Axiom 3: If a transversal intersects two parallel lines, then each pair of corresponding angles is equal. Here, Exterior angles are ∠1, ∠2, ∠7 and ∠8Interior angles are ∠3, ∠4, ∠5 and ∠6Corresponding angles are ∠(i) ∠1 and ∠5(ii) ∠2 and ∠6(iii) ∠4 and ∠8 (iv) ∠3 and ∠7 Axiom 4 If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel to each other. Thus, (i) ∠1 = ∠5, (ii) ∠2 = ∠6, (iii) ∠4 = ∠8 and (iv) ∠3 = ∠7Alternate Interior Angles: (i) ∠4 and ∠6 and (ii) ∠3 and ∠5Alternate Exterior Angles: (i) ∠1 and ∠7 and (ii) ∠2 and ∠8If a transversal intersects two parallel lines then each pair of alternate interior and exterior angles are equal.Alternate Interior Angles: (i) ∠4 = ∠6 and (ii) ∠3 = ∠5Alternate Exterior Angles: (i) ∠1 = ∠7 and (ii) ∠2 = ∠8 Interior angles on the same side of the transversal line are called the consecutive interior angles or allied angles or co-interior angles. They are as follows: (i) ∠4 and ∠5, and (ii) ∠3 and ∠6 Theorem 2 If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal. Solution: Given: Let PQ and RS are two parallel lines and AB be the transversal which intersects them on L and M respectively. To Prove: ∠PLM = ∠SML Proof: ∠PLM = ∠RMB ………….equation (i) (Corresponding ngles)∠RMB = ∠SML ………….equation (ii) (vertically opposite angles)From equation (i) and (ii) ∠PLM = ∠SML Similarly, ∠LMR = ∠ALP ……….equation (iii) (corresponding angles)∠ALP = ∠MLQ …………equation (iv) (vertically opposite angles)From equation (iii) and (iv) ∠LMR = ∠MLQ Proved Theorem 3: If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel. Solution: Given: - A transversal AB intersects two lines PQ and RS such that To Prove: PQ ||RSUse same figure as in Theorem 2.Proof: ∠PLM = ∠SML ……………equation (i) (Given)∠SML = ∠RMB …………equation (ii) (vertically opposite angles)From equations (i) and (ii); ∠PLM = ∠RMB But these are corresponding angles.We know that if a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines ate parallel to each other. Hence, PQ║RS Proved. Theorem 4: If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary. Solution:Solution: Given: Transversal EF intersects two parallel lines AB and CD at G and H respectively. Proof: ∠2 + ∠5 = 180° ………equation (i) (Linear pair of angles)But ∠5 = ∠3 ……………equation (ii) (corresponding angles)From equations (i) and (ii),∠2 + ∠3 = 180°Also, ∠3 + ∠4 = 180° ………equation (iii) (Linear pair)But ∠3 = ∠1 …………..equation (iv) (Alternate interior angles)From equations (iii) and (iv) ∠1 + ∠4 = 180° and ∠2 + ∠3 = 180° Proved Theorem 5: If a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the two lines are parallel. Solution: Given: A transversal EF intersects two lines AB and CD at P and Q respectively. Proof: ∠1 + ∠2 = 180° ………..equation (i) (Given)∠1 + ∠3 = 180° …………..equation (ii) (Linear Pair)From equations (i) and (ii)∠1 + ∠2 = ∠1 + ∠3Or, ∠1 + ∠2 - ∠1 = ∠3 Or, ∠2 = ∠3 But these are alternate interior angles. We know that if a transversal intersects two lines such that the pair of alternate interior angles are equal, then the lines are parallel. Theorem 6: Lines which are parallel to the same line are parallel to each other. Solution: Given: Three lines AB, CD and EF are such that AB║CD, CD║EF. ∠1 = ∠2 ………….equation (i) (corresponding angles)Similarly, CD ||EF and GH is transversal. Therefore;∠2 = ∠3 ……………equation (ii) (corresponding angles)From equations (i) and (ii) ∠1 = ∠3 But these are corresponding angles.We know that if a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines ate parallel to each other. Hence, AB║ EF Proved. Angle Sum Property of Triangle:Theorem 7: The sum of the angles of a triangle is 180º. Solution: Given: Δ ABC.To Prove: ∠1 + ∠2 + ∠3 = 180° Construction: Let us draw a line m though A, parallel to BC. Proof: BC ||m and AB and AC are its transversal.Hence, ∠1 = ∠4 …………….equation (i) (alternate interior angles)∠2 = ∠5 ………..equation (ii) (alternate interior angles)By adding equation (i) and (ii)∠1 + ∠2 = ∠4 + ∠5 ………..equation (iii)Now, by adding ∠3 to both sides of equation (iii), we get∠1 + ∠2 + ∠3 = ∠4 + ∠5 + ∠3Since, ∠4 + ∠5 + ∠ = 180° (Linear group of angle)Hence, ∠1 + ∠2 + ∠3 = 180° Hence Proved. Theorem 8: If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles. Solution: Given: ΔABDC in which side BC is produced to D forming exterior angle ∠ACD of ΔABC. Proof: Since, ∠1 + ∠2 + ∠3 = 180°…………equation (i) (angle sum of triangle)∠2 + ∠4 = 180° ………….equation (ii) (Linear pair)From equations (i) and (ii)∠1 + ∠2 + ∠3 = ∠3 + ∠4Or, ∠1 + ∠2 + ∠3 - ∠3 = ∠4Or, ∠1 + ∠2 = ∠4 Hence, ∠4 = ∠1 + ∠2 Proved Copyright © excellup 2014
In geometry, a transversal is a line that intersects two or more other (often parallel ) lines. In the figure below, line n is a transversal cutting lines l and m .
When two or more lines are cut by a transversal, the angles which occupy the same relative position are called corresponding angles . In the figure the pairs of corresponding angles are: ∠ 1 and ∠ 5 ∠ 2 and ∠ 6 ∠ 3 and ∠ 7 ∠ 4 and ∠ 8 When the lines are parallel, the corresponding angles are congruent . When two lines are cut by a transversal, the pairs of angles on one side of the transversal and inside the two lines are called the consecutive interior angles . In the above figure, the consecutive interior angles are: ∠ 3 and ∠ 6 ∠ 4 and ∠ 5 If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles formed are supplementary . When two lines are cut by a transversal, the pairs of angles on either side of the transversal and inside the two lines are called the alternate interior angles . In the above figure, the alternate interior angles are: ∠ 3 and ∠ 5 ∠ 4 and ∠ 6 If two parallel lines are cut by a transversal, then the alternate interior angles formed are congruent . When two lines are cut by a transversal, the pairs of angles on either side of the transversal and outside the two lines are called the alternate exterior angles . In the above figure, the alternate exterior angles are: ∠ 2 and ∠ 8 ∠ 1 and ∠ 7 If two parallel lines are cut by a transversal, then the alternate exterior angles formed are congruent .
Example 1:
In the above diagram, the lines j and k are cut by the transversal l . The angles ∠ c and ∠ e are… A. Corresponding Angles B. Consecutive Interior Angles C. Alternate Interior Angles D. Alternate Exterior Angles The angles ∠ c and ∠ e lie on either side of the transversal l and inside the two lines j and k . Therefore, they are alternate interior angles. The correct choice is C .
Example 2:
In the above figure if lines A B ↔ and C D ↔ are parallel and m ∠ A X F = 140 ° then what is the measure of ∠ C Y E ? The angles ∠ A X F and ∠ C Y E lie on one side of the transversal E F ↔ and inside the two lines A B ↔ and C D ↔ . So, they are consecutive interior angles. Since the lines A B ↔ and C D ↔ are parallel, by the consecutive interior angles theorem , ∠ A X F and ∠ C Y E are supplementary. That is, m ∠ A X F + m ∠ C Y E = 180 ° . But, m ∠ A X F = 140 ° . Substitute and solve. 140 ° + m ∠ C Y E = 180 ° 140 ° + m ∠ C Y E − 140 ° = 180 ° − 140 ° m ∠ C Y E = 40 ° |