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Congruent Triangles

What are congruent triangles?

When all corresponding sides and corresponding angles of two triangles are of the same measure, then it is said to be congruent triangles. Triangles can be moved, rotated, flipped, and turned to look exactly the same. More so, they coincide with each other when they are moved in the same position. To show congruence between two triangles, we use the symbol ≅. 

The figure shows two congruent △ABC and △DEF where all corresponding sides and corresponding angles are equal.

Working through the lesson below will help your child to understand that congruent figures can be determined by a figure rotation, reflection, or translation or any combination of the three. It will also help them to identify the types of transformations in a sequence.

What are the corresponding parts of triangles?

When we learn about the congruent triangle, we hear the word CPCT. CPCT stands for “Corresponding Parts of Congruent Triangles.” Hence, if all parts of two triangles have the same measure, we use the term CPCT. 

How to identify corresponding parts of triangles?

For us to say that two triangles are congruent, we need to identify all the corresponding parts of the triangle. So, how do we do it?

We will list the corresponding vertices, sides, and angles using the same figure.

Corresponding VerticesA corresponds with D
B corresponds with E
C corresponds with F
Corresponding SidesAB = DE
BC = EF
AC = DF
Corresponding Anglesm∠A = m∠D
m∠B = m∠E
m∠C = m∠F

Note that you cannot interchange the letters because they should always correspond with the other vertex.

Example #1

Identify the corresponding parts of the two triangles.

Given the figure and its side measure and angle measure, we will list the corresponding vertices, corresponding sides, and corresponding angles.

Corresponding VerticesG corresponds with J
H corresponds with K
I corresponds with L
Corresponding SidesGH = 10; JK = 10; GH = JK
HI = 10; KL = 10; HI = KL
GI = 10√2; JL = 10√2; GI = JL
Corresponding Anglesm∠G = 45°; m∠J = 45°; m∠G = m∠J
m∠H = 90°; m∠K = 90°; m∠H = m∠K
m∠I = 45°; m∠L = 45°; m∠I = m∠L

Example #2

Identify the corresponding parts of the two triangles.

Given the figure and its side measure and angle measure, we will list the corresponding vertices, corresponding sides, and corresponding angles.

Corresponding VerticesM corresponds with R
N corresponds with P
O corresponds with Q
Corresponding SidesMN = 15; RP = 15; MN = RP
NO = 12.1; PQ = 12.1; NO = PQ
MO = 13.8; RQ = 13.8; MO = RQ
Corresponding Anglesm∠M = 49.4°; m∠R = 49.4°; m∠M = m∠R
m∠N = 60.26°; m∠P = 60.26°; m∠N = m∠P
m∠O = 70.35°; m∠Q = 70.35°; m∠O = m∠Q

Example #3

If △STU is congruent with △VWX, which parts of the two triangles are equal?

Solution

Since △STU ≅ △VWX, then, we first need to identify the corresponding parts of the triangles.

Corresponding VerticesS corresponds with V
T corresponds with W
U corresponds with X
Corresponding SidesST corresponds with VW
TU corresponds with WX
SU corresponds with VX
Corresponding Angles∠S corresponds with ∠V
∠T corresponds with ∠W
∠U corresponds with ∠X

Since they are congruent, then the corresponding sides and angles must also be equal in measure. Hence, 

Equal SidesST = VW
TU = WX
SU = VX
Equal Anglesm∠S = m∠V
m∠T = m∠W
m∠U = m∠X

What are the tests for congruence?

If two triangles are of the same size and shape, then they are said to be congruent. However, we do not need to find all three pairs of corresponding sides and all three pairs of corresponding angles to say that they are congruent. There are some criterion, tests, or postulates that we can easily follow to say that two triangles are indeed congruent. 

SSS Triangle Congruence Postulate

SSS Triangle Congruence Postulate means Side-Side-Side Triangle Congruence Postulate. When three sides of one triangle are congruent with the three corresponding sides of another triangle, the two triangles are congruent under the SSS Triangle Congruence Postulate. 

In the given figure, OG = CT, OD = CA, and DG = AT, then by SSS Triangle Congruence Postulate, △DOG ≅ △ACT.

Example #1

Determine if the two triangles are congruent.

Solution

Identify all the corresponding vertices and sides. 

Corresponding VerticesA corresponds with X
B corresponds with Y
C corresponds withW
Corresponding sidesAB = 17; XY = 17; AB = XY
BC = 12; YW = 12; BC = YW
AC = 21; XW = 21; AC = XW
Since the AB = XY, BC = YW, and AC = XW, then we can say that △ABC △XYW by SSS Triangle Congruence Postulate.

