Vertically Opposite & Parallel Lines

Section 1 of 2

Vertically Opposite Angles

Cross → equal

When two straight lines cross, the angles opposite each other are equal.

Worked example — straight line and opposite

Find $A$ and $B$.

$A$ is opposite $50^\circ$, so $A = 50^\circ$… here $A$ sits on a straight line with $50^\circ$:

$A = 180 - 50 = 130^\circ$; $B = 50^\circ$ (vertically opposite)

$A = 130^\circ,\ B = 50^\circ$

You try

Two lines cross. $3x$ is vertically opposite $30^\circ$, and $5y$ sits on a straight line with $30^\circ$. Find $x$ and $y$.

Vertically opposite are equal; a straight line adds to $180$.

$3x = 30 \Rightarrow x = 10$

$5y + 30 = 180 \Rightarrow 5y = 150 \Rightarrow y = 30$

$x = 10,\ y = 30$

Section 2 of 2

Parallel Lines

Corresponding & alternate

A line crossing two parallel lines is a transversal. Corresponding angles (same position, an F-shape) are equal. Alternate angles (a Z-shape) are equal.

Worked example — corresponding & straight line

A transversal cuts two parallel lines, making an angle of $50^\circ$. Find $A$, $B$ and $C$.

$B = 50^\circ$ (corresponding to the $50^\circ$)

$A = 180 - 50 = 130^\circ$ (straight line)

$C = 130^\circ$ (vertically opposite $A$)

$A = 130^\circ,\ B = 50^\circ,\ C = 130^\circ$

Worked example — alternate angles

Find $A$ (a Z-shape between two parallel lines, with $53^\circ$ given).

$A$ and $53^\circ$ are alternate angles

$A = 53^\circ$

That’s Class 2.

Vertically opposite (equal), corresponding (F) and alternate (Z) angles. Class 3: angles inside a triangle.