Math Problem Statement

∠RQT is a straight angle. What are m∠RQS and m∠TQS?

Solution

The problem involves determining the measures of angles RQS\angle RQS and TQS\angle TQS, given that RQT\angle RQT is a straight angle (180°). Additionally, you are provided with expressions for the angles:

  • mRQS=15x+1m\angle RQS = 15x + 1
  • mTQS=10x+4m\angle TQS = 10x + 4

The total measure of the straight angle is:

mRQS+mTQS=180m\angle RQS + m\angle TQS = 180^\circ

Step 1: Set up the equation

Substitute the given expressions for mRQSm\angle RQS and mTQSm\angle TQS:

(15x+1)+(10x+4)=180(15x + 1) + (10x + 4) = 180

Step 2: Simplify the equation

Combine like terms:

25x+5=18025x + 5 = 180

Step 3: Solve for xx

Subtract 5 from both sides:

25x=17525x = 175

Divide by 25:

x=7x = 7

Step 4: Find the measures of the angles

Substitute x=7x = 7 back into the expressions for mRQSm\angle RQS and mTQSm\angle TQS:

  1. mRQS=15(7)+1=105+1=106m\angle RQS = 15(7) + 1 = 105 + 1 = 106^\circ
  2. mTQS=10(7)+4=70+4=74m\angle TQS = 10(7) + 4 = 70 + 4 = 74^\circ

Step 5: Verify the result

Check that the sum equals 180°:

106+74=180106^\circ + 74^\circ = 180^\circ

The solution is correct.

Final Answer:

  • mRQS=106m\angle RQS = 106^\circ
  • mTQS=74m\angle TQS = 74^\circ

Would you like further clarification or additional help?


Here are 5 related questions to deepen your understanding:

  1. What is the definition of a straight angle, and how is it applied in such problems?
  2. How would you set up equations for other types of angles (e.g., complementary or supplementary)?
  3. If mTQSm\angle TQS were given instead, how would you solve for mRQSm\angle RQS?
  4. What happens if you receive no algebraic expressions, just numerical values for these angles?
  5. Can this problem be extended to a triangle? If so, how?

Tip: Always double-check your solution by adding up all angle measures to verify the problem's conditions are met!

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Math Problem Analysis

Mathematical Concepts

Algebra
Angles in Geometry
Straight Angles

Formulas

Sum of angles in a straight angle: m∠RQS + m∠TQS = 180°
Expression substitution and solving equations

Theorems

Straight Angle Theorem

Suitable Grade Level

Grades 7-10