Congruent Triangles Practice Problems & Worksheets

Master congruent triangles with step-by-step practice problems. Learn SSS, SAS, ASA, AAS, and HL congruence rules through interactive exercises and solutions.

📚What You'll Master in This Practice Session
  • Identify congruent triangles using SSS, SAS, ASA, AAS, and HL postulates
  • Write congruence statements with proper vertex correspondence order
  • Prove triangle congruence using two-column and paragraph proofs
  • Apply CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
  • Solve for unknown angles and sides in congruent triangles
  • Recognize when triangles are NOT congruent and explain why

Understanding Congruent Triangles

Complete explanation with examples

Congruent triangles are identical triangles.

That means in triangles whose angles and sides are equal, their area and perimeter will also be equal.

But keep in mind that this case is different from when triangles are similar, that is, when the angles are equal but the side lengths are different in the corresponding ratio.

Congruence Criteria

To prove that 2 triangles are congruent we can use one of the following criteria:

By verifying one of the triangle congruence criteria, we can affirm that the triangles are congruent.

Diagram illustrating the congruence of two triangles, showing equal sides and angles marked correspondingly. This visual representation demonstrates the concept of triangle congruence in geometry, featured in a guide on understanding and proving triangle congruence.

Detailed explanation

Practice Congruent Triangles

Test your knowledge with 38 quizzes

Are the triangles in the image congruent?

If so, according to which theorem?

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Examples with solutions for Congruent Triangles

Step-by-step solutions included
Exercise #1

Look at the triangles in the diagram.

Determine which of the statements is correct.

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Step-by-Step Solution

Let's consider that:

AC=EF=4

DF=AB=5

Since 5 is greater than 4 and the angle equal to 34 is opposite the larger side in both triangles, the angle ACB must be equal to the angle DEF

Therefore, the triangles are congruent according to the SAS theorem, as a result of this all angles and sides are congruent, and all answers are correct.

Answer:

All of the above.

Exercise #2

Look at the triangles in the diagram.

Which of the following statements is true?

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Step-by-Step Solution

According to the existing data:

EF=BA=10 EF=BA=10 (Side)

ED=AC=13 ED=AC=13 (Side)

The angles equal to 53 degrees are both opposite the greater side (which is equal to 13) in both triangles.

(Angle)

Since the sides and angles are equal among congruent triangles, it can be determined that angle DEF is equal to angle BAC

Answer:

Angles BAC is equal to angle DEF.

Exercise #3

Look at the triangles in the diagram.

Which of the following statements is true?

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Step-by-Step Solution

This question actually has two steps:

In the first step, you must define if the triangles are congruent or not,

and then identify the correct answer among the options.

Let's look at the triangles: we have two equal sides and one angle,

But this is not a common angle, therefore, it cannot be proven according to the S.A.S theorem

Remember the fourth congruence theorem - S.A.A
If the two triangles are equal to each other in terms of the lengths of the two sides and the angle opposite to the side that is the largest, then the triangles are congruent.

But the angle we have is not opposite to the larger side, but to the smaller side,

Therefore, it is not possible to prove that the triangles are congruent and no theorem can be established.

Answer:

It is not possible to calculate.

Exercise #4

Choose the pair of triangles that are congruent according to S.S.S.

Step-by-Step Solution

In answer A, we are given two triangles with different angles, therefore the sides are also different and they are not congruent according to S.S.S.

In answer B, we are given two right triangles, but their angles are different and so are the sides. Therefore, they are not congruent according to S.S.S.

In answer D, we do not have enough data, therefore it is not possible to determine that they are congruent according to S.S.S.

In answer C, we see that all the sides are equal to each other in both triangles and therefore they are congruent according to S.S.S.

Answer:

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Exercise #5

Given: ΔABC isosceles

and the line AD cuts the side BC.

Are ΔADC and ΔADB congruent?

And if so, according to which congruence theorem?

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Step-by-Step Solution

Since we know that the triangle is isosceles, we can establish that AC=AB and that

AD=AD since it is a common side to the triangles ADC and ADB

Furthermore given that the line AD intersects side BC, we can also establish that BD=DC

Therefore, the triangles are congruent according to the SSS (side, side, side) theorem

Answer:

Congruent by L.L.L.

Frequently Asked Questions

Everything you need to know about Congruent Triangles

What are the 5 ways to prove triangles are congruent?

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The five triangle congruence postulates are: SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), and HL (Hypotenuse-Leg for right triangles). Each requires specific combinations of equal corresponding parts.

How do you write a congruence statement for triangles?

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Write triangle congruence statements by listing vertices in corresponding order. For example, if triangle ABC is congruent to triangle DEF, write △ABC ≅ △DEF, ensuring A corresponds to D, B to E, and C to F.

What is CPCTC and when do you use it?

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CPCTC stands for 'Corresponding Parts of Congruent Triangles are Congruent.' Use it AFTER proving triangles are congruent to show that specific angles or sides are equal. It's the second step in many geometry proofs.

Why doesn't SSA prove triangle congruence?

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SSA (Side-Side-Angle) doesn't work because it can create two different triangles with the same measurements. The angle opposite the longer side could be acute or obtuse, creating an ambiguous case that doesn't guarantee congruence.

What's the difference between ASA and AAS congruence?

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ASA has the side between the two angles (Angle-Side-Angle), while AAS has the side not between the angles (Angle-Angle-Side). Both are valid congruence postulates, but the position of the side relative to the angles differs.

When can you use HL to prove triangle congruence?

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HL (Hypotenuse-Leg) only applies to right triangles. You need: 1) Both triangles must be right triangles, 2) The hypotenuses must be congruent, 3) One pair of corresponding legs must be congruent.

How do you solve for missing parts in congruent triangles?

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First prove the triangles are congruent using one of the five postulates. Then use CPCTC to set up equations where corresponding parts are equal. Solve these equations to find missing angle measures or side lengths.

What are common mistakes students make with congruent triangles?

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Common errors include: mixing up the order of vertices in congruence statements, trying to use SSA as a valid postulate, forgetting to prove congruence before using CPCTC, and incorrectly identifying corresponding parts when triangles are rotated or reflected.

More Congruent Triangles Questions

Click on any question to see the complete solution with step-by-step explanations

Conditions of Congruency

Triangle Congruence: Proving ΔADB≅ΔCBD with Parallel Lines

Congruent Triangles

Triangle Congruence with Parallel Lines: Proving CDA ≅ ABC Triangle Congruence Theorem: Identifying the Proof for ADE ≅ BCE

Practice by Question Type

Choose your preferred learning style and practice format for fraction problems
Ascertaining whether or not the triangles are congruent Identifying and defining elements