Unit 4 Congruent Triangles Homework 3: Chapter 4
By James Charles
Hello everyone and welcome to my blog post on Unit 4 Congruent Triangles Homework 3: Chapter 4. In this post, I will explain what congruent triangles are, how to identify them using different criteria, and how to use them to prove other geometric facts. I hope you will find this post helpful and interesting.
What are congruent triangles?
Congruent triangles are triangles that have the same size and shape. This means that all their corresponding angles and sides are equal. For example, if triangle ABC is congruent to triangle DEF, then we can write ABC ≅ DEF and we know that ∠A = ∠D, ∠B = ∠E, ∠C = ∠F, AB = DE, BC = EF, and AC = DF.
How to identify congruent triangles?
There are several ways to show that two triangles are congruent. The most common ones are:
- SSS (Side-Side-Side): If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent. For example, if AB = DE, BC = EF, and AC = DF, then ABC ≅ DEF by SSS.
- SAS (Side-Angle-Side): If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent. For example, if AB = DE, ∠B = ∠E, and BC = EF, then ABC ≅ DEF by SAS.
- AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle, then the triangles are congruent. For example, if ∠A = ∠D, ∠B = ∠E, and BC = EF, then ABC ≅ DEF by AAS.
- ASA (Angle-Side-Angle): If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent. For example, if ∠A = ∠D, AB = DE, and ∠B = ∠E, then ABC ≅ DEF by ASA.
- HL (Hypotenuse-Leg): If the hypotenuse and a leg of a right triangle are equal to the hypotenuse and a leg of another right triangle, then the triangles are congruent. For example, if AC = DF and BC = EF in right triangles ABC and DEF, then ABC ≅ DEF by HL.
How to use congruent triangles?
Congruent triangles can be used to prove other geometric facts or properties. For example:
- If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. We can prove this by constructing congruent triangles using SAS or ASA criteria.
- If a line is perpendicular to two other lines at their point of intersection, then it is the angle bisector of the angle formed by those lines. We can prove this by constructing congruent triangles using SSS or SAS criteria.
- If two chords intersect inside a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. We can prove this by constructing congruent triangles using HL criterion.
Conclusion
In this blog post, I have explained what congruent triangles are, how to identify them using different criteria, and how to use them to prove other geometric facts. I hope you have learned something new and enjoyed reading this post. If you have any questions or comments, please feel free to leave them below. Thank you for your attention and have a great day!
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