From facts to proofs
In earlier classes you learnt facts about angles by measuring. In Class 9 you begin to prove them, starting from a few basic statements accepted as true, called axioms, and deducing others, called theorems.
Key terms
Acute (< 90°), right (= 90°), obtuse (between 90° and 180°), straight (= 180°) and reflex (between 180° and 360°) angles.
Complementary angles add to 90°; supplementary angles add to 180°.
Adjacent angles share a vertex and an arm, and their other arms lie on opposite sides of the common arm.
A linear pair is a pair of adjacent angles whose non-common arms form a straight line.
The linear pair axiom
Axiom (linear pair). If a ray stands on a line, the sum of the two adjacent angles so formed is 180°.
Axiom (converse). If the sum of two adjacent angles is 180°, their non-common arms form a line.
Together these are the linear pair axioms.
Theorem: vertically opposite angles are equal
Statement. If two lines intersect, the vertically opposite angles are equal.
Proof. Let lines AB and CD intersect at O, forming ∠AOC, ∠COB, ∠BOD and ∠DOA.
Ray OA stands on line CD, so ∠AOC + ∠AOD = 180° (linear pair axiom). … (1)
Ray OD stands on line AB, so ∠AOD + ∠BOD = 180°. … (2)
From (1) and (2), ∠AOC + ∠AOD = ∠AOD + ∠BOD, so ∠AOC = ∠BOD.
Similarly, ∠AOD = ∠BOC. ∎
Parallel lines and a transversal
When a transversal crosses two lines, it forms corresponding angles, alternate interior angles, alternate exterior angles, and interior angles on the same side.
Axiom (corresponding angles). If a transversal intersects two parallel lines, each pair of corresponding angles is equal.
Axiom (converse of corresponding angles). If a transversal intersects two lines so that a pair of corresponding angles is equal, the lines are parallel.
From these, the following theorems are proved:
If the lines are parallel, then … | Conversely, the lines are parallel if … |
|---|---|
alternate interior angles are equal | a pair of alternate interior angles is equal |
interior angles on the same side are supplementary | a pair of same-side interior angles is supplementary |
Proof that alternate interior angles are equal. Let the transversal make corresponding angles ∠1 and ∠5, and let ∠3 be vertically opposite ∠1, so that ∠3 and ∠5 are alternate interior angles. ∠1 = ∠5 (corresponding angles) and ∠1 = ∠3 (vertically opposite). So ∠3 = ∠5. ∎
Theorem. Lines parallel to the same line are parallel to each other.
Angle sum property of a triangle
Theorem. The sum of the angles of a triangle is 180°.
Proof. In triangle PQR, draw a line XY through P parallel to QR.
∠XPQ = ∠PQR (alternate interior angles, XY ∥ QR).
∠YPR = ∠PRQ (alternate interior angles).
At P, ∠XPQ + ∠QPR + ∠YPR = 180° (angles on the straight line XY).
Substituting, ∠PQR + ∠QPR + ∠PRQ = 180°. ∎
Exterior angle theorem. If one side of a triangle is extended, the exterior angle equals the sum of the two interior opposite angles. It follows from two facts: the exterior angle and its adjacent interior angle add to 180° (a linear pair), and the three interior angles also add to 180°. Subtracting the shared adjacent angle from both leaves exterior angle = sum of the other two interior angles.
Worked examples
Example. In the figure, AB ∥ CD and a transversal makes ∠x = 3a + 10° and its same-side interior partner ∠y = 2a + 20°. Find a.
Same-side interior angles are supplementary: 3a + 10 + 2a + 20 = 180, so 5a = 150 and a = 30°. Then ∠x = 100° and ∠y = 80°.
Example. The angles of a triangle are in the ratio 2 : 3 : 4. Then 2k + 3k + 4k = 180°, so k = 20°, and the angles are 40°, 60° and 80°.
Summary
Linear pair axiom: angles on a straight line add to 180°.
Vertically opposite angles are equal (proved from the linear pair axiom).
With parallel lines, corresponding and alternate interior angles are equal and same-side interior angles are supplementary; the converses test for parallel lines.
The angles of a triangle add to 180°, and an exterior angle equals the sum of the two interior opposite angles.