【Grade 5 Math】 Congruent Figures
Hello everyone! Today, let's learn about the term "congruent." "Congruent" might sound a bit intimidating, but it’s actually a very simple concept. Let’s have fun learning by looking for things around us that have the "same shape and same size!"
You might find it tricky at first, but that’s okay. If you take it one step at a time, you’ll master it in no time!
1. What is "Congruence"?
Congruence means that two figures are "exactly the same in shape and size."
If you were to cut out one figure and place it on top of the other, and it fits perfectly without any parts sticking out or missing, those two figures are congruent.
(Example: Pages in a notebook, two 10-yen coins, or the same cards from a deck are all congruent!)
💡 Fun Fact: Flipping them over is okay!
If you can make figures overlap by rotating them or flipping them over, they are still "congruent." Even if they are facing a different direction, as long as the shape and size are the same, they are a match.
【Key Points】
・They must have the same shape.
・They must have the same size.
Only when both of these are met do we call them "congruent."
2. Properties (Rules) of Congruent Figures
When you perfectly overlap congruent figures, the points, sides, and angles that match up have special names.
- Corresponding vertices: The corners that overlap.
- Corresponding sides: The boundary lines that overlap.
- Corresponding angles: The angles that overlap.
Congruent figures follow two very important rules:
1. All corresponding sides are equal in length.
2. All corresponding angles are equal in size.
【Common Mistakes】
Sometimes people call "figures that have the same shape but different sizes" congruent. For example, a small equilateral triangle and a large equilateral triangle have the same shape, but they are not congruent. Don't forget that the size must be identical as well!
3. How to Draw Congruent Triangles
To draw a triangle that is congruent (exactly the same) to another, you don't need to measure every single length. You only need to know one of three conditions. Get your compass and protractor ready!
① When the lengths of all 3 sides are known
1. First, draw one of the sides.
2. Measure the lengths of the remaining two sides with your compass, and draw arcs from each end of the first side.
3. The point where the two arcs cross is your third vertex!
② When the lengths of 2 sides and the angle between them are known
1. First, draw one of the sides.
2. Use a protractor to measure and mark the specific angle at one end.
3. Measure the length of the second side along that angle, draw the line, and you're done!
③ When the length of 1 side and the angles at both ends are known
1. First, draw the known side.
2. From both ends of that side, use your protractor to measure the respective angles and draw lines outward.
3. The point where those two lines meet is your third vertex!
【Tip】
When drawing triangles, it’s smoother to think about "which tool (compass or protractor) to use" first.
・If you know the lengths → Compass
・If you know the angles → Protractor
4. Congruence in Quadrilaterals
Just like triangles, quadrilaterals are congruent if all their corresponding sides and angles are equal.
When drawing a quadrilateral, draw one diagonal to split it into two triangles. Then, you can use the triangle drawing methods we just learned. Whenever it feels difficult, just think, "It’s just two triangles put together!"
🌟 Summary of this Chapter
1. Congruence means figures overlap perfectly in both shape and size.
2. In congruent figures, the lengths of corresponding sides and the sizes of corresponding angles are equal.
3. To draw a triangle, you just need to know one of these: 3 sides, 2 sides and the angle between them, or 1 side and the angles at both ends.
4. Even if the orientation is different, if they overlap perfectly, they are congruent!
Using a compass and protractor might feel tricky at first, but your hands will get used to it with practice. Enjoy that satisfying feeling when you successfully draw a perfect match! I’m rooting for you!