Example #2

Are two equilateral triangles congruent?

Explanation

Yes. By SSS Triangle Congruence Postulate, we can prove that two equilateral triangles are congruent since by definition, all sides of an equilateral are equal. Hence, all three sides of an equilateral are equal to the three corresponding sides of the other equilateral triangle. 

Example #3

If the measures of the side of △BAG are BA = 16, AG = 13, BG = 10 and the side measure of another triangle are LE = 13, ET = 16, LT = 10. Can we say that △BAG and △LET are congruent by SSS Triangle Congruence Postulate?

Solution

Step-by-step ProcessExplanation
B corresponds with L
A corresponds with E
G corresponds with T
Assume the corresponding vertices of △BAG and △LET.
BA corresponds with LE
AG corresponds with ET
BG corresponds with LT
Assume the corresponding sides of △BAG and △LET.
BA = 16; LE = 13; BA ≠ LE
AG = 13; ET = 16; AG ≠ ET
BG = 10; LT = 10; BG ≠ LT
Compare the side measure of each segments.
Therefore, △BAG and △LET are not congruent since not all three corresponding sides are of equal measure. 

SAS Triangle Congruence Postulate

SAS Triangle Congruence Postulate means Side-Angle-Side Triangle Congruence Postulate. When two sides and an included angle of one triangle are congruent to the corresponding two sides and an included angle of another triangle, the two triangles are said to be congruent by SAS Triangle Congruence Postulate. 

In the given figure, HT = JA, TU = AG, and the included angle ∠T ≅ ∠A, then by SAS Triangle Congruence Postulate, △HTU ≅ △JAG.

Example #1

Determine if the two triangles are congruent. 

Solution

Identify all the corresponding vertices, sides, and angles. 

Corresponding VerticesB corresponds with G
N corresponds with O
E corresponds with T
Corresponding SidesBN = 10; GO = 10; BN = GO
NE = 13; OT = 13; NE = OT
Corresponding Anglem∠N = 75°; m∠O = 75°; ∠N ≅ ∠O
Since the BN = GO, NE = OT, and the included angle between them ∠N ≅ ∠O, then we can say that △BNE △GOT by SAS Triangle Congruence Postulate.

Example #2

The measure of the sides of △HOP are HO = 25 and OP = 16, and the angle measure of the included angle is m∠O = 65°. Meanwhile, another triangle △SIT has side measures of SI = 25 and IT = 16 and the measure of the included angle is m∠I = 65°. Can we say that △HOP and △SIT are congruent by SAS Triangle Congruence Postulate?

Solution

Step-by-step ProcessExplanation
H corresponds with S
O corresponds with I
P corresponds with T
Assume the corresponding vertices of △HOP and △SIT.
HO corresponds with SI
OP corresponds with IT
HP corresponds with ST
Assume the corresponding sides of △HOP and △SIT.
HO = 25; SI = 25; HO = SI
OP = 16; IT = 16; OP = IT
Compare the side measure of each segments.
m∠O = 65°; m∠I = 65°; ∠O ≅ ∠ICompare the angle between the segments HO and OP, and SI and IT.
Therefore, by SAS Triangle Congruence Postulate, △HOP and △SIT are congruent since the two sides and the included angle are congruent with the corresponding sides of the other triangle. 

ASA Triangle Congruence Postulate

ASA Triangle Congruence Postulate means Angle-Side-Angle Triangle Congruence Postulate. So, when two angles and the included side of one triangle are congruent with the corresponding two angles and one included side of another triangle, then the two triangles are congruent. 

In the given figure, the angles ∠Y ≅ ∠T and ∠V ≅ ∠I and the included side between them YV = TI. Hence, △YVE and △TIM are congruent by ASA Triangle Congruence Postulate. 

Example #1

Determine if the two triangles are congruent.

Solution

Identify all the corresponding vertices, sides, and angles. 

Corresponding VerticesT corresponds with P
U corresponds with A
G corresponds with R
Corresponding Anglesm∠U = 55°; m∠A = 55°; ∠U ≅ ∠A
m∠G = 82°; m∠R = 82°; ∠G ≅ ∠R
Corresponding SideUG = 21; AR = 21; UG = AR
Since two angles and an included side are congruent, then by ASA Triangle Congruence Postulate, △TUG △PAR.

