Pairs of Angles
Digital Handwritten Lesson
New Millennium Academy
Birauta, Pokhara-17, Kaski
1. Introduction to Lines
Activity- Lines in fig–(I) are apart by a fixed distance.
- If those lines are extended, there is no possibility that they will ever meet.
- Lines in fig–(II) meet at a point O.
- The lines in fig–I are parallel lines.
- The lines in fig–II are intersecting lines.
• A pair of lines which never meet, however far extended, is called a pair of parallel lines.
• A pair of lines which meet (or cross) at a point is called a pair of intersecting lines.
Real-life examples: (i) Railway tracks – parallel lines. (ii) Two roads crossing at a junction – intersecting lines.
2. Transversal Line
Activity- AB and CD are a pair of lines.
- A line EH intersects this pair at points F and G.
- PQ and RS are a pair of lines.
- A line UX intersects this pair at points V and W.
- In each figure, 8 angles are formed at the two points of intersection.
- All 8 angles have distinct names.
- Some angles are formed inside the pair of lines, and some are formed outside the pair of lines.
- The line EH (fig–I) and UX (fig–II) are transversal lines.
- The angles formed inside the pair of lines are interior angles.
- The angles formed outside the pair of lines are exterior angles.
A line which intersects a pair of lines, forming several angles at the two points of intersection, is called a transversal line.
Table Activity: Complete the table below for the two figures of the previous activity.
| Fig. | Pair of lines | Transversal | Interior angles | Exterior angles |
|---|---|---|---|---|
| I | AB and CD | EH | \(\angle AFG, \angle BFG\) \(\angle FGC, \angle FGD\) |
\(\angle AFE, \angle EFB\) \(\angle CGH, \angle DGH\) |
| II | PQ and RS | UX | \(\angle PVW, \angle QVW\) \(\angle RWV, \angle SWV\) |
\(\angle PVU, \angle QVU\) \(\angle RWX, \angle XWS\) |
3. Corresponding Angles
ActivityThe teacher draws these four "F-shaped" cut-outs on the board:
Students fit each shape onto the transversal figure below and write the names of the angles it traces:
- Shape (I) traces \(\angle EFB\) and \(\angle FGD\).
- Shape (II) traces \(\angle AFE\) and \(\angle FGC\).
- Shape (III) traces \(\angle BFG\) and \(\angle DGH\).
- Shape (IV) traces \(\angle AFG\) and \(\angle CGH\).
- In every pair, one angle is interior and the other is exterior.
- In every pair, both angles lie on the same side of the transversal line.
- Each such pair of angles is called a pair of corresponding angles.
A pair of angles formed on the same side of a transversal, one interior and one exterior, and which are not adjacent to each other, is called a pair of corresponding angles.
Corresponding angles trace shapes like F, reversed-F, and similar "F-type" outlines.
4. Alternate Angles
ActivityThe teacher draws these two "Z-shaped" cut-outs on the board:
Students fit each shape onto the transversal figure below and write the names of the angles it traces:
- Shape (I) traces \(\angle AGH\) and \(\angle GHD\).
- Shape (II) traces \(\angle BGH\) and \(\angle GHC\).
- In every pair, both angles are interior angles.
- In every pair, the two angles lie on opposite sides of the transversal line EF.
- Each such pair of angles is called a pair of alternate angles.
A pair of interior angles formed on opposite sides of a transversal, which are not adjacent to each other, is called a pair of alternate angles.
Alternate angles trace a Z-shaped (or reversed Z) outline.
5. Co-interior Angles
ActivityThe teacher, for the last time, draws these two "C-shaped" cut-outs on the board:
Students fit each shape onto the transversal figure below and write the names of the angles it traces:
- Shape (I) traces \(\angle BFG\) and \(\angle FGD\).
- Shape (II) traces \(\angle AFG\) and \(\angle FGC\).
- In every pair, both angles are interior angles.
- In every pair, the two angles lie on the same side of the transversal line EH.
- Each such pair of angles is called a pair of co-interior angles.
A pair of interior angles formed on the same side of a transversal, which are not adjacent to each other, is called a pair of co-interior angles.
Co-interior angles trace a "C-shaped" (or reversed C) outline – like a bracket pair [ ].
6. Experiment – Angles on Parallel Lines
Objective: To examine the relation between the pairs of angles formed when a transversal cuts a pair of parallel lines.
Using a protractor, measure all eight angles formed at G and H in each figure, and record the results below.
| Fig. No. | \(\angle AGE\) | \(\angle EGB\) | \(\angle AGH\) | \(\angle BGH\) | \(\angle GHC\) | \(\angle GHD\) | \(\angle CHF\) | \(\angle FHD\) |
|---|---|---|---|---|---|---|---|---|
| I | ||||||||
| II | ||||||||
| III |
(a) Relation between corresponding angles on parallel lines
- (i) \(\angle EGB\) and \(\angle GHD\) are corresponding angles, and are found to be equal.
- (ii) \(\angle AGE\) and \(\angle GHC\) are corresponding angles, and are found to be equal.
- (iii) \(\angle AGH\) and \(\angle CHF\) are corresponding angles, and are found to be equal.
- (iv) \(\angle BGH\) and \(\angle FHD\) are corresponding angles, and are found to be equal.
(b) Relation between alternate angles on parallel lines
- (i) \(\angle AGH\) and \(\angle GHD\) are alternate angles, and are found to be equal.
- (ii) \(\angle BGH\) and \(\angle GHC\) are alternate angles, and are found to be equal.
(c) Relation between co-interior angles on parallel lines
- (i) \(\angle AGH\) and \(\angle GHC\) are co-interior angles, and their sum is found to be \(180^{\circ}\).
- (ii) \(\angle BGH\) and \(\angle GHD\) are co-interior angles, and their sum is found to be \(180^{\circ}\).
Course material curated by Mr. Nripendraswar Acharya