Lesson Guide
Explanation
Two straight lines crossing make four angles round the point, so they add to three hundred and sixty. Two of them sit across the crossing from each other — opposite angles, touching only at the point. The others sit side by side along one of the lines. Before any arithmetic, name the pairs in the picture: which two are opposite, and which two are side by side.
Your Turn
In the picture, which angle is opposite b? Check: d.
Watch Out
"Opposite" means across the point, not the one next to it. a is opposite c, never b.
Explanation
One angle is enough to find all four. Given seventy degrees, the angle opposite is also seventy, and each of the two beside it is a hundred and eighty minus seventy, a hundred and ten. The four add back to three hundred and sixty, which is the check. There are only ever two different sizes at a crossing, and they add to a hundred and eighty — unless the lines are at right angles, when all four are ninety.
Your Turn
One angle at a crossing is 130°. What are the other three? Check: 130°, 50°, 50°.
Watch Out
Two sizes, alternating round the point. Do not make the opposite angle 180 minus — that is the one beside it.
Explanation
Here is why opposite angles are equal, and it is short. Angle a and angle b sit on a straight line, so they add to a hundred and eighty. Angle b and angle c sit on the other straight line, so they add to a hundred and eighty too. Both a and c are a hundred and eighty minus b, so they are the same. It is the both-sides rule from the number-sentence chapter: take b off both sentences and what is left is a equals c.
Your Turn
If b = 115°, what are a and c? Check: both 180 − 115 = 65°.
Watch Out
The proof needs BOTH lines to be straight. A bent line gives no 180° fact and the angles need not match.
Explanation
Most diagrams need the two facts together. Work along a straight line first, where the angles add to a hundred and eighty, and use the crossing to copy angles across. With forty degrees and a hundred and ten above the line, the third angle there is thirty, and the angle opposite the forty is forty again. Write each angle on the picture as you find it, so the next step has something to build on.
Your Turn
Angles on a line: 40°, 110° and x. Find x. Check: 180 − 150 = 30°.
Watch Out
Only copy an angle across if it is truly opposite. Check that both arms are straight lines through the point.
Explanation
The whole lesson is two lines: opposite angles are equal, side-by-side angles are a hundred and eighty minus. Two roads crossing at fifty-five degrees make angles of fifty-five, a hundred and twenty-five, fifty-five, a hundred and twenty-five. And if all four angles are equal, each is three hundred and sixty divided by four, ninety, which means the lines are perpendicular. Quiz and exercises now.
Your Turn
All four angles at a crossing are equal. How big is each? Check: 360 ÷ 4 = 90°.
Watch Out
Read which angle the question wants — the opposite one or the one beside it. They are different unless both are 90°.
📌 Key Points — this lesson in a nutshell
- Two straight lines crossing make four angles: two pairs of equal opposite angles.
- Opposite angles are equal. Angles that are side by side add to 180°.
- Why: if a + b = 180 and b + c = 180, then a and c must be the same.
- One angle gives all four: the opposite one is equal, the two beside it are 180 minus it.
Step 1: Introduction
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