Chapter 5.6 Proving Triangle Congruence by ASA and AAS

5.6 Proving Triangle Congruence by ASA and AAS Exercise 1 (a) The AAS Congruence Theorem and ASA Congruence Theorem are similar in the sense that they are used to prove that two triangles are congruent. Both AAS and ASA require two pairs of corresponding angles to be congruent. (b) When applying AAS Congruence Theorem, we consider a non-included side to prove that triangles are congruent. But when applying ASA Congruence Theorem, we consider an included side to prove that triangles are congruent. Exercise 2 To show that triangles are congruent using AAS Congruence Theorem or ASA Congruence, we need to know the pair of corresponding congruent side in addition to the pairs of congruent corresponding angles. Exercise 3 To prove that ∆ ABC ≅ ∆ QRS, we need to know the corresponding congruent parts. From the diagram, we can see that they have equal corresponding sides and angles: ( 𝐴𝐴𝐴𝐴 ���� ) ≅ ( 𝑄𝑄𝑄𝑄 ���� ) ∠ B ≅∠ R ∠ A ≅∠ Q Hence, enough information is given to prove that the triangles are congruent by AAS Congruence Theorem. Exercise 4 To prove that ∆ ABC ≅ ∆ DBC, we need to know the corresponding congruent parts. From the diagram, we can see that they have equal corresponding sides and angles: ( 𝐴𝐴𝐶𝐶 ���� ) ≅ ( 𝐴𝐴𝐶𝐶 ���� ) (common sides) ∠ ACB ≅∠ DCB (right angles) ∠ A ≅∠ D Hence, enough information is given to prove that the triangles are congruent by AAS Congruence Theorem. Exercise 5 To prove that ∆ XYZ ≅ ∆ JKL, we need to know the corresponding congruent parts. From the diagram, we can see that they have equal corresponding sides and angles: ( 𝑋𝑋𝑋𝑋 ���� ) ≅ ( 𝐽𝐽𝐽𝐽 ��� ) ( 𝑋𝑋𝑌𝑌 ���� ) ≅ ( 𝐽𝐽𝐾𝐾 ���� ) ∠ Z ≅∠ L The triangle is congruent based on SSA Congruent Theorem Hence, there is not enough information is given to prove that the triangles are congruent by AAS Congruence Theorem. Exercise 6 To prove that ∆ RSV ≅ ∆ UTV, we need to know the corresponding congruent parts. From the diagram, we can see that they have equal corresponding sides and angles: ( 𝑄𝑄𝑆𝑆 ���� ) ≅ ( 𝑇𝑇𝑆𝑆 ���� ) ∠ S ≅∠ T ∠ RVS ≅∠ UVT(cross angles) The triangle is congruent based on ASA Congruence Theorem. Hence, enough information is given to prove that the triangles are congruent by ASA

Congruence Theorem. Exercise 7 To prove that ∆ FGH and ∆ LMN are congruent using AAS Congruence Theorem, they must have two angles and a non-included side. We are given that: ( 𝐺𝐺𝐺𝐺 ���� ) ≅ ( 𝑀𝑀𝑀𝑀 ����� ) ∠ G ≅∠ M The 3rd congruence statement needed is ∠ F ≅∠ L Exercise 8 To prove that ∆ FGH and ∆ LMN are congruent using ASA Congruence Theorem, they must have two angles and the included side. We are given that: ( 𝐹𝐹𝐺𝐺 ���� ) ≅ ( 𝐾𝐾𝑀𝑀 ���� ) ∠ G ≅∠ M The 3rd congruence statement needed is ∠ F ≅∠ L Exercise 9 To prove that ∆ ABC and ∆ DEF are congruent we are given that : ∠ A ≅∠ D ∠ C ≅∠ F ( 𝐴𝐴𝐴𝐴 ���� ) ≅ ( 𝐷𝐷𝐹𝐹 ���� ) When we compare the two triangles, we see that they have equal corresponding parts. The corresponding congruent angles are: ∠ A ≅∠ D ∠ B ≅∠ E ∠ C ≅∠ F The corresponding congruent sides are: ( 𝐴𝐴𝐶𝐶 ���� ) ≅ ( 𝐷𝐷𝐷𝐷 ���� ) ( 𝐶𝐶𝐴𝐴 ���� ) ≅ ( 𝐷𝐷𝐹𝐹 ���� ) ( 𝐴𝐴𝐴𝐴 ���� ) ≅ ( 𝐷𝐷𝐹𝐹 ���� ) Using ASA Congruence Theorem, we can prove that ∆ ABC and ∆ DEF are congruent. This is because; two angles and the included sides given are congruent Hence, the given information ∠ A ≅∠ D, ∠ C ≅∠ F,( 𝐴𝐴𝐴𝐴 ���� ) ≅ ( 𝐷𝐷𝐹𝐹 ���� ) is enough to prove that the two triangles are congruent. Exercise 10 To prove that ∆ ABC and ∆ DEF are congruent we are given that : ∠ C ≅∠ F ( 𝐴𝐴𝐶𝐶 ���� ) ≅ ( 𝐷𝐷𝐷𝐷 ���� ) ( 𝐶𝐶𝐴𝐴 ���� ) ≅ ( 𝐷𝐷𝐹𝐹 ���� ) When we compare the two triangles, we see that they have equal corresponding parts. The corresponding congruent angles are: ∠ A ≅∠ D ∠ B ≅∠ E ∠ C ≅∠ F The corresponding congruent sides are: ( 𝐴𝐴𝐶𝐶 ���� ) ≅ ( 𝐷𝐷𝐷𝐷 ���� )

