Look at these animals opening their mouths. Do you see any angles here? If yes, mark the arms and vertex of each one. Can you arrange the angles in this picture from smallest to largest?
Answer
Yes. Every open mouth makes an angle.
Vertex: the hinge of the jaw, where the upper and lower jaws are joined.
Arms: the upper jaw and the lower jaw.
To arrange them, look at how much each jaw has been turned: the wider the mouth is open, the greater the turn and the larger the angle. Trace the angles on tracing paper and place them one over the other to be sure — the widest gape goes last in the list.
Careful: a crocodile's jaw is long and a bird's beak is short, but length has nothing to do with it. A small bird with a wide-open beak can make a bigger angle than a big animal with a slightly open mouth.
Q2.
Is it always easy to compare two angles?
Answer
No. Comparing is easy only when the two angles are clearly different.
A 30° angle and a 120° angle can be told apart at a glance.
But an 89° angle and a 91° angle look almost the same — the eye cannot decide.
Two angles of nearly the same size, drawn with arms of different length. The eye alone cannot decide which is bigger.
What to do then: use one of two reliable methods — superimposition (trace one angle and place it exactly over the other, vertex on vertex) or measurement with a protractor. Long arms can also trick the eye into thinking an angle is bigger, which is one more reason not to trust a quick look.
Q3.
Here are some angles. Label each of the angles. How will you compare them? Draw a few more angles; label them and compare.
Answer
First give every angle a name using three letters, with the vertex in the middle — for example ∠ABC, ∠PQR, ∠XYZ.
Then compare them in any of these ways:
By looking — works only when the difference is large.
By superimposition — trace one angle on tracing paper, place its vertex exactly on the other vertex and one arm along the other arm, then see which second arm lies inside.
Using a circle — place a transparent circle with its centre on the vertex and see how much of the circle each angle covers.
By measuring — read both angles with a protractor and compare the numbers. This is the surest method.
Math Talk: draw four angles of your own, label them, first guess the order from smallest to largest, then check by measuring. How many did you get right?
Q4.
Where else do we use superimposition to compare?
Answer
Superimposition — placing one thing exactly over another — is used to compare many shapes and sizes:
Line segments — put one over the other with the left ends together and see which one sticks out.
Circles — place them centre on centre; the one whose boundary lies outside is bigger.
Squares, triangles and other figures — to check whether two cut-outs are of exactly the same size and shape.
Leaves, footprints, bangles, coins — placed one over the other to compare size.
Tracing a map or a drawing — to check that a copy matches the original.
Did you know? Two figures that fit exactly on each other by superimposition are called congruent figures. You will study them in higher classes.