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

Triangles ABC and CDA are congruent.

Which angle is equal to angle BAC?

AAABBBEEECCCDDD

Examples with solutions for Congruent Triangles

Step-by-step solutions included
Exercise #1

Look at the triangles in the diagram.

Which of the following statements is true?

535353535353101010131313131313101010AAABBBCCCDDDEEEFFF

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 #2

Determine whether the triangles DCE and ABE congruent?

If so, according to which congruence theorem?

AAABBBCCCDDDEEE50º50º

Step-by-Step Solution

Congruent triangles are triangles that are identical in size, meaning that if we place one on top of the other, they will match exactly.

In order to prove that a pair of triangles are congruent, we need to prove that they satisfy one of these three conditions:

  1. SSS - Three sides of both triangles are equal in length.

  2. SAS - Two sides are equal between the two triangles, and the angle between them is equal.

  3. ASA - Two angles in both triangles are equal, and the side between them is equal.

If we take an initial look at the drawing, we can already observe that there is one equal side between the two triangles, as they are both marked in blue,

We don't have information regarding the other sides, thus we can rule out the first two conditions,

And now we'll focus on the last condition - angle, side, angle.

We can observe that angle D equals angle A, both equal to 50 degrees,

Let's proceed to the angles E.

At first glance, one might think that there's no way to know if these angles are equal, however if we look at how the triangles are positioned,
We can see that these angles are actually corresponding angles, and corresponding angles are of course equal.

Therefore - if the angle, side, and second angle are equal, we can prove that the triangles are equal using the ASA condition

Answer:

Congruent according to A.S.A

Exercise #3

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:

879879

Exercise #4

Look at the triangles in the diagram.

Determine which of the statements is correct.

343434343434555444444555AAABBBCCCDDDEEEFFF

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 #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?

AAABBBCCCDDD

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

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.

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