( 𝐶𝐶𝐴𝐴 ���� ) ≅ ( 𝐷𝐷𝐹𝐹 ���� ) ( 𝐴𝐴𝐴𝐴 ���� ) ≅ ( 𝐷𝐷𝐹𝐹 ���� ) Hence, the given information ∠ C ≅∠ F,( 𝐴𝐴𝐶𝐶 ���� ) ≅ ( 𝐷𝐷𝐷𝐷 ���� ),( 𝐶𝐶𝐴𝐴 ���� ) ≅ ( 𝐷𝐷𝐹𝐹 ���� ) is not enough to prove that the two triangles are congruent. This is because, ∠ C ≅∠ F is not the included angle. Exercise 11 To prove that ∆ ABC and ∆ DEF are congruent we are given that : ∠ B ≅∠ E ∠ C ≅∠ F ( 𝐴𝐴𝐴𝐴 ���� ) ≅ ( 𝐷𝐷𝐷𝐷 ���� ) When we compare the two triangles, we see that they have equal corresponding parts. The corresponding congruent angles are: ∠ A ≅∠ D ∠ B ≅∠ E ∠ C ≅∠ F The corresponding congruent sides are: ( 𝐴𝐴𝐶𝐶 ���� ) ≅ ( 𝐷𝐷𝐷𝐷 ���� ) ( 𝐶𝐶𝐴𝐴 ���� ) ̅ ≅ ( 𝐷𝐷𝐹𝐹 ���� ) ( 𝐴𝐴𝐴𝐴 ���� ) ≅ ( 𝐷𝐷𝐹𝐹 ���� ) Hence, the given information ∠ B ≅∠ E, ∠ C ≅∠ F,( 𝐴𝐴𝐴𝐴 ���� ) ≅ ( 𝐷𝐷𝐷𝐷 ���� ) is not enough to prove that the two triangles are congruent. This is because, ( 𝐴𝐴𝐴𝐴 ���� ) ≅ ( 𝐷𝐷𝐷𝐷 ���� ) is not a corresponding side for the given angle. Exercise 12 To prove that ∆ ABC and ∆ DEF are congruent we are given that : ∠ A ≅∠ D ∠ B ≅∠ E ( 𝐶𝐶𝐴𝐴 ���� ) ≅ ( 𝐷𝐷𝐹𝐹 ���� ) When we compare the two triangles, we see that they have equal corresponding parts. The corresponding congruent angles are: ∠ A ≅∠ D ∠ B ≅∠ E ∠ C ≅∠ F The corresponding congruent sides are: ( 𝐴𝐴𝐶𝐶 ���� ) ≅ ( 𝐷𝐷𝐷𝐷 ���� ) ( 𝐶𝐶𝐴𝐴 ���� ) ≅ ( 𝐷𝐷𝐹𝐹 ���� ) ( 𝐴𝐴𝐴𝐴 ���� ) ≅ ( 𝐷𝐷𝐹𝐹 ���� ) Using AAS Congruence Theorem, we can prove that ∆ ABC and ∆ DEF are congruent. This is because; two angles and a non-included side given are congruent. Hence, the given information ∠ A ≅∠ D, ∠ B ≅∠ E,( 𝐶𝐶𝐴𝐴 ���� ) ≅ ( 𝐷𝐷𝐹𝐹 ���� ) is enough to prove that the two triangles are congruent. Exercise 13 1) Construct AB so that it is congruent to DF. 2) Construct ∠ A with vertex A and side AB so that it is congruent to ∠ D. 3) Construct ∠ B with vertex B and side BD so that it is congruent to ∠ F. 4) Label the intersection of the sides of ∠ A and ∠ B that you constructed in Step 2 and Step 3 as C. 5) Join A and C, B and C. \triangle ABC is the required triangle.

