Congruent Triangles Worksheet With Answers For Students
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Congruent triangles are an essential concept in geometry, forming the foundation for various proofs and theorems in both academic and real-world applications. Understanding congruence helps students grasp the properties of shapes and their relationships better. In this article, we will delve into congruent triangles, their properties, and provide a worksheet designed for students, complete with answers to facilitate learning.
What Are Congruent Triangles? 🤔
Congruent triangles are triangles that are identical in shape and size. This means that all corresponding sides and angles are equal. The notation for congruence is denoted by the symbol ( \cong ). For example, if triangle ABC is congruent to triangle DEF, it can be expressed as:
[ \triangle ABC \cong \triangle DEF ]
Properties of Congruent Triangles
Understanding the properties of congruent triangles is vital for students. Here are the main properties:
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Equal Corresponding Sides: If ( \triangle ABC \cong \triangle DEF ), then:
- ( AB = DE )
- ( BC = EF )
- ( AC = DF )
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Equal Corresponding Angles: Similarly, the angles are equal:
- ( \angle A = \angle D )
- ( \angle B = \angle E )
- ( \angle C = \angle F )
Criteria for Triangle Congruence
There are several criteria to establish if two triangles are congruent. The most common ones include:
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Side-Side-Side (SSS): If all three sides of one triangle are equal to the three sides of another triangle, the triangles are congruent.
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Side-Angle-Side (SAS): If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, they are congruent.
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Angle-Side-Angle (ASA): If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, they are congruent.
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Angle-Angle-Side (AAS): If two angles and a non-included side of one triangle are equal to two angles and the corresponding non-included side of another triangle, they are congruent.
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Hypotenuse-Leg (HL): This is specific to right triangles. If the hypotenuse and one leg of one right triangle are equal to the hypotenuse and one leg of another right triangle, they are congruent.
Congruent Triangles Worksheet 📄
Below is a worksheet designed to help students practice the concept of congruent triangles.
Worksheet Questions
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Determine whether the following triangles are congruent. If they are, state the criterion used.
- Triangle ABC: ( AB = 5 ), ( AC = 7 ), ( BC = 6 )
- Triangle DEF: ( DE = 5 ), ( DF = 7 ), ( EF = 6 )
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Given triangles GHI and JKL, where ( \angle G = 45^\circ ), ( \angle H = 75^\circ ), and ( GH = 10 ) cm, ( \angle J = 45^\circ ), ( \angle K = 75^\circ ), and ( JK = 10 ) cm, are the triangles congruent? Explain your answer.
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Prove that triangle MNO is congruent to triangle PQR if:
- ( MN = 8 ), ( NO = 6 ), ( MO = 10 )
- ( PQ = 8 ), ( QR = 6 ), ( PR = 10 )
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For triangle STU and triangle VWX:
- ( \angle S = 60^\circ )
- ( \angle T = 50^\circ )
- ( SU = 12 ) cm
- ( \angle V = 60^\circ )
- ( \angle W = 50^\circ )
- ( VW = 12 ) cm Are the triangles congruent? If yes, state the criterion.
Table of Criteria for Triangle Congruence
<table> <tr> <th>Criterion</th> <th>Definition</th> <th>Notation Example</th> </tr> <tr> <td>SSS</td> <td>All three sides are equal</td> <td>AB = DE, AC = DF, BC = EF</td> </tr> <tr> <td>SAS</td> <td>Two sides and the included angle are equal</td> <td>AB = DE, AC = DF, &angle A = &angle D</td> </tr> <tr> <td>ASA</td> <td>Two angles and the included side are equal</td> <td>&angle A = &angle D, &angle B = &angle E, AB = DE</td> </tr> <tr> <td>AAS</td> <td>Two angles and a non-included side are equal</td> <td>&angle A = &angle D, &angle B = &angle E, BC = EF</td> </tr> <tr> <td>HL</td> <td>For right triangles, hypotenuse and one leg are equal</td> <td>AB = DE, AC = DF</td> </tr> </table>
Answers to the Worksheet
- Answer: Yes, the triangles are congruent by SSS (Side-Side-Side).
- Answer: Yes, the triangles are congruent by ASA (Angle-Side-Angle).
- Answer: The triangles are congruent by SSS (Side-Side-Side).
- Answer: Yes, the triangles are congruent by AAS (Angle-Angle-Side).
Conclusion
Congruent triangles play a vital role in the study of geometry, providing crucial insights into the properties of shapes and their relationships. By understanding congruence, students can enhance their problem-solving skills and apply these concepts in more advanced mathematical situations. This worksheet aims to aid students in their learning process, offering them a practical understanding of congruent triangles. Practice makes perfect, so keep exploring this topic, and don't hesitate to ask for help whenever needed! 🏆📐