Parallel Lines and Angles Problems

Parallel lines and angles problems with detailed solutions are presented below.

Problem 1: In the figure below, AA' is parallel to CC'. The size w of angle A'AB is equal to 135 degrees and the size z of angle C'CB is equal to 147 degrees. Find angle ABC.

parallel lines and angles problem 1


Solution to Problem 1:

  • Draw BB' parallel to AA' and CC'as shown in the figure below.

    parallel lines and angles, solution to problem 1


  • Note that angle ABC is given by

    angle ABC = angle ABB' + angle CBB'

  • Angle w' and angle ABB' are alternate interior angles and their sizes are equal.

    angle ABB' = angle w'

  • Angle z' and angle CBB' are alternate interior angles and their sizes are equal.

    angle CBB' = angle z'

  • Angles w and w' are supplementary which gives

    w' = 180 - w = 180 - 135 = 45 0

  • Angles z and z' are also supplementary which gives

    z' = 180 - z = 180 - 147 = 33 0

  • We now substitute angle ABB' by w' and angle CBB' by z' in angle ABC = angle ABB' + angle CBB' found above.

    angle ABC = w' + z' = 45 + 33 = 78 0

Problem 2: In the figure below lines A'A" and C'C" are parallel. AB is the bisector of angle CAA" and BC is the bisector of angle ACC". Show that the size of angle ABC is equal to 90 degrees.

parallel lines and angles problem 2


Solution to Problem 2:

  • Angles A'AC and angle ACC" are alternate interior angles and their sizes are equal.

    angle A'AC = angle ACC"

  • Angles A'AC and angle ACC" are alternate interior angles and their sizes are equal.

    angle A'AC = angle ACC"

  • Angles A'AC and angle A"AC are supplementary so that

    angle A"AC = 180 - angle A'AC = 180 - angle ACC"

  • Rearrange the above to obtain

    angle A"AC + angle ACC" = 180 0

  • Because AB and CB are bisectors(they divide the angle into two equal angles), angle ABC in triangle ABC is given by

    angle ABC = 180 - (angle A"AC + angle ACC") / 2

    = 180 - 180 / 2 = 90 0

Problem 3: In the figure below lines BC and DD' are parallel. The size of angle x is equal to 127 degrees and the size of angle y is equal to 115 degrees. Find all interior angles of triangle ADD'.

parallel lines and angles problem 3


Solution to Problem 3:

  • Angle x and angle ABC are supplementary hence

    angle ABC = 180 - x = 180 - 127 = 53 0

  • Angle y and angle ACB are supplementary hence

    angle ACB = 180 - y = 180 - 115 = 65 0

  • Angle ADD' and angle ABC are corresponding angles and their sizes are equal

    angle ADD' = angle ABC = 53 0

  • Angle AD"D and angle ACB are corresponding angles and their sizes are equal

    angle AD'D = angle ACB = 65 0

  • Angle DAD' is given by

    angle DAD' = 180 - (angle ADD' + angle AD'D)

    = 180 - 118 = 62 0

More references on geometry tutorials and problems.


Geometry Tutorials and Problems

Triangle Problems