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Parallel and Intersecting Lines - Alternate Angles

Grade 7CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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When a transversal tt intersects two parallel lines ll and mm, the alternate interior angles formed on opposite sides of the transversal and between the lines are equal. For example, ∠4=∠6\angle 4 = \angle 6 and ∠3=∠5\angle 3 = \angle 5.

Diagram showing parallel lines l and m with transversal t, highlighting interior angles 3, 4, 5, and 6.
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Alternate exterior angles are those on opposite sides of the transversal and outside the parallel lines. These angles are also equal when the lines are parallel (∠1=∠7 \angle 1 = \angle 7 and ∠2=∠8\angle 2 = \angle 8).

Diagram showing exterior angles 1, 2, 7, and 8 outside the parallel lines.
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The 'Z' shape rule: A quick way to identify alternate interior angles is to look for a 'Z' or 'N' shape. The angles inside the corners of the 'Z' are the alternate interior angles.

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Converse property: If two lines are intersected by a transversal and a pair of alternate interior (or alternate exterior) angles are equal, then the two lines must be parallel.

📐Formulae

If l∥m, then ∠Alternate Interior 1=∠Alternate Interior 2\text{If } l \parallel m, \text{ then } \angle \text{Alternate Interior 1} = \angle \text{Alternate Interior 2}

If l∥m, then ∠Alternate Exterior 1=∠Alternate Exterior 2\text{If } l \parallel m, \text{ then } \angle \text{Alternate Exterior 1} = \angle \text{Alternate Exterior 2}

Condition for Parallelism: ∠3=∠5 or ∠4=∠6  ⟹  l∥m\text{Condition for Parallelism: } \angle 3 = \angle 5 \text{ or } \angle 4 = \angle 6 \implies l \parallel m

💡Examples

Problem 1:

In a figure, two parallel lines ll and mm are intersected by a transversal tt. If one of the alternate interior angles is 125∘125^{\circ}, find the measure of the other alternate interior angle xx.

Solution:

x=125∘x = 125^{\circ}

Explanation:

According to the property of parallel lines, when two lines are parallel (l∥ml \parallel m), the alternate interior angles are always equal. Therefore, the value of the other angle is also 125∘125^{\circ}.

Problem 2:

Two parallel lines are cut by a transversal. The alternate interior angles are (2x+10)∘(2x + 10)^{\circ} and 70∘70^{\circ}. Find the value of xx.

Solution:

2x+10=702x=70−102x=60x=30\begin{array}{r} 2x + 10 = 70 \\ 2x = 70 - 10 \\ 2x = 60 \\ x = 30 \end{array}

Explanation:

Since the lines are parallel, the alternate interior angles must be equal. We set up the equation (2x+10)=70(2x + 10) = 70 and solve for xx by subtracting 1010 from both sides and then dividing by 22.

Problem 3:

One angle in a pair of alternate exterior angles is 180∘−75∘180^{\circ} - 75^{\circ}. Calculate the value of the other alternate exterior angle yy using vertical arithmetic for the subtraction.

Solution:

The first angle is: 180−75105\begin{array}{r} 180 \\ - 75 \\ \hline 105 \end{array} Therefore, y=105∘y = 105^{\circ}.

Explanation:

Alternate exterior angles are equal when lines are parallel. First, we calculate the value of the given angle using subtraction: 180−75=105180 - 75 = 105. Since they are alternate exterior angles, yy must also be 105∘105^{\circ}.

Problem 4:

In the given figure, p∥qp \parallel q. If ∠1=65∘\angle 1 = 65^{\circ}, find the value of the alternate interior angle ∠x\angle x.

A transversal crossing two parallel lines p and q, showing an angle of 65 degrees and its alternate interior angle x.

Solution:

  1. Given that p∥qp \parallel q.
  2. ∠1\angle 1 and ∠x\angle x are alternate interior angles because they lie on opposite sides of the transversal and between the parallel lines.
  3. By the property of parallel lines, alternate interior angles are equal.
  4. Therefore, ∠x=∠1=65∘\angle x = \angle 1 = 65^{\circ}.

Explanation:

Since the lines are parallel, we apply the alternate interior angle theorem which states that such angles are equal in measure.

Problem 5:

Identify if lines aa and bb are parallel if the alternate exterior angles are 110∘110^{\circ} and (150−40)∘(150 - 40)^{\circ}.

Two lines a and b with a transversal showing exterior angles of 110 and 150-40 degrees.

Solution:

  1. First, calculate the value of the second angle using vertical subtraction: 150−40110\begin{array}{r} 150 \\ - 40 \\ \hline 110 \end{array}
  2. The first alternate exterior angle is 110∘110^{\circ}.
  3. The second alternate exterior angle is 110∘110^{\circ}.
  4. Since the pair of alternate exterior angles are equal (110∘=110∘110^{\circ} = 110^{\circ}), the lines aa and bb must be parallel.

Explanation:

By the converse of the alternate exterior angles theorem, if the angles are equal, the lines are parallel.

Alternate Angles Class 7 Notes & Examples | CBSE Maths