Geometry: Supplementary and Complementary Angles

Definitions of Complementary and Supplementary Angles

• Complementary Angles: The sum of complementary angles is equal to $90^\circ$.

  • This is known as a right angle.

• Supplementary Angles: The sum of supplementary angles is equal to $180^\circ$.

  • This is known as a straight angle.

Angle Construction via Compass

• Instructions are provided to draw specific angles using a compass rather than a protractor. The required angles for construction include:

  • $40^\circ$

  • $75^\circ$

  • $30^\circ$

Geometrical Calculations for Unknown Angles

• Scenario Background: A diagram involves intersecting lines where one angle is given as $110^\circ$, and variables $x$, $y$, and $z$ represent unknown angles.

• Finding Angle x:

  • $x = 110^\circ$

  • Reasoning: This is based on the property of vertically opposite angles. When two lines intersect, the angles opposite each other at the vertex are equal.

• Finding Angle y:

  • The relationship between the given angle ($110^\circ$) and $y$ is that they lie on a straight line, making them supplementary.

  • Equation: $x + y = 180^\circ$

  • Substitution: $110^\circ + y = 180^\circ$

  • Calculation: $y = 180^\circ - 110^\circ$

  • Result: $y = 70^\circ$

• Finding Angle z:

  • Result: $y = z = 70^\circ$

  • Reasoning: Angle $z$ is vertically opposite to angle $y$, and since $y$ was calculated as $70^\circ$, $z$ must also be $70^\circ$.