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Congruent Triangle Theorems

4 min readdecember 13, 2021

Sitara H

Sitara H

Sitara H

Sitara H

The 5 Congruent Triangle Theorems and How to Use Them

What is Congruency?

Any amount of triangles are said to be congruent if they have the same shape and dimensions.

Congruence is the term used to describe two objects with the same shape and size. When two objects have the same shape but are not necessarily the same size, we call them similar. When looking at any triangle diagram, it is important to recognize which parts are congruent to each other. We use small tick marks to indicate the sets of congruent angles or congruent sides. There are different postulates that we can use to prove that two triangles are congruent. 

Writing Congruence Statements

The mathematical symbol used to show congruence is ≅, and the mathematical way to refer to a triangle using its three points is ΔFIG, using 'F,' 'I,' and 'G' as its three points. 

Therefore, to show the congruence of two triangles, take ΔFIG and another triangle ΔABC, you would write the statement ΔFIG ≅ ΔABC.

SSS: Side-Side-Side

This rule states that two triangles are congruent if all 3️⃣of their corresponding side lengths are equal.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-UrCo6p1afR1z.png?alt=media&token=50794909-1886-4a7e-93da-a0ee617e13df

According to the SSS Postulate, triangles ΔFGH & ΔABC have three sets of corresponding, congruent sides (F̅H̅ and A̅C̅, G̅H̅ and C̅B̅, & F̅H̅ and A̅B̅) and therefore ΔFGH ≅ ΔABC.

SAS: Side-Angle-Side

This rule states that two triangles are congruent if 2️⃣sides and 1️⃣angle in a given triangle have the same length and angle measure as the corresponding two sides and another angle in another triangle. The two sides must form the included angle for the triangles to be congruent.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-f1aEQiahlRFK.png?alt=media&token=aa6e30c8-c179-4290-a5dd-a14faf7ec067

Triangles FGH & ABC have two sets of corresponding, congruent sides (F̅G̅ and A̅C̅, & G̅H̅ and B̅C̅) and one set of included, corresponding, congruent angles (∠G and ∠C). Therefore ΔFGH ≅ ΔABC, according to the SAS Postulate.

ASA: Angle-Side-Angle

This rule states that two triangles are congruent if 2️⃣of their corresponding angles and 1️⃣included side are equal.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-70n1evuX0Ii5.png?alt=media&token=7318a763-4044-4b22-92ed-d35c3f2599ce

Triangles ΔFGH & ΔABC have two sets of corresponding, congruent angles (∠G and ∠C, & ∠H and ∠B) and one set of included, corresponding, congruent sides (G̅H̅ and C̅B̅). Therefore ΔFGH ≅ ΔABC, according to the ASA Postulate.

AAS: Angle-Angle-Side

This rule states that two triangles are congruent if 2️⃣of their corresponding sides and 1️⃣non-included ❌ side are equal. We cannot use the side formed between the two angles.

The main difference between this theorem and ASA is whether the side is between the two corresponding angles - you just have to think of the names of the two theorems to remember this! AAS uses the side that isn't between the two angles, and the 'S' isn't in between the two 'A's.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-giojd9MIF9ag.png?alt=media&token=c5b928a7-0440-442b-bbe1-3e99f5b14cfc

Triangles ΔFGH & ΔABC have two sets of corresponding, congruent angles (∠G and ∠C, & ∠H and ∠B) and one set of non-included, corresponding, congruent sides (F̅H̅ and A̅B̅). Therefore ΔFGH ≅ ΔABC, according to the AAS Postulate.

Hypotenuse Leg Theorem

This is slightly different from the other theorems, which can apply to all triangles, as this one only applies specifically to right triangles.

This rule states that if the hypotenuse and one leg of a given right triangle are equal to the hypotenuse and corresponding leg of another right triangle, then the two right triangles are congruent.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-Q2kHlwvJrjt0.png?alt=media&token=774641d2-176a-4e95-be9c-ee0438aae4b7

Triangles ΔFGH & ΔABC have corresponding, congruent hypotenuses (G̅H̅ and C̅B̅) and one set of corresponding, congruent legs (F̅H̅ and A̅B̅). Therefore ΔFGH ≅ ΔABC, according to the AAS Postulate.

Examples

For each pair of triangles below, state the postulate or theorem that can be used to prove that the triangles are congruent.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-1QBFXJG0sqsk.png?alt=media&token=e42a44a4-79a7-4a46-a515-e5c839b92611

For #1, both corresponding sides FG and GH and corresponding angles ∠FGI and ∠HGI are marked as congruent by the matching tick marks. In addition, both triangles share the side GI. Thus, both triangles ΔFIG and ΔHIG have two corresponding congruent sides and one included angle, so we can use the SAS Postulate ✔️ to prove their congruence.

For #2, both sets of sides OP and PQ and ON and QN are congruent. In addition, similarly to Triangle 1, both triangles share the side PN. Thus, both triangles have three corresponding and congruent sets of sides, so we can use the SSS Postulate ☑️ to show that both triangles ΔPNO and ΔPNQ are congruent.

#3 has a corresponding set of marked congruent sides, BC and CE, and a corresponding set of angles ∠ABC and ∠CDE. However, we can still find a 3rd congruent set of corresponding angles, despite no other marks or the triangles having a common side.

When two lines, like BE and AD, intersect to make an X, angles on the opposite side of the X are called vertical angles, and therefore they are congruent. Thus, we can conclude that angles ∠BCA and ∠DCE are congruent because they are vertical angles. In this way, we have found two sets of corresponding and congruent angles and 1 set of corresponding, congruent sides. We can use the ASA postulate ✅ to show that triangles ΔABC and ΔCED are congruent.

