Geometry & Measures
Grade 4-5Angles in Parallel Lines
Once a transversal crosses a pair of parallel lines, a whole new set of angle facts opens up, and examiners regularly test whether you can name the correct angle rule as well as calculate the answer. This lesson covers corresponding, alternate and co-interior angles, using these facts together to solve multi-step problems, and testing whether two lines are actually parallel.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Corresponding and alternate angles
Corresponding angles are in matching positions at each intersection, forming an F shape, and are equal. Alternate angles are on opposite sides of the transversal between the two parallel lines, forming a Z shape, and are also equal.
If one angle where a transversal crosses the first parallel line is 75°, the corresponding angle at the second line is also 75°, and the alternate angle on the opposite side of the transversal between the two lines is also 75°.
Co-interior angles
Co-interior angles (sometimes called allied angles) are between the two parallel lines, on the same side of the transversal, forming a C or U shape, and they sum to 180° rather than being equal.
If one angle between the parallel lines is 105°, the co-interior angle on the same side is 180 - 105 = 75°.
Combining angle facts
Many exam questions require more than one angle fact in sequence, such as using vertically opposite angles to find one angle before applying co-interior angles to find another.
If y is vertically opposite a 70° angle, then y = 70°. If x is co-interior with y, then x = 180 - 70 = 110°.
Testing whether lines are parallel
To test whether two lines cut by a transversal are parallel, check whether the angle facts actually hold, for example whether a pair of co-interior angles genuinely sums to 180°.
If two angles that would need to be co-interior are 115° and 70°, they sum to 185°, not 180°, so the lines are not parallel.
Worked Examples
Three exam-style questions, fully solved.
The diagram shows two parallel lines crossed by a straight line. One angle where the line crosses the first parallel line is 75°. Find the corresponding angle a at the second parallel line.
Easy- 1.Identify that a is in the corresponding position to the 75° angle
- 2.Corresponding angles are equal
Answer: a = 75°
The diagram shows two parallel lines crossed by a straight line, forming a 70° angle. Angle y is vertically opposite the 70° angle, and angle x is co-interior with y. Find the values of x and y.
Medium- 1.Vertically opposite angles are equal, so y = 70°
- 2.Co-interior angles sum to 180°: x + y = 180
- 3.Substitute y = 70°: x = 180 - 70
Answer: y = 70°, x = 110°
The diagram shows two parallel lines crossed by a straight line. One angle is 4x - 10, and the corresponding angle at the other parallel line is 70°. Find the value of x.
Hard- 1.Corresponding angles are equal, so set up an equation: 4x - 10 = 70
- 2.Add 10 to both sides: 4x = 80
- 3.Divide both sides by 4: x = 80 ÷ 4
Answer: x = 20
Avoid These
The most common mistakes students make.
Confusing alternate angles (Z shape, equal) with co-interior angles (C shape, sum to 180°).
Assuming two angles are corresponding or alternate just because they look similar in the diagram, without checking they are in the correct matching or opposite positions.
Forgetting to combine angle facts in multi-step questions, stopping after the first angle fact instead of using it to find the next required angle.
When testing whether lines are parallel, forgetting to state the angle fact that would need to hold, and only giving a numerical comparison without the reasoning.
Not stating the angle rule used as a reason, such as "alternate angles are equal", which can cost a mark even when the numerical answer is correct.
FAQ
Questions parents and students ask.
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