Angles Created When Parallel Lines are Cut by a Transversal

Angles Created When Parallel Lines are Cut by a Transversal exercise

• Vertically opposite angles.

  Hints

  Vertically opposite angles are congruent (the same size). They form an X shape.

  These angles are vertically opposite.

  • $x = x$
  • $y = y$

  Solution

  $c$ is vertically opposite to $a$, $a = c$.

  The opposite angles in a cross (X shape) are the same size or congruent.

• Find the supplemetary angles.

  Hints

  Supplementary angles form a straight line as their sum is $180^\circ$. Therefore, we are looking for angles which would add to $a$ and make a straight line.

  We can see on the image $x$ and $y$ are supplementary.

  $x + y = 180$.

  There are $3$ correct answers here.

  Solution

  The supplementary angles to $a$ are $b$, $d$ and $f$.

  • $a + b = 180$
  • $a + d = 180$
  • $a + f = 180$

• What are the values of the angles?

  Hints

  Angles in a straight line are supplementary and have a sum of $180^\circ$. We can find a missing angle by doing the calculation $180 - 50 = 130$.

  Interior alternate angles are congruent and can be found forming a Z shape on the opposite sides of the transversal line.

  Solution

  • $110^\circ$ is corresponding with the angle of $110^\circ$ at the bottom.
  • $70^\circ$ is in a straight line with $110^\circ$.
  • $95^\circ$ is interior alternate with the angle of $95^\circ$.
  • $85^\circ$ is a straight line with $95^\circ$.

• What is the angle?

  Hints

  Angles are congruent if they are the same. All the angles marked in green are congruent to $106^\circ$.

  Angles are supplementary if their sum is $180^\circ$. All the angles marked in green are supplementary with $106^\circ$.

  Solution

  $106^\circ$ (Exterior alternate angles)

  • The angle is congruent to $106^\circ$
  • Both angles are on the exterior of the parallel lines
  • Both angles are on the opposite side of the transverse line.

• Congruent angles.

  Hints

  Alternate angles are on the opposite side of the transverse line. The interior ones form a Z shape.

  Corresponding angles are in the same position and can form an F shape with each other.

  Exterior alternate angles are on the opposite side of the transverse line and the exterior of the parallel lines.

  Vertically opposite angles are on the opposite side forming an X shape.

  Solution

  • Interior alternate angles form a Z shape on the opposite side of the transverse line.
  • Exterior alternate angles are on the exterior opposite sides of the transverse line.
  • Corresponding angles angles form an F shape in the same positions.
  • Vertically opposite angles form an X shape and are opposite each other.

• Find the missing angle.

  Hints

  To help with this question, we need to use the fact that the sum of angles in a triangle is $180^\circ$. For example, in the image here, we need to find the values of $b$ and $c$ in order to find $a$.

  To find the angles at $b$ and $c$, we are using the angle facts we know. For example, congruent angles that have the same value as a given angle and supplementary angles, which sum to $180^\circ$.

  Solution

  $a = 32^\circ$

  • $65^\circ$ is in a straight line with $115^\circ$, this is supplementary $(180^\circ - 115^\circ = 65^\circ)$
  • $83^\circ$ is vertically opposite to $83^\circ$, this is congruent
  • $a = 32^\circ$, a triangle with $65^\circ$ and $83^\circ$ (angles in a triangle sum to $180^\circ$)
  • $180^\circ - 65^\circ - 83^\circ = 32^\circ$.