Examkin

Question

Difficulty: HardLines and Angles

Angles ∠P\angle P and ∠Q\angle Q are supplementary angles. The measure of ∠P\angle P is given by the expression (7x+20)∘(7x + 20)^\circ, and the measure of ∠Q\angle Q is given by (3x−10)∘(3x - 10)^\circ. What is the measure of the angle that is complementary to ∠Q\angle Q?

  1. 49∘49^\circAnswer
  2. B
    139∘139^\circ
  3. C
    73∘73^\circ
  4. D
    29∘29^\circ

Answer

The complementary angle measures 49∘49^\circ.
The sum of supplementary angles is 180∘180^\circ. Setting (7x+20)+(3x−10)=180(7x + 20) + (3x - 10) = 180 yields 10x+10=18010x + 10 = 180, which simplifies to 10x=17010x = 170, meaning x=17x = 17. Substituting x=17x = 17 into the expression for ∠Q\angle Q gives 3(17)−10=41∘3(17) - 10 = 41^\circ. The complement of an angle sums to 90∘90^\circ, so the complementary angle is 90∘−41∘=49∘90^\circ - 41^\circ = 49^\circ.

Step-by-Step Solution

1
Set up an equation based on the definition of supplementary angles.
(7x+20)+(3x−10)=180(7x + 20) + (3x - 10) = 180
Supplementary angles have a sum of 180∘180^\circ.
2
Combine like terms and solve for xx.
10x+10=180⇒10x=170⇒x=1710x + 10 = 180 \Rightarrow 10x = 170 \Rightarrow x = 17
This isolates the variable xx to determine its numerical value.
3
Substitute the value of xx into the expression for ∠Q\angle Q.
∠Q=3(17)−10=51−10=41∘\angle Q = 3(17) - 10 = 51 - 10 = 41^\circ
This finds the exact degree measure of ∠Q\angle Q.
4
Calculate the complement of ∠Q\angle Q.
90∘−41∘=49∘90^\circ - 41^\circ = 49^\circ
Complementary angles have a sum of 90∘90^\circ.

Key Concept

Solving multi-step algebraic equations derived from the properties of supplementary and complementary angles.
Rate this question

Topics

Lines and Angles

More questions from this topic