3.3.5. Angles of a parallelogram

We studied a property of parallelograms concerning the (opposite) sides. What can we say about the angles?
DO THIS
Let ABCD be a parallelogram (Fig 3.17). Copy it on a tracing sheet. Name this copy as A′B′C′D′. Place A′B′C′D′ on ABCD. Pin them together at the point where the diagonals meet. Rotate the transparent sheet by 180°. The parallelograms still coincide; but you now find A′ lying exactly on C and vice-versa; similarly B′ lies on D and vice-versa.
Does this tell you anything about the measures of the angles A and C? Examine the same for angles B and D. State your findings.

Property: The opposite angles of a parallelogram are of equal measure.

In a parallelogram, opposite angles have the same measure. To understand this, imagine taking a parallelogram ABCD and tracing it onto another sheet as A'B'C'D'. By pinning them together at the intersection of the diagonals and rotating the sheet, we can see that the points A' and C coincide, indicating that angles A and C are equal. The same applies for angles B and D. Therefore, opposite angles in a parallelogram must always be equal.

Think of a parallelogram like a pair of opposing magnets; when you turn one magnet to line it up with the other, they match perfectly. Just as those magnets hold their positions when they're aligned, the angles at opposite corners of the parallelogram remain equal, no matter how you turn it.

TRY THESE
Take two identical 30° – 60° – 90° set-squares and form a parallelogram as before. Does the figure obtained help you to confirm the above property?
You can further justify this idea through logical arguments.
If AC and BD are the diagonals of the parallelogram, (Fig 3.18) you find that ∠1 =∠2 and ∠3 = ∠4 (Why?)
Property: The adjacent angles in a parallelogram are supplementary.

Adjacent angles in a parallelogram are supplementary, meaning that the sum of each pair of adjacent angles equals 180 degrees. When you look at the diagonals of the parallelogram, angles ∠1 and ∠2 are formed by one pair of adjacent sides, while ∠3 and ∠4 form another pair. Since they share a side and are on opposite sides of the diagonal, they must sum up to 180 degrees. Thus, adjacent angles in a parallelogram are always supplementary.

Imagine sitting at a rectangular table. If you're at one corner and your friend is sitting at the opposite corner, the angles formed by the corners around you add up to make a straight line (180 degrees). Every time you turn to face another corner, those angles you see are still together making a straight line. This is similar to how adjacent angles in a parallelogram relate to each other.

Example 4: In Fig 3.20, BEST is a parallelogram. Find the values x, y and z.
Solution: S is opposite to B. So, x =100° (opposite angles property) y =100° (measure of angle corresponding to ∠x) z =80° (since ∠y, ∠z is a linear pair).

To find the angles in parallelogram BEST, we apply the properties we discussed. Knowing that opposite angles are equal, we determine that if angle S is opposite to B and equals 100°, then ∠B must also equal 100°. Since angles y and z are adjacent, they must add up to 180° because of the supplementary property. Therefore, if angle y is 100°, then angle z = 180° − 100° = 80°.

Think of following a set of rules in a board game. If you know one player has 100 points (angle of x), then by the rules of the game, the player across from them must also have 100 points (angle of y). The points next to the first player then must total to the whole set of 180 points available, meaning they lose some of their points (angle of z). This reflects how angles work in a parallelogram!

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