Congruence Criterion: Angle, Side, Angle

In this article, we will study the second criterion of congruence:

Angle, Side, Angle

Definition:

Two triangles in which two angles and the included side between them are equal are congruent triangles.

Attention: The two angles must be adjacent to the equal and corresponding side in both triangles!

To prove that two triangles are congruent we can use one of the following postulates:

Example 1 - Congruent Triangles

Given the triangles ΔABCΔ ABC and ΔDEFΔ DEF such that:
A=D\sphericalangle A=\sphericalangle D
AB=DEAB = DE
B=E\sphericalangle B=\sphericalangle E

2 congruent triangles

From this, we deduce that the triangles ΔABCΔ ABC and ΔDEFΔ DEF are congruent, therefore, we will write:

ΔDEFΔABCΔ DEF ≅ Δ ABC according to the congruence criterion: Angle, Side, Angle (ASA)

Consequently, we will deduce that:

BC=EFBC = EF

AC=DFAC = DF

since these are corresponding and equal sides in congruent triangles.

Then, we will also deduce that:

\\sphericalangle C=\sphericalangle F\)

since these are corresponding and equal angles in congruent triangles.


Example 2 - Triangle Congruence Exercise

Given two parallel lines. The line AC AC and the line BD BD pass between them in such a way that they intersect at point O O . Also, we are informed that AO=OCAO = OC

Example 1

Prove that AB=DCAB = DC


Proof:

First, we must show that triangles ΔABOΔ ABO and ΔDOCΔ DOC are congruent. We will base this on the previous criterion.

Note that AOB=COD \sphericalangle AOB = \sphericalangle COD (Because they are vertical angles)

Since AE=ECAE = EC (Side)

Remember that the two given lines are parallel lines.

Therefore, ​​OAB=OCD​​\sphericalangle OAB=\sphericalangle OCD since they are alternate interior angles between parallel lines (angle).

We observe that now we have 2 2 triangles in which 2 2 of their angles and the side included between them are equal.

Consequently, triangles ΔABOΔ ABO and ΔDOCΔ DOC are congruent

and we write it as ΔABOΔDOCΔ ABO ≅ Δ DOC according to the Angle, Side, Angle (ASA) congruence criterion

Therefore, we can deduce that AB=DCAB=DC (Corresponding sides of congruent triangles).

QED QED


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Angle-Side-Angle Congruence Exercises

Exercise 1

Assignment

Given that point K K bisects AC AC .

and also A=C ∢A=∢C

According to which theorem of congruence do the triangles ΔAMKΔCBK ΔAMK≅ΔCBK coincide?

Given that point K bisects AC

Solution

Given that the angles A=C A=C

AK=KC AK=KC

Given that point K K cuts AC AC

Angles AKM=CKB AKM=CKB

Opposite angles by the vertex are equal

The triangles are congruent according to the A.S.A A.S.A theorem

Answer

A.S.A A.S.A


Exercise 2

Assignment

Given: the quadrilateral ABCD ABCD is a rectangle.

By which theorem of congruence do the triangles ΔADOΔCBO ΔADO≅ΔCBO coincide?

the quadrilateral ABCD is a parallelogram.

Solution

BC=AD BC=AD

Since the quadrilateral ABCD ABCD is a rectangle and in a rectangle there are two pairs of parallel and equal sides

Angles BCO=ADO \sphericalangle BCO=\sphericalangle ADO are alternate angles between equal parallel lines.

Angles O1=O2 O_1=O_2

opposite angles at the vertex are equal

Therefore, we say that the triangles are congruent according to the A.A.S. A.A.S. theorem

Answer

According to the A.A.S. A.A.S. theorem


Exercise 3

Assignment

In the given figure: DEAB DE || AB

and point C C bisects segment BE BE .

According to which theorem of congruence do the triangles match?

ΔABCΔDEC ΔABC≅ΔDEC

Exercise 3 Assignment In the given figure

Solution

Given that DE DE is parallel to AB AB

Angles D=A D=A

Alternate angles are equal between parallel lines

BC=CE BC=CE Point C C bisects the line BE BE

Angles C1=C2 C_1=C_2

Vertex opposite angles

Triangles congruent according to the superposition theorem A.A.S..A.A.S..

Answer

Superimposed according to A.A.S. A.A.S.


Exercise 4

Assignment

Given the isosceles triangle ΔEDC ΔEDC .

ADE=BCE ∢ADE=∢BCE

AC=BD AC=BD

According to which theorem of congruence do the triangles coincide?

ΔADEΔBCE ΔADE≅ΔBCE

Exercise 4 Given the isosceles triangle EDC.

Solution

Triangle ΔEDC ΔEDC is an isosceles triangle

DE=EC DE=EC

In an isosceles triangle, two sides are equal

Angles D=C D=C given

angles EDC=ECDEDC=ECD the base angles of an isosceles triangle are equal

angles ADE=BCE ADE=BCE

DEDC=CECD \sphericalangle D-\sphericalangle EDC=\sphericalangle C-\sphericalangle ECD

Subtraction of angles

E1=E2 \sphericalangle E_1=\sphericalangle E_2

Triangles are congruent according to the ASA ASA theorem

Answer

ASA ASA


Exercise 5

Assignment

Given: quadrilateral ABCD ABCD square.

And within it is enclosed the kite KBPD KBPD .

According to which theorem of congruence do the triangles ΔBAKΔBCP ΔBAK≅ΔBCP coincide?

Exercise 5 quadrilateral ABCD square.

Solution

ABCD ABCD is a square

AB=BC AB=BC

In a square all sides are equal

Given that KBPD KBPD is a kite

BK=BP BK=BP

In a kite two pairs of adjacent sides are equal

C=A \sphericalangle C=\sphericalangle A

Equal angles in a square are 90° 90° degrees

BPC=BKA \sphericalangle BPC=\sphericalangle BKA

The angles of the kite are equal K=P \sphericalangle K=\sphericalangle P

therefore 180°K=180°P 180° -\sphericalangle K=180° -\sphericalangle P

ABK=PBC \sphericalangle ABK=\sphericalangle PBC

if two angles are equal then the third is also equal

The triangles are congruent according to S.A.S S.A.S

Answer

S.A.S S.A.S


Review Questions

What is a criterion in geometry?

In mathematics, a criterion is a guideline that allows us to determine certain characteristics depending on the case or topic being studied. In the case of geometry, it allows us to judge certain characteristics for given figures.


What is the triangle congruence criterion?

There are 4 4 triangle congruence criteria, which help us determine when two triangles are congruent, that is, they help us determine when two triangles have the same dimensions and corresponding angles, thus having the same shape and side measurements regardless of the orientation of these triangles.


What is the ASA criterion?

We say that two triangles are congruent with the ASA criterion when two of their angles and a non-included side are congruent.


What is the angle-side-angle criterion?

This criterion tells us that two triangles are congruent when two angles and the side between them are congruent.


How do you determine the criterion of a triangle?

This depends on the corresponding angles and sides of both triangles.

For example, given two triangles:

If we have three corresponding sides of two triangles that are congruent, then we are talking about the SSS criterion.

If it is found that the two angles and the included side between the corresponding angles are congruent, we refer to the ASA criterion.

When observing that two of their angles and the non-included side are congruent respectively, then we are discussing the AAS criterion.

And finally, when two pairs of corresponding sides and the included angle between these sides are congruent, it will be the SAS criterion.

Based on this, we can determine which congruence criterion we are referring to and deduce whether or not they are congruent triangles.