Lines and Angles · Lesson 3 of 9
Comparing Angles
“Compare openings fairly and discover why longer arms do not make a larger angle.”
• Compare two angles by aligning their vertices and one arm. • Recognise equality when both arms coincide after superimposition. • Use a transparent circle to transfer and compare an opening. • Explain why arm length and the orientation of a drawing do not determine angle size. • Interpret the rotating-arm and slit investigations.
What Makes One Opening Larger?
Two animals can open their mouths by different amounts. The directions of the upper and lower jaws model angle arms, and the place where the jaws turn models a vertex. A wider turn gives a larger angle. However, pictures may show animals of different sizes or facing different directions, which makes a quick visual judgement unreliable.
Before comparing, imagine the same starting direction for both angles. Compare how far the second arm turns away from it. The distance between the tips of two drawn arms is not a fair measure, because the tips may be at different distances from the vertex. We need a method that compares the openings themselves.
Superimposition: Put One Angle over the Other
Superimposition means placing one figure on top of another. Trace an angle onto transparent paper and move the tracing onto the second angle. First match the vertices. Then turn the tracing so that one arm lies along one arm of the other angle, with both openings on the same side. The positions of the other arms now show which turn is larger.
- Trace one angle, including its vertex and the directions of both arms.
- Place its vertex directly on the other vertex.
- Rotate the tracing until one corresponding pair of arms coincides.
- Compare the other arms: the one turned farther from the shared baseline gives the larger angle.
Angles of the same size. After matching their vertices and one pair of arms, the other arms also coincide, so the required turns are equal.
Equal angles do not need equal visible arm lengths. An arm is a ray, so it continues beyond the part drawn. If a long drawn arm lies in the same direction as a short drawn arm, they can represent the same ray direction. Likewise, tilting the whole angle on the page does not change its opening.
Problem
One angle has long drawn arms and points upward. Another has shorter drawn arms and is tilted. After tracing and alignment, both pairs of arms coincide. Which angle is larger?
- 1.Align the vertices, then place one arm of each angle on the same baseline.
- 2.The second arms coincide as well, so the turns from the baseline are equal.
- 3.Neither is larger. Different visible arm lengths and different orientations do not change the angle size.
A wider-looking gap between distant arm tips can come from longer drawn arms. Angle size changes only when one arm turns relative to the other. Shortening an arm without changing its direction does not reduce the angle.
Comparing with a Transparent Circle
A movable transparent circle can record an opening without tracing the entire angle. Put the circle centre on the vertex. Record the two places where the arms meet its edge, calling them A and B. These locations preserve the directions from the centre to the two arms.
Move the same circle to the other angle. Keep its centre on the new vertex and align the recorded direction OA with one arm. Look at the other arm: does it reach before, at, or beyond the recorded direction OB on the intended side? This answers whether the second angle is smaller, equal, or larger. Using the same circle matters because it keeps the comparison consistent.
Problem
You record the first crane’s angle on a transparent circle. After aligning it over the second crane’s angle, the second jaw direction lies beyond the recorded second direction. What can you conclude?
- 1.The circle centre and one recorded direction have been aligned, giving a common starting point and baseline.
- 2.The other direction for the second crane requires more turning than the recorded opening.
- 3.The second crane makes the larger angle. This conclusion compares the jaw directions rather than the birds’ sizes.
Investigating with Rotating Arms
Make a simple angle model using two paper straws and the two sides of a paper clip. Fit a straw over each side so that the straws can open around the clip. The joint models the vertex and the straw directions model the arms. Make several different openings and order them using superimposition.
For the slit investigation, trace one fixed pair of arms onto cardboard and cut an angle-shaped slit. Mix several pairs of fixed arms, keeping their openings unchanged while testing them. Match the vertex and one arm to the slit, then check the other arm. In this flat matching test, a smaller opening misses one slit branch, and a larger opening misses it on the other side.
If both arms align with the slit, the openings are equal. Their lengths can differ as long as the straws fit within the slit’s lengths and widths. This condition separates angle equality from practical size restrictions. Use the model to test a claim, rather than treating any failed physical fit as proof that the angle differs.
Problem
A student is asked to reduce the opening of a fixed straw model and cuts the straws shorter. Has the angle been reduced?
- 1.Identify what changed: the visible arm lengths became shorter.
- 2.The directions of the two arms relative to one another did not change.
- 3.The turn is unchanged, so the angle has not been reduced. The student must rotate one arm toward the other to reduce it.
You can also fold a rectangular sheet diagonally and compare the angles between the crease and the paper edges. Try several different folds. Trace an angle if moving the sheet alone would make comparison confusing. Explain which vertex and arm you align; a reasoned comparison is more useful than a guess.
Quiz
What must be matched first when superimposing two angles?
After vertices and one pair of arms coincide, the other pair also coincides. The angles are:
A drawn arm is lengthened along the same direction. What happens to the angle?
Why should both openings lie on the same side of the shared baseline when comparing?
What does a transparent-circle comparison preserve?
Fixed arms match both branches of an angle-shaped slit. What follows, assuming their lengths and widths fit?
Practice Problems
- Draw two angles facing different directions. Trace one and describe how you would compare them fairly.
- Draw equal angles with visibly different arm lengths. Explain why their turns are equal.
- Use a transparent circle to compare two angle drawings. State what you align at each step.
- Make three rotating-arm models. Order them by opening and check by superimposition.
- Explain why both a smaller and a larger fixed angle can fail the angle-shaped slit test.
- Fold a rectangular sheet in three different ways. Compare the crease-and-edge angles you produce.
- A student compares only the gap between two arm tips. Give a drawing or explanation showing why that can be misleading.
- Describe two changes that preserve angle size and one change that alters it.
Key Takeaways
• Compare angles by matching the vertices and one arm, then comparing the other arms. • Both arm directions coincide when aligned angles are equal. • Arm length and the orientation of the whole drawing do not determine angle size. • A transparent circle records directions and helps transfer an opening. • The rotating-arm model changes its angle only when one arm turns relative to the other.