Properties of Congruent Triangles

Figures having the same shape and size are called congruent figures. Are the following pairs of figures the same? Congruence They are the same!

If two triangles have the same shape and size, they are called congruent triangles. Congruent Triangles and their Properties A X B Y CZ For the congruent triangles △ ABC and △ XYZ above,  A =  X,  B =  Y,  C =  Z AB = XY,BC = YZ,CA = ZX

 A and  X,  B and  Y,  C and  Z are called AB and XY, BC and YZ, CA and ZX are called A and X, B and Y, C and Z are called AX B Y C Z corresponding vertices. corresponding sides. corresponding angles.

is congruent to  △XYZ△XYZ △ABC△ABC The corresponding vertices of congruent triangles should be written in the same order. In the above example, we can also write △ BAC  △ YXZ, but NOT △ CBA  △ XYZ. A X B Y CZ The properties of congruent triangles are as follows:  AB = XY, BC = YZ,  A =  X,  B =  Y,  C =  Z CA = ZX (ii)All their corresponding sides are equal. (i)All their corresponding angles are equal,

If △ ABC  △ XYZ … X Y Z 4 cm A B C 4.5 cm 40° According to the properties of congruent triangles, AB = 4.5 cm AC =4 cm  B = 40° △  △ A B C XYZ XY XZ Y == = 

In the figure, △ ABC  △ PQR. Find the unknowns. Follow-up question 1 A B C 130° 30° x cm 4 cm 7 cm P Q R y 9 cm z cm According to the properties of congruent triangles, AC  PR CBy   180    20  4  z BCQR  9  xAP 

Example 1 In the figure, AB = 5 cm, AC = 4 cm and BC = 7 cm. If △ ABC  △ DFE, find DE, EF and DF. Solution

Example 2 In the figure, AB = 7 cm, ∠ A = 50° and ∠ B = 30°. If △ ABC  △ PRQ, find PR, ∠ P and ∠ R. Solution

Conditions for Congruent Triangles

Yes, because AB = XY, BC = YZ, CA = ZX, ∠ A = ∠ X, ∠ B = ∠ Y, ∠ C = ∠ Z. But, can we say that two triangles are congruent when only some of the properties of congruence are satisfied? Are these two triangles congruent? A C B X Y Z Yes… let’s see the following 5 conditions for congruent triangles first.

Condition ISSS In △ ABC and △ XYZ, if AB = XY, BC = YZ and CA = ZX, then △ ABC  △ XYZ. [Abbreviation: SSS] C A B Z X Y

For example, T U V 2 cm 4 cm 5 cm D E F 2 cm 5 cm 4 cm TU = FE,UV = ED, TV = FD △ TUV  △ FED (SSS)

In △ ABC and △ XYZ, if AB = XY, AC = XZ and  A =  X, then △ ABC  △ XYZ. [Abbreviation: SAS] Condition IISAS C A B Z X Y Note that ∠ A and ∠ X are the included angles of the 2 given sides.

For example, UV = FE,TV = DE,  V =  E △ TUV  △ DFE (SAS) T U V 120° 2 cm 2.5 cm D E F 2 cm 120°

Follow-up question 2 Determine whether each of the following pairs of triangles are congruent and give the reason. (a) A B C E G F 3 cm 3.2 cm 3 cm 3.2 cm 2.5 cm △ ABC  △ EGF (SSS) ◄ AB = EG, BC = GF, AC = EF

(b) I J K 2 cm 2.5 cm 50° N M L 2 cm 2.5 cm 50° △ IJK  △ MNL (SAS) ◄ IJ = MN, ∠ J = ∠ N, JK = NL

Example 3 Are △ MNP and △ YZX in the figure congruent? If they are, give the reason. Yes, △ MNP  △ YZX. (SSS) Solution

Example 4 In the figure, WX = WY and ZX = ZY. Are △ WXZ and △ WYZ congruent? If they are, give the reason. Solution Yes, △ WXZ  △ WYZ. (SSS)

Example 5 Which two of the following triangles are congruent? Give the reason. Solution △ PQR  △ WUV (SAS)

Example 6 In the figure, AB = CD = 8 cm and ∠ ABD = ∠ CDB = 30°. Are △ ABD and △ CDB congruent? If they are, give the reason. Yes, △ ABD  △ CDB. (SAS) Solution

Example 7 Which two of the following triangles are congruent? Give the reason. Solution △ DEF  △ ZYX (ASA)

In △ ABC and △ XYZ, if  A =  X,  B =  Y and AB = XY, then △ ABC  △ XYZ. [Abbreviation: ASA] Condition IIIASA C A B Z X Y Note that AB and XY are the included sides of the 2 given angles.