Example #2

The measure of the angles of △BLK are m∠B = 30° and m∠K = 100°, and the side in between them measures BK = 40. Another triangle, △WYT has angle measures of m∠W = 30° and m∠Y = 100° and a side of WT = 40, is it true that △BLK is congruent with △WYT by ASA Triangle Congruence Postulate?

Solution

Step-by-step ProcessExplanation
B corresponds with W
L corresponds with Y
K corresponds with T
Assume the corresponding vertices of BLK and WYT.
B corresponds with W
L corresponds with Y
K corresponds with T
Assume the corresponding angles of BLK and WYT.
BL corresponds with WY
LK corresponds with YT
BK corresponds with WT
Assume the corresponding sides of BLK and WYT.
m∠L = 180° – (m∠B + m∠K)
m∠L = 180° – (30° + 100°)
m∠L = 180° – 130°
m∠L = 50°
Find the measure of the unknown angle L.  
m∠T = 180° – (m∠W + m∠Y)
m∠T = 180° – (30° + 100°)
m∠T = 180° – 130°
m∠T = 50°
Find the measure of the unknown angle T.
m∠B = 30°; m∠W = 30°; ∠B ≅ ∠W
m∠L = 50°; m∠Y = 100°; m∠L ≠ m∠Y
m∠K = 100°; m∠T = 50°; m∠K ≠ m∠T
Compare the angle measure of each corresponding angle. 
BK = 40; WT = 40; BK = WTCompare the given side measure to its corresponding side. 
Therefore, we △BLK and △WYT are not congruent because even though both triangles have one equal corresponding side, no two corresponding angles are of equal measure. Hence, we cannot say that that it is congruent by ASA Triangle Congruence Postulate.

AAS Triangle Theorem

AAS Triangle Theorem means Angle-Angle-Side Triangle Theorem. The AAS Triangle Theorem states that if the two angles and a non-included side of one triangle are congruent to the corresponding two angles and a non-included side, then the two triangles are said to be congruent. 

Two triangles △PAT and △FGO are said to be congruent since ∠T ≅ ∠O and ∠A ≅ ∠G and the AP = GF which is the corresponding non-included side are also congruent with each other. Hence, by AAS Triangle Theorem, △PAT ≅ △FGO. 

Example

The angle measures of △LEN are m∠L = 95° and m∠E = 20° where a side not between them EN measures 6 centimeters. Meanwhile, △CAM has angles that measures m∠C = 95° and m∠A = 20° with a non-included side that measures AM = 6 cm. Can we say that △LEN and △CAM are congruent?

Solution

Step-by-step ProcessExplanation
L corresponds with C
E corresponds with A
N corresponds with M
Assume the corresponding vertices of △LEN and △CAM.
L corresponds with C
E corresponds with A
N corresponds with M
Assume the corresponding angles of △LEN and △CAM.
LE corresponds with CA
EN corresponds with AM
LN corresponds with CM
Assume the corresponding sides of △LEN and △CAM. 
m∠L = 95°; m∠C = 95°; ∠L ≅ ∠C
m∠E = 20°; m∠A = 20°; ∠E ≅ ∠A
Compare the angle measure of each corresponding angle.
EN = 6; AM = 6; EN = AMCompare the side measure of the corresponding non-included side of the two triangles. 
Since the two corresponding angles of the two triangles are congruent, and the non-included sides are also congruent. Then, by AAS Triangle Theorem, △LEN △CAM

RHS Triangle Theorem

RHS Triangle Theorem means Right-Hypotenuse-Side Triangle Theorem. In this theorem, it states that if the hypotenuse and the side of a right triangle are congruent to the corresponding hypotenuse and the side of the other right triangle, then the two right triangles are said to be congruent by RHS Triangle Theorem. 

In the given right triangle △GWY and △LUN,  ∠G ≅ ∠L because they both measure 90°. More so, the hypotenuse of △GWY and △LUN which is WY and UN are also congruent, respectively. Lastly, the sides GY of △GWY and NL of △LUN are also congruent. Hence, by RHS Triangle Theorem, △GWY ≅ △LUN. 

Example #1

Determine if the two triangles are congruent. 

