=1803x - 30 = 180. Add 3030 to both sides: 3x=2103x = 210. Divide by 33: x=70∘x = 70^\circ.
Isolating the variable determines the exact measure of the smaller angle.
4
Calculate the measure of the larger angle.
2(70)−30=140−30=110∘2(70) - 30 = 140 - 30 = 110^\circ.
The question specifically asks for the measure of the larger angle, not the smaller one.

Anahtar Kavram

Properties of Parallelograms and Algebraic Translation

Alternatif Yöntem

You can solve this by testing the answer choices. If the larger angle is 110∘110^\circ, the smaller consecutive angle must be 180∘−110∘=70∘180^\circ - 110^\circ = 70^\circ. Check if 110∘110^\circ is 30∘30^\circ less than twice 70∘70^\circ: 2(70)−30=140−30=110∘2(70) - 30 = 140 - 30 = 110^\circ. The values perfectly match the problem statement.
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Quadrilaterals and Polygons

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