So, the image of the construction, Exercise 14 1) Construct AB so that it is congruent to DF. 2) Construct ∠ A with vertex A and side AB so that it is congruent to ∠ D. 3) Construct ∠ B with vertex B and side BD so that it is congruent to ∠ F. 4) Label the intersection of the sides of ∠ A and ∠ B that you constructed in Step 2 and Step 3 as C. 5) Join A and C, B and C. \triangle ABC is the required triangle. So, the image of the construction, Exercise 15 Using ASA Congruence Theorem, we can prove that ∆ JKL and ∆ FGH are congruent if two angles and the included sides given are congruent. From the diagram, we can see that the corresponding congruent parts given are: ( 𝐽𝐽𝐽𝐽 ��� ) ≅ ( 𝐺𝐺𝐺𝐺 ���� ) ∠ K ≅∠ G ∠ J ≅∠ F Based on ASA Congruence Theorem, ∆ JKL and ∆ FGH are congruent. But, the error is in that the vertices of ∆ FGH is written in a wrong order. It was written as

∆ FHG instead of ∆ FGH. Exercise 16 Using AAS Congruence Theorem, we can prove that ∆ QRS and ∆ VWX are congruent if two angles and the non-included sides given are congruent. From the diagram, we can see that the corresponding congruent parts given are: ( 𝑄𝑄𝑄𝑄 ���� ) ≅ ( 𝑆𝑆𝑉𝑉 ����� ) ∠ Q ≅∠ V ∠ R ≅∠ W The error is that ( 𝑄𝑄𝑄𝑄 ���� ) ≅ ( 𝑆𝑆𝑉𝑉 ����� ) is not the non-included sides based on AAS Congruence Theorem. Hence, ∆ QRS and ∆ VWX are not congruent using AAS Congruence Theorem. But, the triangles are congruent based on ASA Congruence Theorem. Exercise 17 Using ASA Congruence Theorem, we can prove that ∆ NQM and ∆ MPL are congruent if two angles and the included sides given are congruent We are given that: M is the midpoint of ( 𝑀𝑀𝐾𝐾 ���� ) ( 𝑀𝑀𝐾𝐾 ���� ) ⊥ ( 𝑀𝑀𝑄𝑄 ���� ) ( 𝑀𝑀𝐾𝐾 ���� ) ⊥ ( 𝑀𝑀𝑀𝑀 ����� ) (QM) ⊥ (PL) From the diagram, we can see that ∠ QNM and ∠ PML are right angles. Hence, ∠ QNM ≅ ∠ PML because all right angles are congruent Recall that if two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. ⟹∠ QMN ≅ ∠ PLM Therefore, ∆ NQM and ∆ MPL are congruent based on ASA Congruence Theorem. Exercise 18 Using ASA Congruence Theorem, we can prove that ∆ ABK and ∆ CBJ are congruent if two angles and the included sides given are congruent We are given that: M is the midpoint of ( 𝑀𝑀𝐾𝐾 ���� ) ( 𝐴𝐴𝐽𝐽 ��� ) ≅ ( 𝐽𝐽𝐴𝐴 ���� ) ∠ BJK ≅ ∠ BKJ ∠ A ≅ ∠ C From the diagram, we can see that: ( 𝐽𝐽𝐽𝐽 ��� ) ≅ ( 𝐽𝐽𝐽𝐽 ��� ) (By Reflexive property of equality) ( 𝐴𝐴𝐽𝐽 ���� ) ≅ ( 𝐴𝐴𝐽𝐽 ��� ) (By definition of congruent segments) Therefore, ∆ ABK and ∆ CBJ are congruent based on ASA Congruence Theorem. Exercise 19 Using AAS Congruence Theorem, we can prove that ∆ XWV and ∆ ZWU are congruent if two angles and the non-included sides given are congruent We are given that: ( 𝑆𝑆𝑉𝑉 ����� ) ≅ ( 𝑈𝑈𝑉𝑉 ����� ) ∠ X ≅ ∠ Z