Solution

Step-by-step ProcessExplanation
Y corresponds with C
W corresponds with A
X corresponds with B
Identify the corresponding vertices of △YWX and △CAB.
∠Y corresponds with ∠C
∠W corresponds with ∠A
∠X corresponds with ∠B
Identify the corresponding angles of △YWX and △CAB.
YW corresponds with CA
WX corresponds with AB
YX corresponds with CB
Identify the corresponding sides of △YWX and △CAB.
m∠W = 90°; m∠A = 90°; ∠W ≅ ∠ACompare the angle measure of each corresponding angle.
YW = 20; CA = 20; YW = CA
BC = 35; XY = 35; BC = XY
Compare the side measure of each corresponding side.
Since we have two right-angled triangles where the hypotenuse are of equal measure and the corresponding sides are also congruent, then we can say that △YWX and △CAB are congruent by RHS Triangle Theorem. 

Example #2

△KRC is a right triangle where the hypotenuse RC measures 15 units and the other side KC measures 10 units. Meanwhile, another triangle △SMJ is also a right triangle with the same hypotenuse measure as KRC, and SM measures 10 units. Is it true that △KRC and △SMJ are congruent by RHS Triangle Theorem?

Solution

Step-by-step ProcessExplanation
K corresponds with S
R corresponds with M
C corresponds with J
Assume the corresponding vertices of △KRC and △SMJ.
∠K corresponds with ∠S
∠R corresponds with ∠M
∠C corresponds with ∠J
Assume the corresponding angles of △KRC and △SMJ.
KR corresponds with SM
RC corresponds with MJ
KC corresponds with SJ
Assume the corresponding sides of △KRC and △SMJ. 
m∠R = 90°; m∠M = 90°; ∠R ≅ ∠MCompare the angle measure of each corresponding angle.
RC = 15; MJ = 15; RC = MJCompare the hypotenuse of each triangle.
KR = $\sqrt{RC^2- KC^2}$
KR = $\sqrt{15^2- 10^2}$
KR = $\sqrt{225- 100}$
KR = $\sqrt{125}$
KR = 5$\sqrt{5}$
Find the measure of the other segment of △KRC using the Pythagorean Theorem. 
SJ = $\sqrt{MJ^2- SM^2}$
SJ = $\sqrt{15^2- 10^2}$
SJ = $\sqrt{225- 100}$
SJ = $\sqrt{125}$
SJ = 5$\sqrt{5}$
Find the measure of the other segment of △SMJ using the Pythagorean Theorem.
KR = 5$\sqrt{5}$; SM = 10; KR ≠ SM
KC = 10; SJ = 5$\sqrt{5}$; KC ≠ SJ
Compare the side measure of each corresponding side.
Hence, even if both triangles are right-angled and have the same hypotenuse measure, △KRC and △SMJ are not congruent since we need a hypotenuse and another side that have the same measure.

Congruent Triangles (and other figures)

This section will help your child to identify the characteristics of congruent figures.

Two figures are congruent if they are the:

  • Exact same shape
  • Exact same size
    • Angle measures are equal
    • Line segments are equal

Look at the example below.

Discuss the examples and questions below with your child regarding whether the figures are congruent.

Which figure is congruent to figure C shown below?

Figure b. is congruent.

Transformations : Rotations, Reflections, & Translations

This section will help your child to perform a transformation (rotation, reflection, and translation) on a figure .

Make sure your child is familiar with the vocabulary below:

  • Transformation moves a figure from its original place to a new place.
  • Angle of Rotation: How big the angle is that you rotate a figure.  Common angle rotations are 45°, 90°, 180°.
  • Isometric Transformation: A transformation that does not change the size of a figure.

There are three types of transformations. Alternative names are in parenthesis:

  1. Rotation (Turn): Turns a figure around a fixed point. 
  2. Reflection (Flip): Flip of figure over a line where a mirror image is created.
  3. Translation (Slide or glide): Sliding a shape to a new place without changing the figure.

Rotations, reflections, and translations are isometric.  That means that these transformations do not change the size of the figure.  If the size and shape of the figure is not changed, then the figures are congruent.

Explore and discuss the examples of transformations below with your child.

Try It! Find a flat object in your home that can easily be moved (small book, calculator, drink coaster, coin, etc.)  Perform each transformation using that object.

Multiple Transformations

This section will help your child to understand that congruent figures can have more than one transformation.

Make sure your child is familiar with the vocabulary below:

  • Sequence: A group of things arranged in a certain order.  Commonly known as a pattern.

Recapping from earlier in his lesson, there are three types of transformation:

  1. Rotation (Turn): Turns a figure around a fixed point. 
  2. Reflection (Flip): Flip of figure over a line where a mirror image is created.
  3. Translation (Slide or glide): Sliding a shape to a new place without changing the figure.

Two Transformations

Try It! Look at the figure below. What transformations does parallelogram Z perform?

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