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Zorluk: KolayLines and Angles

Translate the following geometric relationships into algebraic equations and perform the first step of simplification. Pair each geometric description on the left with its corresponding simplified equation on the right.

  • An angle measuring (3x)∘(3x)^\circ and its complement measuring 30∘30^\circ.3x=603x = 60
  • An angle measuring (3x)∘(3x)^\circ and its supplement measuring 30∘30^\circ.3x=1503x = 150
  • Two intersecting lines forming vertical angles of (3x)∘(3x)^\circ and 30∘30^\circ.x=10x = 10
  • Two adjacent angles on a straight line measuring (3x)∘(3x)^\circ and (2x+30)∘(2x + 30)^\circ.5x=1505x = 150

Cevap

Match the complementary angles with 3x=603x = 60; the supplementary angles with 3x=1503x = 150; the vertical angles with x=10x = 10; and the adjacent angles on a straight line with 5x=1505x = 150.
Each geometric term dictates a specific mathematical equality: 'complement' means the sum is 90∘90^\circ, 'supplement' and 'straight line' mean the sum is 180∘180^\circ, and 'vertical angles' means the measures are equal to each other. Applying these definitions and performing basic algebraic simplification yields the correct matches.

Adım Adım Çözüm

1
Formulate and simplify the equation for the complementary angles.
The setup is 3x+30=903x + 30 = 90, which simplifies to 3x=603x = 60.
Complementary angles always add up to 90∘90^\circ.
2
Formulate and simplify the equation for the supplementary angles.
The setup is 3x+30=1803x + 30 = 180, which simplifies to 3x=1503x = 150.
Supplementary angles always add up to 180∘180^\circ.
3
Formulate and simplify the equation for the vertical angles.
The setup is 3x=303x = 30, which simplifies to x=10x = 10.
Vertical angles (angles opposite each other at an intersection) are geometrically equal.
4
Formulate and simplify the equation for the adjacent angles on a straight line.
The setup is 3x+(2x+30)=1803x + (2x + 30) = 180. Combining the xx terms gives 5x+30=1805x + 30 = 180, and subtracting 30 gives 5x=1505x = 150.
Angles that form a linear pair on a straight line must sum to 180∘180^\circ.

Anahtar Kavram

Translating Geometric Angle Relationships into Algebraic Equations
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Konular

Lines and Angles

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