Angles at intersecting lines

What is it about?

Two lines intersect at the point $S$ . This forms four angles $\alpha$ , $\beta$ , $\gamma$ , $\delta$ (anticlockwise).

Vertical angles lie opposite each other at the intersection. They are equal: $\alpha = \gamma$ and $\beta = \delta$ .
Adjacent angles lie side by side on a line and together form a straight angle. They add up to $180^\circ$ : $\alpha + \beta = 180^\circ$ .

Example: if the lines intersect at $\alpha = 65^\circ$ , then $\gamma = 65^\circ$ and $\beta = \delta = 180^\circ - 65^\circ = 115^\circ$ . All four angles together make $360^\circ$ . If the lines are perpendicular, all four angles are $90^\circ$ .

Transversal: two lines $g$ and $h$ are crossed by a third line $s$ . There are four angles at each crossing: $\alpha_1, \beta_1, \gamma_1, \delta_1$ at $g$ and $\alpha_2, \beta_2, \gamma_2, \delta_2$ at $h$ .

Pair of angles	Position	Shape	if $g \parallel h$
corresponding angles	same position at both crossings	F	equal
alternate angles	opposite sides of $s$ , both between $g$ and $h$ (or both outside)	Z	equal
co-interior angles	same side of $s$ , both between $g$ and $h$ (or both outside)	C (in German books: E)	add up to $180^\circ$

Why? If you slide the crossing at $g$ along $s$ until it lies on the crossing at $h$ , then $g$ covers $h$ exactly when $g \parallel h$ – and then the corresponding angles cover each other too. Alternate angles are vertical angles of corresponding angles: $\gamma_2 = \alpha_2 = \alpha_1$ . Conversely: if corresponding (or alternate) angles are equal, then $g$ and $h$ are parallel.

g \parallel h \;\Longrightarrow\; \alpha_1 = \alpha_2, \qquad \alpha_1 = \gamma_2, \qquad \alpha_1 + \delta_2 = 180^\circ

Try it

Drag the handles on $g$ and $h$ . Which angles always stay equal? What do two neighbouring angles add up to? Load example
Press “Join adjacent angles” and “Turn vertical angle”. Tap a different angle first.
Set up perpendicular lines and almost parallel lines. What happens to the adjacent angles?
Switch to the transversal. Press “Slide along s”: the crossing at $g$ slides upwards. What do you notice?
Choose alternate angles and press “Turn by 180°”.
Switch off “g ∥ h” (example) and slide again. How big is the difference?
Give a reason for every angle in the exercise mode. Compare with the reason in the side panel.

Tasks

Task 1: Four angles

Two lines intersect at $\alpha = 38^\circ$ . Find $\beta$ , $\gamma$ and $\delta$ and give reasons. Load task

Show solution

$\gamma = 38^\circ$ (vertical angle to $\alpha$ ), $\beta = 180^\circ - 38^\circ = 142^\circ$ (adjacent angle to $\alpha$ ), $\delta = 142^\circ$ (vertical angle to $\beta$ ). Check: $38^\circ + 142^\circ + 38^\circ + 142^\circ = 360^\circ$ . Show in the simulation

Task 2: Three times as big

An angle is three times as big as its adjacent angle. How big are the two angles?

Show solution

Adjacent angle $x$ , angle $3x$ : $x + 3x = 180^\circ$ , so $4x = 180^\circ$ and $x = 45^\circ$ . The angles are $45^\circ$ and $135^\circ$ . Show in the simulation

Task 3: Eight angles at parallels

$g \parallel h$ , the transversal $s$ crosses $g$ at $\alpha_1 = 62^\circ$ . Find all eight angles. Load task

Show solution

$\gamma_1 = 62^\circ$ (vertical angle), $\beta_1 = \delta_1 = 180^\circ - 62^\circ = 118^\circ$ (adjacent angles). Because $g \parallel h$ , corresponding angles are equal: $\alpha_2 = \gamma_2 = 62^\circ$ , $\beta_2 = \delta_2 = 118^\circ$ . So every angle is either $62^\circ$ or $118^\circ$ . Show in the simulation

Task 4: A chain of reasons

$g \parallel h$ and $\beta_2 = 70^\circ$ . Find $\alpha_1$ and give a reason for every step. Load task

Show solution

$\alpha_2 = 180^\circ - 70^\circ = 110^\circ$ (adjacent angle to $\beta_2$ ), then $\alpha_1 = \alpha_2 = 110^\circ$ (corresponding angles, $g \parallel h$ ). Other ways are just as correct, e.g. $\beta_1 = \beta_2 = 70^\circ$ (corresponding angles) and $\alpha_1 = 180^\circ - 70^\circ = 110^\circ$ (adjacent angles).

Task 5: Parallel or not?

The transversal $s$ crosses $g$ at $\alpha_1 = 50^\circ$ and $h$ at $\alpha_2 = 58^\circ$ . Are $g$ and $h$ parallel? Give a reason using a pair of angles. Load task

Show solution

No. $\alpha_1$ and $\alpha_2$ are corresponding angles. If $g$ and $h$ were parallel, they would have to be equal; but they differ by $58^\circ - 50^\circ = 8^\circ$ . So the lines intersect somewhere (here outside the picture). Sliding shows the difference

Task 6: The converse (extension)

At a transversal, $\alpha_1 = 75^\circ$ and $\delta_2 = 105^\circ$ . Explain why $g \parallel h$ .

Show solution

$\alpha_2 = 180^\circ - \delta_2 = 75^\circ$ (adjacent angles). So the corresponding angles $\alpha_1$ and $\alpha_2$ are equal, and therefore $g \parallel h$ . Shorter: the co-interior angles $\alpha_1$ and $\delta_2$ add up to $75^\circ + 105^\circ = 180^\circ$ . Show in the simulation

Ideas for the classroom

Use: grade 7, angles in figures: vertical, adjacent, corresponding and alternate angles as groundwork for proving the angle sum of a triangle. The animations work well as a whole-class introduction on the whiteboard, the exercise mode for individual or pair work with spoken reasons.
Reasoning: In the exercise mode, every tapped angle shows a short chain of reasons. You could let students give their own reasons first and use the chain shown for comparison – there are often several correct ways.
Terms: Several names are common for co-interior angles (same-side interior angles, allied angles, C or U shape; German books often say E shape). Some textbooks write $\alpha'$ , $\beta'$ , … instead of $\alpha_2$ , $\beta_2$ , … Please check which terms your textbook uses.
Model: Angles are rounded to whole degrees; $g \parallel h$ here means that $s$ crosses both lines at the same angle.
Common stumbling blocks:
- “Corresponding angles are always equal” – only if $g$ and $h$ are parallel. Sliding with non-parallel lines makes the difference visible.
- Mixing up vertical and adjacent angles: vertical angles lie opposite and share only the vertex, adjacent angles share an arm.
- Looking for alternate angles on the same side of $s$ – the Z shape helps.
- Reasons that leave out the condition “because $g \parallel h$ ”.