Examkin

Question

Difficulty: HardProperties of Quadrilaterals

In parallelogram ABCDABCD, the ratio of the measure of angle AA to the measure of angle BB is 2:32:3. The measure of angle CC is 1212 degrees less than 33 times the value of xx. What is the value of xx?

  1. 28Answer
  2. B
    40
  3. C
    -20
  4. D
    52
  5. E
    44

Answer

28
The correct answer is 28. In a parallelogram, consecutive angles are supplementary, so the measure of angle AA and the measure of angle BB must sum to 180∘180^\circ. Based on the given ratio of 2:32:3, the measure of angle AA is calculated as 25×180∘=72∘\frac{2}{5} \times 180^\circ = 72^\circ. Since opposite angles in a parallelogram are equal, the measure of angle CC is also 72∘72^\circ. The description '12 degrees less than 3 times the value of xx' translates to the expression 3x−123x - 12. Equating this to 72∘72^\circ gives 3x−12=723x - 12 = 72, which simplifies to 3x=843x = 84 and yields x=28x = 28.

Step-by-Step Solution

1
Determine the measures of angles AA and BB using their ratio and the properties of a parallelogram.
The measure of angle AA is 72∘72^\circ and the measure of angle BB is 108∘108^\circ.
Adjacent angles in a parallelogram are supplementary, meaning their sum is 180∘180^\circ. Given the ratio of the measure of angle AA to the measure of angle BB is 2:32:3, we can express their measures as 2y2y and 3y3y respectively. Solving 2y+3y=180∘2y + 3y = 180^\circ gives 5y=180∘  ⟹  y=36∘5y = 180^\circ \implies y = 36^\circ. Therefore, the measure of angle AA is 2(36∘)=72∘2(36^\circ) = 72^\circ and the measure of angle BB is 3(36∘)=108∘3(36^\circ) = 108^\circ.
2
Relate the measure of angle CC to the calculated angle measures.
The measure of angle CC is 72∘72^\circ.
Opposite angles of a parallelogram are equal in measure. Since the measure of angle AA is 72∘72^\circ, the measure of the opposite angle CC must also be 72∘72^\circ.
3
Set up and solve the algebraic equation to find the value of xx.
x=28x = 28
We are given that the measure of angle CC is 1212 degrees less than 33 times the value of xx, which translates to 3x−123x - 12. Setting this expression equal to 72∘72^\circ gives the equation 3x−12=723x - 12 = 72. Adding 1212 to both sides yields 3x=843x = 84, and dividing by 33 gives x=28x = 28.

Key Concept

Properties of Parallelograms
Estimated Time:1m 30s
Rate this question

Topics

Properties of Quadrilaterals

More questions from this topic