Definition properties and triangle congruence rules with examples
How to Draw Congruent Figures?
You can take a sheet of paper and draw two similar figures on it. Cut the figures and place them on one another. Both the statistics will be congruent when you put one picture on top of the other image.
In Geometry, when we say that one (A) figure is equal to figure (B), we write down as figure A ≅ to figure B. Below, you will learn more about congruent shapes, congruent line segments, and corresponding shapes and angles.
How to find The Congruence of two Figures?
To know what are congruent figures you have to keep two plane figures in front of you, and you have to find if they are identical with each other or not. Use the following method to discover if the two figures are congruent figures or not:
Take a tracing paper and trace the outline of figure A.
Cut out the shape of figure A and place it over the figure B.
You can set the paper over figure B or flip it and then put it over figure B.
If the two figures completely cover each other, they are congruent to each other. You can symbolically represent it, as figure A is ≅ to figure B.
Congruent Lines Segments
Like two figures can be congruent, so can two lines and lines segments. Two line segments are congruent if they have equal or the same length.
If you want to find out if two figures are corresponding to each other or not, you can use the same method as given above. Take two line segments and place them on top of each other. So, two line segments are congruent if they are equal or the same.
What are Congruent Angles?
Besides, line segments, angles can be congruent to each other too. If angle A and angle B are 36 degrees each, they are congruent. However, if the corners are different, the angles are not congruent with each other.
When two angles have the same measure in degree, they are congruent to each other. The angles may differ in position and orientation on the plane. You can use the method given above to find out the congruency of the angles.
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Here are two triangles with congruent angles.
If you have two figures, ∠ABC and ∠QPR, with the same angle, which you can assume as 45 degrees, when you use the method given above, you will see that the ∠ABC ≅ ∠QPR means they are congruent to each other.
How to prove if two Angles are Congruent to each other?
There is another way to check for the congruence of two angles. Follow the below-given method to prove that two angles are congruent:
When two angles and the side between them are equal or similar in both the triangles, it is an ASA congruence.
When two angles and a side that is not between them is identical or identical in both the triangles, it’s an AAS congruence.
If three sides of a triangle are the same or equal to each other, it is SSS congruence.
When two hands and one angle between them makes the two triangles congruent, it’s the SAS congruence.
Congruent Angles Solved Examples
Here are a few congruent angles solved examples that will help you understand how to find the congruence of two angles.
Example 1:
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Here are two figures which you need to prove as congruent.
The angles G and S of the diagram above are 42 degrees, which means that they are equal. When two angles of a figure are identical or similar, it means that they are congruent.
Example 2:
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Here are two figures which you have to prove as congruent.
The angles of the two figures are 155 degrees each, so It means that R and Q are equal to each other. If two aspects are similar to each other, it means that they are congruent to each other.
Fun Facts About Congruent Shapes
Here are some fun facts about the corresponding figures:
Squares that have equal length on all four sides are congruent shapes.
An equilateral triangle which has equal distances on all the three sides are congruent.
Slices of pizza are a perfect example of a congruent triangle and congruent angles.
When you combine two congruent triangles, they form vertical angles at a corner.
Most buildings in the United States are congruent triangles.
FAQs on Congruent Figures in Geometry Explained Clearly
1. What are congruent figures?
Congruent figures are figures that have exactly the same shape and the same size. This means:
- All corresponding sides are equal in length.
- All corresponding angles are equal in measure.
- They can be mapped onto each other using rigid transformations like translation, rotation, or reflection.
For example, two triangles with the same side lengths and angle measures are congruent, even if one is flipped or rotated.
2. How do you know if two figures are congruent?
Two figures are congruent if all corresponding sides and angles are equal and one can be superimposed on the other. To check congruence:
- Compare corresponding side lengths.
- Compare corresponding angle measures.
- Verify that a rigid transformation (slide, flip, or turn) makes them overlap exactly.
If every pair of matching parts is equal, the figures are congruent.
3. What is the symbol for congruent figures?
The symbol for congruent figures is ≅. For example:
- If triangle ABC is congruent to triangle DEF, we write △ABC ≅ △DEF.
The order of letters shows corresponding vertices, meaning A corresponds to D, B to E, and C to F.
4. What are the triangle congruence rules?
The main triangle congruence rules are SSS, SAS, ASA, AAS, and RHS (HL). These criteria prove two triangles are congruent:
- SSS: All three sides are equal.
- SAS: Two sides and the included angle are equal.
- ASA: Two angles and the included side are equal.
- AAS: Two angles and a non-included side are equal.
- RHS (HL): Right angle, hypotenuse, and one side are equal (for right triangles).
5. What is the difference between congruent and similar figures?
Congruent figures have the same shape and the same size, while similar figures have the same shape but different sizes. The key differences are:
- Congruent: Corresponding sides are equal and angles are equal.
- Similar: Corresponding angles are equal, but sides are proportional.
All congruent figures are similar, but not all similar figures are congruent.
6. Can two rectangles be congruent?
Yes, two rectangles are congruent if their corresponding lengths and widths are equal. For example:
- Rectangle A: length = 8 cm, width = 5 cm
- Rectangle B: length = 8 cm, width = 5 cm
Since both dimensions match, the rectangles are congruent, even if one is rotated.
7. Are all circles congruent?
No, circles are congruent only if their radii (or diameters) are equal. For example:
- Circle A: radius = 4 cm
- Circle B: radius = 4 cm
These circles are congruent because their radii are equal. If the radii differ, the circles are not congruent.
8. How do rigid transformations relate to congruent figures?
Rigid transformations preserve length and angle measure, which means they produce congruent figures. The three rigid transformations are:
- Translation (slide)
- Rotation (turn)
- Reflection (flip)
If one figure can be moved onto another using these transformations, the figures are congruent.
9. Why is AAA not a congruence rule?
AAA is not a congruence rule because equal angles alone do not guarantee equal side lengths. Two triangles can have:
- The same three angles
- Different side lengths
Such triangles are similar but not necessarily congruent, since their sizes may differ.
10. Can you give an example of proving two triangles congruent using SSS?
Two triangles are congruent by SSS if all three pairs of corresponding sides are equal. Example:
- Triangle ABC: AB = 5 cm, BC = 7 cm, AC = 6 cm
- Triangle DEF: DE = 5 cm, EF = 7 cm, DF = 6 cm
Since AB = DE, BC = EF, and AC = DF, the triangles are congruent by SSS, so △ABC ≅ △DEF.
