Angles (I): Right angles

1Summary 重點摘要

Two lines meeting = an angle; 90° is right angle.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: What is an angleAn angle is made of one vertex and two arms starting from it. On a clock, for example, the two hands open from the centre: the centre is the vertex, the hands are the two arms, and their opening forms an angle. Angles are everywhere in life: open scissors, a pushed-open door, an open book all show angles. First learn to name the parts of an angle: one vertex and two arms, nothing more and nothing less. A common mistake is to treat the numbers on the clock face as parts of the angle. An angle consists only of a vertex and two arms; point at the vertex, then trace along each arm once, and the angle is confirmed.
2. Step 2: Angles have sizeAngles have size: the wider the two arms open, the bigger the angle; the narrower, the smaller. At 3 o'clock, for example, the hour hand points at 3 and the minute hand at 12, and the two hands open into a right angle; at 6 o'clock they form a straight line, a far bigger angle. When comparing two angles, compare how wide they open, not how long the arms are drawn. A common mistake is to think the angle with longer arms is bigger. To compare, put the two vertices together, lay one arm of each on top of the other, and look at the other arms: the wider opening is the bigger angle.
3. Step 3: Meet the right angleA right angle is a special angle measuring exactly 90 degrees. The corner of a book, a door or a blackboard is a right angle. Right angles are the most common in daily life; learn them well and measuring and drawing angles become easy later. A right angle looks square, opening over exactly a quarter of a circle. A common mistake is to call any angle close to 90 degrees a right angle, such as 85 or 95 degrees; the standard is strict, and only exactly 90 degrees counts. Confirm with a set square: looking like one is not the same as being one.
4. Step 4: Test with a set squareTo test a right angle, use a set square: place its right angle on the corner to be checked, vertex on vertex, with the two edges lined up. If the edges fit exactly with no gap, the angle is a right angle; if the angle is narrower than the square's, a gap appears and it is less than 90 degrees; if it is wider, the arms stick out past the square and it is more than 90 degrees. A common mistake is to conclude too early with the vertices not aligned or one edge not held against an arm. Hold the set square still during the check, examine both edges, and only finish when no gap can be seen.
5. Step 5: Right angle markTo mark a right angle, maths books draw a small square symbol at the corner. When you see this symbol, you know the angle is 90 degrees without testing with a set square, which makes reading diagrams easier. When drawing angles, we can also use this symbol to mark a right angle; if a diagram has no such symbol, measure first with a protractor or a set square before marking. A common mistake is to add the symbol to ordinary angles casually, or to draw it so large that it fills half the angle. The symbol belongs inside the angle, a small square fitting snugly against the vertex and both arms.
6. Step 6: Size does not depend on arm lengthThe size of an angle depends only on how wide the two arms open, not on how long they are drawn. The same angle drawn with a long ruler or a short ruler is the same size; lengthening the arms does not make the angle bigger. For example, two 30-degree angles, one with 3-centimetre arms and one with 8-centimetre arms, are still equal. A common mistake is to choose the angle with longer arms in a test, believing it bigger. Judge an angle only by how wide its arms open, or measure it with a set square and a protractor, and never be fooled by the size of the drawing.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• An angle = 1 vertex + 2 arms.
• A right angle = 90 degrees.
• Check right angles with a set square.
• Angle size does not depend on arm length.
• Clock hands make angles too.