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Difficulty: MediumLines and Angles

Match each angle relationship description with the correct calculated value for its variable.

  • Angles AA and BB are vertical angles. m∠A=(4x−12)∘m\angle A = (4x - 12)^\circ and m∠B=(2x+28)∘m\angle B = (2x + 28)^\circ. Find xx.20
  • Angles CC and DD are complementary. m∠C=(4y+10)∘m\angle C = (4y + 10)^\circ and m∠D=(2y+20)∘m\angle D = (2y + 20)^\circ. Find yy.10
  • Angles EE and FF are same-side interior angles between parallel lines. m∠E=(3z+10)∘m\angle E = (3z + 10)^\circ and m∠F=(z+70)∘m\angle F = (z + 70)^\circ. Find zz.25
  • Angles GG and HH are corresponding angles formed by parallel lines. m∠G=(5w−10)∘m\angle G = (5w - 10)^\circ and m∠H=(3w+50)∘m\angle H = (3w + 50)^\circ. Find ww.30

Answer

The correct matches are: x=20x = 20, y=10y = 10, z=25z = 25, and w=30w = 30.
Each relationship dictates a specific equation setup. Vertical and corresponding angles require setting the expressions equal to each other. Complementary angles require their sum to equal 90∘90^\circ, while same-side interior angles require their sum to equal 180∘180^\circ.

Step-by-Step Solution

1
Determine the relationship and setup the equation for the vertical angles.
4x−12=2x+284x - 12 = 2x + 28
Vertical angles are congruent (equal in measure).
2
Solve the equation for x.
x=20x = 20
Subtract 2x2x from both sides (2x−12=282x - 12 = 28), add 12 to both sides (2x=402x = 40), and divide by 2.
3
Determine the relationship and setup the equation for the complementary angles.
(4y+10)+(2y+20)=90(4y + 10) + (2y + 20) = 90
Complementary angles have a sum of 90∘90^\circ.
4
Solve the equation for y.
y=10y = 10
Combine like terms (6y+30=906y + 30 = 90), subtract 30 (6y=606y = 60), and divide by 6.
5
Determine the relationship and setup the equation for the same-side interior angles.
(3z+10)+(z+70)=180(3z + 10) + (z + 70) = 180
Same-side interior angles between parallel lines are supplementary (sum to 180∘180^\circ).
6
Solve the equation for z.
z=25z = 25
Combine like terms (4z+80=1804z + 80 = 180), subtract 80 (4z=1004z = 100), and divide by 4.
7
Determine the relationship and setup the equation for the corresponding angles.
5w−10=3w+505w - 10 = 3w + 50
Corresponding angles between parallel lines are congruent (equal in measure).
8
Solve the equation for w.
w=30w = 30
Subtract 3w3w from both sides (2w−10=502w - 10 = 50), add 10 (2w=602w = 60), and divide by 2.

Key Concept

Geometric angle relationships (vertical, complementary, supplementary, corresponding, same-side interior) can be modeled and solved using linear algebraic equations.
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Topics

Lines and Angles

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