U T V 130° 20° 4 cm D E F 130° 20° 4 cm For example,  U =  F, UV = FD,  V =  D △ TUV  △ EFD (ASA)

Condition IVAAS In △ ABC and △ XYZ, if  A =  X,  B =  Y and AC = XZ, then △ ABC  △ XYZ. [Abbreviation: AAS] C A B Z X Y Note that AC and XZ are the non-included sides of the 2 given angles.

T U V D E F 130° 20° 7 cm For example,  U =  F, TV = ED  V =  D, △ TUV  △ EFD (AAS) 130°

Follow-up question 3 In each of the following, name a pair of congruent triangles and give the reason. (a) A B C E F G 45° 40° 45° 5.25 cm △ ABC  △ FEG (ASA) ◄ ∠ B = ∠ E, BC = EG, ∠ C = ∠ G

(b) △ IJK  △ MNL (AAS) K I L J M N 100° 20° 100° 12 cm B A C 100° 12 cm 20° ◄ ∠ J = ∠ N, ∠ K = ∠ L, IK = ML

Example 8 In the figure, ∠ BAD = ∠ CAD and AD ⊥ BC. Are △ ABD and △ ACD congruent? If they are, give the reason. Solution Yes, △ ABD  △ ACD. (ASA)

Example 9 Which two of the following triangles are congruent? Give the reason. Solution △ PQR  △ ZYX (AAS)

Example 10 In the figure, ∠ ABC = ∠ CDA and ∠ ACB = ∠ CAD. Are △ ABC and △ CDA congruent? If they are, give the reason. Solution Yes, △ ABC  △ CDA. (AAS)

In △ ABC and △ XYZ, if  C =  Z = 90°, AB = XY and BC = YZ (or AC = XZ), then △ ABC  △ XYZ. [Abbreviation: RHS] Condition VRHS A B C X Y Z

For example, 2 cm 5 cm T U V D E F  U =  F = 90°, TV = ED,TU = EF △ TUV  △ EFD (RHS)

Are there any congruent triangles? Give the reason. Follow-up question 4 A B C D 6 cm Yes, △ ABC  △ ADC. (RHS) ◄ ∠ B = ∠ D = 90°, AC = AC, BC = DC

C A B Z X Y 1. SSS C A B Z X Y 2. SAS C A B Z X Y 3. ASA C A B Z X Y 4. AAS A BC X YZ 5. RHS To sum up, two triangles are said to be congruent if any ONE of the following FIVE conditions is satisfied.

Example 11 Are △ ABC, △ RPQ and △ XYZ in the figure congruent? If they are, give the reasons. Solution △ ABC  △ RPQ (RHS) △ XYZ  △ RPQ (SAS) ∴△ ABC, △ RPQ and △ XYZ are congruent.

Example 12 In the figure, AB ⊥ BC, DC ⊥ BC and AC = DB. Are △ ABC and △ DCB congruent? If they are, give the reason. Solution Yes, △ ABC  △ DCB. (RHS)

Properties of Similar Triangles

Similar figures have the same shape but not necessarily the same size. The following pairs of figures have the same shape, they are called similar figures. Similarity

Similar Triangles and their Properties If two triangles have the same shape, they are called similar triangles. For the similar figures △ ABC and △ XYZ above,  A =  X,  B =  Y,  C =  Z ABXYABXY = BCYZBCYZ CAZXCAZX = A X B Y C Z

A X B Y C Z  A and  X,  B and  Y,  C and  Z are called AB and XY, BC and YZ, CA and ZX are called A and X, B and Y, C and Z are calledcorresponding vertices. corresponding sides. corresponding angles.

A X B Y C Z The properties of similar triangles are as follows:  A =  X,  B =  Y,  C =  Z ABXYABXY = BCYZBCYZ CAZXCAZX = (ii)All their corresponding sides are proportional. (i)All their corresponding angles are equal, ~is similar to △XYZ△XYZ △ABC△ABC Note: The corresponding vertices of congruent triangles should be written in the same order.

If △ ABC ~ △ XYZ... 4 cm A B C 4.5 cm 40° X Y Z 2 cm According to the properties of similar triangles, B  Y   40  AC XZ AB XY  cm  XY cm 25.2  △ ~ △ A BCXYZ ◄ XZ and AC are corresponding sides.