🤝Connect with other students studying geometry with Hours

Congruent Triangle Theorems

4 min readdecember 13, 2021

Sitara H

Sitara H

Sitara H

Sitara H

The 5 Congruent Triangle Theorems and How to Use Them

What is Congruency?

Any amount of triangles are said to be congruent if they have the same shape and dimensions.

Congruence is the term used to describe two objects with the same shape and size. When two objects have the same shape but are not necessarily the same size, we call them similar. When looking at any triangle diagram, it is important to recognize which parts are congruent to each other. We use small tick marks to indicate the sets of congruent angles or congruent sides. There are different postulates that we can use to prove that two triangles are congruent. 

Writing Congruence Statements

The mathematical symbol used to show congruence is ≅, and the mathematical way to refer to a triangle using its three points is ΔFIG, using 'F,' 'I,' and 'G' as its three points. 

Therefore, to show the congruence of two triangles, take ΔFIG and another triangle ΔABC, you would write the statement ΔFIG ≅ ΔABC.

SSS: Side-Side-Side

This rule states that two triangles are congruent if all 3️⃣of their corresponding side lengths are equal.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-UrCo6p1afR1z.png?alt=media&token=50794909-1886-4a7e-93da-a0ee617e13df

According to the SSS Postulate, triangles ΔFGH & ΔABC have three sets of corresponding, congruent sides (F̅H̅ and A̅C̅, G̅H̅ and C̅B̅, & F̅H̅ and A̅B̅) and therefore ΔFGH ≅ ΔABC.

SAS: Side-Angle-Side

This rule states that two triangles are congruent if 2️⃣sides and 1️⃣angle in a given triangle have the same length and angle measure as the corresponding two sides and another angle in another triangle. The two sides must form the included angle for the triangles to be congruent.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-f1aEQiahlRFK.png?alt=media&token=aa6e30c8-c179-4290-a5dd-a14faf7ec067

Triangles FGH & ABC have two sets of corresponding, congruent sides (F̅G̅ and A̅C̅, & G̅H̅ and B̅C̅) and one set of included, corresponding, congruent angles (∠G and ∠C). Therefore ΔFGH ≅ ΔABC, according to the SAS Postulate.

ASA: Angle-Side-Angle

This rule states that two triangles are congruent if 2️⃣of their corresponding angles and 1️⃣included side are equal.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-70n1evuX0Ii5.png?alt=media&token=7318a763-4044-4b22-92ed-d35c3f2599ce

Triangles ΔFGH & ΔABC have two sets of corresponding, congruent angles (∠G and ∠C, & ∠H and ∠B) and one set of included, corresponding, congruent sides (G̅H̅ and C̅B̅). Therefore ΔFGH ≅ ΔABC, according to the ASA Postulate.

AAS: Angle-Angle-Side

This rule states that two triangles are congruent if 2️⃣of their corresponding sides and 1️⃣non-included ❌ side are equal. We cannot use the side formed between the two angles.

The main difference between this theorem and ASA is whether the side is between the two corresponding angles - you just have to think of the names of the two theorems to remember this! AAS uses the side that isn't between the two angles, and the 'S' isn't in between the two 'A's.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-giojd9MIF9ag.png?alt=media&token=c5b928a7-0440-442b-bbe1-3e99f5b14cfc

Triangles ΔFGH & ΔABC have two sets of corresponding, congruent angles (∠G and ∠C, & ∠H and ∠B) and one set of non-included, corresponding, congruent sides (F̅H̅ and A̅B̅). Therefore ΔFGH ≅ ΔABC, according to the AAS Postulate.

Hypotenuse Leg Theorem

This is slightly different from the other theorems, which can apply to all triangles, as this one only applies specifically to right triangles.

This rule states that if the hypotenuse and one leg of a given right triangle are equal to the hypotenuse and corresponding leg of another right triangle, then the two right triangles are congruent.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-Q2kHlwvJrjt0.png?alt=media&token=774641d2-176a-4e95-be9c-ee0438aae4b7

Triangles ΔFGH & ΔABC have corresponding, congruent hypotenuses (G̅H̅ and C̅B̅) and one set of corresponding, congruent legs (F̅H̅ and A̅B̅). Therefore ΔFGH ≅ ΔABC, according to the AAS Postulate.

Examples

For each pair of triangles below, state the postulate or theorem that can be used to prove that the triangles are congruent.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-1QBFXJG0sqsk.png?alt=media&token=e42a44a4-79a7-4a46-a515-e5c839b92611

For #1, both corresponding sides FG and GH and corresponding angles ∠FGI and ∠HGI are marked as congruent by the matching tick marks. In addition, both triangles share the side GI. Thus, both triangles ΔFIG and ΔHIG have two corresponding congruent sides and one included angle, so we can use the SAS Postulate ✔️ to prove their congruence.

For #2, both sets of sides OP and PQ and ON and QN are congruent. In addition, similarly to Triangle 1, both triangles share the side PN. Thus, both triangles have three corresponding and congruent sets of sides, so we can use the SSS Postulate ☑️ to show that both triangles ΔPNO and ΔPNQ are congruent.

#3 has a corresponding set of marked congruent sides, BC and CE, and a corresponding set of angles ∠ABC and ∠CDE. However, we can still find a 3rd congruent set of corresponding angles, despite no other marks or the triangles having a common side.

When two lines, like BE and AD, intersect to make an X, angles on the opposite side of the X are called vertical angles, and therefore they are congruent. Thus, we can conclude that angles ∠BCA and ∠DCE are congruent because they are vertical angles. In this way, we have found two sets of corresponding and congruent angles and 1 set of corresponding, congruent sides. We can use the ASA postulate ✅ to show that triangles ΔABC and ΔCED are congruent.

🤝Connect with other students studying geometry with Hours



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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.


© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.