Congruent Triangles Practice Problems - Right Triangle Congruence

Master congruent triangles with practice problems covering SAS, ASA, SSS congruence theorems. Includes right triangle congruence rules and step-by-step solutions.

📚What You'll Master in This Practice Session
  • Apply SAS, ASA, and SSS congruence theorems to prove triangles are congruent
  • Identify corresponding sides and angles in congruent right triangles
  • Use the Pythagorean theorem to find missing sides in right triangle congruence
  • Solve problems involving perpendicular sides and right angles in triangle congruence
  • Determine which congruence theorem applies to specific triangle pairs
  • Work with isosceles triangles, squares, and deltoids in congruence proofs

Understanding Congruence of Right Triangles (using the Pythagorean Theorem)

Complete explanation with examples

In right triangles, we have a condition that already exists in the first place. It refers to the right angle that iss given and that turns a triangle into a right triangle.

In the second stage, we will move on to the sides. In every right triangle we have two perpendiculars (two sides between which the right angle is comprised) and the other (the larger side of the triangle that faces the right angle).

When there are two right triangles in front of us, in which one size is perpendicular and the size of the rest is equal to each other, then we can conclude that these are congruent triangles.

Diagram demonstrating the congruence of two right-angled triangles, with equal sides and angles marked correspondingly. The visual highlights the concept of right-angled triangle congruence, emphasizing equal hypotenuses and legs. Featured in a guide on proving the congruence of right-angled triangles.

Detailed explanation

Practice Congruence of Right Triangles (using the Pythagorean Theorem)

Test your knowledge with 16 quizzes

What data must be added so that the triangles are congruent?

414141393939777999999777AAABBBCCCDDDEEEFFF

Examples with solutions for Congruence of Right Triangles (using the Pythagorean Theorem)

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

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 #4

The triangles ABO and CBO are congruent.

Which side is equal to BC?

AAABBBCCCDDDOOO

Step-by-Step Solution

Let's consider the corresponding congruent triangles letters:

CBO=ABO CBO=ABO

That is, from this we can determine:

CB=AB CB=AB

BO=BO BO=BO

CO=AO CO=AO

Answer:

Side AB

Video Solution
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

Everything you need to know about Congruence of Right Triangles (using the Pythagorean Theorem)

What makes right triangle congruence different from regular triangle congruence?

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Right triangle congruence already has one condition met - the 90° angle. This means you only need to prove two additional corresponding parts are equal (like two sides or one side and another angle) rather than three parts like in regular triangles.

What are the three main congruence theorems for triangles?

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The three main congruence theorems are: 1) SAS (Side-Angle-Side) - two sides and the included angle are equal, 2) ASA (Angle-Side-Angle) - two angles and the included side are equal, 3) SSS (Side-Side-Side) - all three corresponding sides are equal.

How do you prove two right triangles are congruent using SAS?

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To prove right triangles congruent using SAS, show that two corresponding sides are equal and the right angles (90°) are equal. Since both triangles have right angles, you need to identify two pairs of equal sides with the right angle between them.

Can you use the Pythagorean theorem in triangle congruence proofs?

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Yes, the Pythagorean theorem helps find missing sides in right triangles during congruence proofs. If two right triangles have equal perpendicular sides, you can use a²+b²=c² to show their hypotenuses are also equal, proving SSS congruence.

What is the difference between perpendicular sides and hypotenuse in right triangles?

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Perpendicular sides are the two sides that form the 90° angle (also called legs). The hypotenuse is the longest side opposite the right angle. In congruence problems, you often compare perpendicular sides first, then use them to find the hypotenuse.

How do you identify corresponding parts in congruent triangles?

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Corresponding parts are matched based on position and function. In congruent triangles, corresponding sides are equal in length and corresponding angles are equal in measure. Use proper notation like △ABC ≅ △DEF to show which vertices correspond.

What are common mistakes when proving triangle congruence?

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Common mistakes include: 1) Not clearly stating which congruence theorem is being used, 2) Incorrectly identifying corresponding parts, 3) Assuming triangles are congruent without proving all required conditions, 4) Mixing up the order of vertices in congruence statements.

When working with isosceles triangles in congruence proofs, what special properties help?

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Isosceles triangles have two equal sides and two equal base angles. When an isosceles triangle has a median to its base, it creates two congruent right triangles, making it easier to apply ASA or SAS congruence theorems.

More Congruence of Right Triangles (using the Pythagorean Theorem) Questions

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