In the figure, △ ABC ~ △ PQR. Find the unknowns. Follow-up question 5 A B C 132° 25° 10 cm 4 cm x cm According to the properties of similar triangles, P Q R y 5 cm z cm 4 cm PR AC PQ AB   x 8  x 2  z AP  CBy   180    23  AC PR BC QR   z

Example 13 In △ ABC and △ RQP, BC = 1 cm, PQ = 2 cm, QR = 5 cm and PR = 4 cm. If △ ABC ~ △ RQP, find AB and AC.

Solution

Example 14 In the figure, AD = 3 cm, AC = 2 cm, CE = 4 cm, ∠ A = 60° and BC ⊥ AE. If △ ABC ~ △ AED, find ∠ E and AB.

Solution

Conditions for Similar Triangles

(i)All their corresponding angles are equal. (ii)All their corresponding sides are proportional. A B C X Y Z We have learnt that if two triangles are similar, then Two triangles are similar if any one of the following three conditions is satisfied.

In △ ABC and △ XYZ, if  A =  X,  B =  Y and  C =  Z, then △ ABC ~ △ XYZ. [Abbreviation: AAA] Condition IAAA A B C X Y Z

 U =  F,  T =  E,  V =  D △ TUV ~ △ EFD (AAA) For example, T U V 127° 25° 28° D E F 127° 25° 28°

Condition II3 sides prop. In △ ABC and △ XYZ, if then △ ABC ~ △ XYZ. [Abbreviation: 3 sides prop.], ZX CA YZ BC XY AB  A B C X Y Z

T U V 4 cm 2 cm 3 cm D E F 1.5 cm 1 cm 2 cm For example, △ TUV ~ △ DFE (3 sides prop.) UV 2 cm FE 1 cm == 2, TV 3 cm DE 1.5 cm == 2, TU 4 cm DF 2 cm == 2

Condition IIIratio of 2 sides, inc.  A B C In △ ABC and △ XYZ, if and  B =  Y, then △ ABC ~ △ XYZ. [Abbreviation: ratio of 2 sides, inc.  ] X Y Z

For example, D E F 120° 2 cm 1.5 cm V 4 cm U T 120° 3 cm △ TUV ~ △ EFD (ratio of 2 sides, inc.  ) UV 4 cm FD 2 cm == 2, UT 3 cm FE 1.5 cm == 2,  U =  F

Follow-up question 6 Determine whether each of the following pairs of triangles are similar and give the reason. (a) E G F 2.4 cm 2.8 cm 2 cm A B C 3 cm 3.5 cm 2.5 cm △ ABC ~ △ EGF (3 sides prop.) AB EG BC GF ◄ = AC EF =

(b) △ ABC ~ △ ZXY (AAA) C A B 95° 40° 45° X Y Z 40° 95° I J K 2.4 cm 3 cm 50° N M L 1.6 cm 2 cm 50° △ IJK ~ △ MNL (ratio of 2 sides, inc.  ) (c) ◄ ∠ A = ∠ Z, ∠ B = ∠ X, ∠ C = ∠ Y IJ MN JK NL ◄ =, ∠ J = ∠ N

Example 15 Are △ ABC and △ XZY in the figure similar? If they are, give the reason. Yes, △ ABC ~ △ XZY. (AAA) Solution

Example 16 Are △ ABC and △ QRP in the figure similar? If they are, give the reason. Solution

Example 17 Which two of the following triangles are similar? Give the reason. Solution

Example 18 In the figure, AB = 15 cm, BC = 12 cm, AC = 9 cm, BD = 20 cm and CD = 16 cm. Are △ ABC and △ BDC similar? If they are, give the reason. Solution

Example 19 Which two of the following triangles are similar? Give the reason.

Solution

Example 20 In the figure, PT = 2.7 cm, TR = 3.3 cm, QR = 3 cm, TS = 4.8 cm, RS = 2.4 cm and ∠ PRQ = ∠ TSR. Are △ PQR and △ TRS similar? If they are, give the reason. Solution

Example 2 (Extra) In the figure, △ ABC  △ EDF and △ FED  △ IHG. Find GH, HI and IG. According to the properties of congruent triangles, Solution

Example 10 (Extra) In the figure, AB = PQ, ∠ ABC = ∠ QRP and ∠ ACB = ∠ PQR. Are △ ABC and △ QRP congruent? If they are, give the reason. Solution Cannot be determined. Since the length of PR may not equal to AB.

Example 12 (Extra) In the figure, AGE, CGF and BCD are straight lines. (a)Are △ ABC and △ CDE congruent? If they are, give the reason. (b)Are △ FAC and △ FEC congruent? If they are, give the reason. Solution

Example 20 (Extra)

Solution