From the diagram, we can see that by the reflexive property of congruence: ∠ W ≅ ∠ W Hence, based on AAS Congruence Theorem ∆ XWV ≅ ∆ ZWU. Exercise 20 Using AAS Congruence Theorem, we can prove that ∆ NMK and ∆ LKM are congruent if two angles and the non-included sides given are congruent We are given that: ∠ L ≅ ∠ N ∠ NKM ≅ ∠ LMK From the diagram, we can see that by the reflexive property of congruence: ( 𝐽𝐽𝑀𝑀 ����� ) ≅ ( 𝑀𝑀𝐽𝐽 ����� ) Hence, based on AAS Congruence Theorem ∆ NMK ≅ ∆ LKM . Exercise 21 Hypotenuse Angle Theorem is true for the right triangles only because, the hypotenuse is one of the right-angled triangle legs. It is a criterion used to prove whether a given set of right triangles are congruent. Given that a triangle is a right triangle, only 1 angle out of the 3 angles is a right angle. Recall that the sum of interior angle s of a triangle =180° Since a right triangle has 1 right angle, the other 2 angles will be acute. To find the 3rd angle in a right triangle =180°-90°-m ∠ 2 Hence all the corresponding angles will be congruent. Given that the hypotenuse is congruent, then the 2 other legs of the triangle will also be congruent. Exercise 23 Given that ∆ ABC and ∆ XYZ are right triangles, we will know that: ( 𝐴𝐴𝐶𝐶 ���� ) ≅ ( 𝑋𝑋𝑋𝑋 ���� ) (leg),( 𝐶𝐶𝐴𝐴 ���� ) ≅ ( 𝑋𝑋𝑌𝑌 ���� ) (leg) and ∠ B ≅ ∠ Y(right angle) Since we have 2 legs, it will help us prove that the triangles are congruent based on leg- leg theorem. We can conclude that leg-leg congruence theorem is equivalent to SAS Congruence Theorem. Exercise 23 Assuming that ∆ ABC and ∆ DEF are right angles, we will know that: ∠ B ≅ ∠ E(right angle),( 𝐴𝐴𝐶𝐶 ���� ) ≅ ( 𝐷𝐷𝐷𝐷 ���� ) (leg) and ∠ A ≅ ∠ D(acute angle) Since the sum of the interior angles of a triangle is 180, we know that C and F must also be congruent to each other. Hence, angle-leg congruence theorem is equivalent to ASA Congruence Theorem Exercise 24 Given that ∆ ABC ≅ ∆ DEF using ASA Congruence Theorem, the corresponding congruent parts from the diagram are: ∠ L ≅∠ L(common angle) ( 𝐾𝐾𝐽𝐽 ���� ) ≅ ( 𝐾𝐾𝑀𝑀 ���� ) The additional information we need to prove that he triangles are congruent using ASA Congruence Theorem is: ∠ LKJ ≅∠ LNM

Hence, the correct option is D Exercise 25 To prove that ∆ ABC ≅ ∆ DBC, one of the equal corresponding parts will be: ( 𝐶𝐶𝐴𝐴 ���� ) ≅ ( 𝐶𝐶𝐴𝐴 ���� ) (common side) ∠ ABC ≅∠ DBC Step 1 From the diagram, the values of the angles are: m ∠ ABC=(8x-32)° m ∠ DBC=(4y-24)° Step 2 Recall that m ∠ ABC ≅ m ∠ DBC ⟹ m ∠ ABC=m ∠ DBC Equate the values to solve the equation (8x-32)°=(4y-24)° 4y=8x-32+24 4y=8x-8 Divide both sides by 4 y=2x-2----equation (1) Step 3 Assuming that m ∠ BCA ≅ m ∠ BCD ⟹ m ∠ BCA=m ∠ BCD Equate the values to solve the equation (5x+10)°=(3y+2)°-----equation (2) Substitute y=2x-2 in equation (2) 5x+10=3(2x-2)+2 Clear the brackets 5x+10=6x-6+2 5x+10=6x-4 Collect like terms together 5x-6x=-4-10 -x=-14 x=14 Replace x with 14 iin equation (2) y=2x-2 ⟹ y=2(14)-2 ⟹ y=28-2 ⟹ y=26 Step 4 To calculate the measures of the given angles, we replace x with 14 and y with 26 m ∠ ABC=(8x-32)°=(8(14)-32)°=(112-32)°=80° m ∠ BCA=(5x+10)°=(5(14)+10)°=(70+10)°= 80° m ∠ DBC=(4y-24)°= (4(26)-24)°=(104-24)°=80° m ∠ BCD= (3y+2)°= (3(26)+2)°=(78+2)°=80° m ∠ CAB= (2x-8)°=(2(14)-8)°=(28-8)°=20° m ∠ CDB=(y-6)°= (26-6)°=20° Hence, ∆ ABC ≅ ∆ DBC using ASA Congruence Theorem and AAS Congruence Theorem Exercise 26 From the diagram, we can see that ∆ TVS and ∆ VWU are right triangles and their equal corresponding parts are; ( 𝑄𝑄𝑇𝑇 ���� ) ≅ ( 𝑈𝑈𝑆𝑆 ���� ) (included side)

