LINES AND ANGLES
Frequently Asked Questions
A line is a straight, endless path extending in both directions. It is defined by two points and has only length .
An angle is formed when two rays diverge from a common point called the vertex. Angles are measured in degrees .
A ray starts from one point and extends infinitely in one direction, whereas a line segment connects two points and has a definite length .
Collinear points lie on the same line; non-collinear points do not lie on the same line .
Acute (0°–90°), Right (90°), Obtuse (90°–180°), Straight (180°), Reflex (180°–360°), Complete (360°) .
Complementary angles sum to 90°, supplementary angles sum to 180° .
Adjacent: Share a common arm and vertex; Vertically opposite: Formed when two lines intersect, equal in measure; Linear pair: Two adjacent angles whose non-common arms form a line (sum = 180°) .
The sum of the angles in any triangle is \(180^\circ\) .
The sum of interior angles of an n-sided polygon is \((n-2) \times 180^\circ(n-2)\)
A transversal is a line that intersects two or more lines at distinct points .
Corresponding, Alternate Interior, Alternate Exterior, Co-interior (Consecutive Interior) angles .
Corresponding angles are equal; Alternate interior angles are equal; Co-interior angles are supplementary .
The sum of the angles forming a linear pair is \(180^\circ\)
Vertically opposite angles formed by intersecting lines are equal .
Used in engineering, architectural design, navigation, and geographical surveys .
Use a protractor for measuring and drawing angles accurately .
Use angle relationships: If alternate interior angle \(x\), then corresponding angle also \(x\); Co-interior angles sum to 180° .
\(90^\circ - 36^\circ = 54^\circ\)
Let angles be \(3x,\,7x,\,8x\); so
\[\begin{aligned}3x + 7x + 8x &= 180\\ \Rightarrow 18x &= 180\\ \Rightarrow x&=10
\end{aligned}\] angles are 30°, 70°, 80° .
\[\begin{aligned}3x + 7x + 8x &= 180\\ \Rightarrow 18x &= 180\\ \Rightarrow x&=10
\end{aligned}\] angles are 30°, 70°, 80° .
The two lines are parallel to each other .
Ruler, protractor, compass for drawing and measuring lines and angles .
Place compass at vertex, draw arcs cutting both arms, measure width between intersections, draw arc from each intersection, join vertex to intersecting point of new arcs .
Linear pair axiom, vertically opposite angles, parallel lines with transversal properties, angle sum property of triangle .
Vertically opposite angles, linear pairs; both properties tested in board and competitive exams .
They form a linear pair (sum = 180°) .
Let angle \(=x\). Complement =\[\begin{aligned}90°-x\\\Rightarrow x&=(1/2)(90°-x)+30°\\x& = (1/2)(90^\circ - x) + 30^\circ\\\Rightarrow 2x&=90°-x+60°\\2x &= 90^\circ - x + 60^\circ\\\Rightarrow 3x &= 150^\circ\\3x&=150°\\\Rightarrow x &= 50^\circ\end{aligned}\] .