∠ STV ≅∠ UVW(right angle) ∠ TSV ≅∠ VUW Since we are given an included sided and 2 angles , we can conclude that ∆ TVS and ∆ VWU are congruent using ASA Congruence Theorem. The correct option is C. Exercise 27 Converse of the base angles theorem: If two angles in a triangle are congruent, then the sides opposite those angles are also congruent. Statement & Reasons ∠ 𝐶𝐶 ≅ ∠ 𝐴𝐴 & Given Draw a bisector AD of ∠ A & Construction ∠ 𝐴𝐴𝐴𝐴𝐷𝐷 ≅ ∠ 𝐶𝐶𝐴𝐴𝐷𝐷 & 𝐴𝐴𝐷𝐷 is the bisector of ∠ A 𝐴𝐴𝐷𝐷 ≅ 𝐴𝐴𝐷𝐷 & 𝑄𝑄𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 property of Congruence ∆ ABD ≅ ∆ ACD & AAS congruence theorem AB ≅ AC & Corresponding sides of two congruent triangles Exercise 28 It is not possible to interchange AAS Congruence Theorem with ASA Congruence Theorem. This is because the congruent side in AAS is not between the two congruent angles and ASA has the congruent side between the two angles. Hence, the claim is wrong Exercise 29

The required sketch, a) Statement & Reasons ∠ 𝐴𝐴𝐷𝐷𝐶𝐶 ≅ ∠ 𝐴𝐴𝐷𝐷𝐶𝐶 , 𝐷𝐷𝐶𝐶 ⊥ 𝐴𝐴𝐴𝐴 & Given ∠ 𝐴𝐴𝐶𝐶𝐷𝐷 ≅ ∠ 𝐴𝐴𝐶𝐶𝐷𝐷 & Both are right angles 𝐶𝐶𝐷𝐷 ≅ 𝐶𝐶𝐷𝐷 & Same side △ 𝐴𝐴𝐶𝐶𝐷𝐷 ≅△ 𝐴𝐴𝐶𝐶𝐷𝐷 & AAS congruence theorem b) As ∠ 𝐴𝐴𝐶𝐶𝐷𝐷 ≅ ∠ 𝐴𝐴𝐶𝐶𝐷𝐷 , So AD = AC, corresponding sides of △ 𝐴𝐴𝐶𝐶𝐷𝐷 and \triangle CBD. So, △ 𝐴𝐴𝐷𝐷𝐴𝐴 is an isosceles triangle. c) No, because the person is already seeing the complete reflected image of his or her body. But the size of the image will get smaller. Exercise 30 The pairs of congruent triangles that can be formed from the triangle are: ∆ PQS ≅ ∆ RSQ (Based on ASA Congruence Theorem) ∆ PSR ≅ ∆ RQP (Based on ASA Congruence Theorem) ∆ PTS ≅ ∆ RTQ(Based on SAS Congruence Theorem) ∆ PQT ≅ ∆ RST (Based on SAS Congruence Theorem) Exercise 31 The required image, From the picture, we see that all the angles in both triangles have the same measure, but they are not congruent. Exercise 32

In a graph theory, any 2congruent triangles can show isomorphism. This is because the corresponding vertices can be mapped on to each other to completely overlap each other as required for isomorphism. Exercise 33 (a) The combination of three given statements that would provide enough information to prove that ∆ TUV ≅ ∆ XYZ are: ASA Congruence Theorem AAS Congruence Theorem SAS Congruence Theorem SSS Congruence Theorem (b) If we choose 3 statements at random, the population will be: n=6×5×4=120 Probability of choosing ASA =3 Probability of choosing AAS =0(This is because,AAS leads to ASA) Probability of choosing SAS =3 Probability of choosing SSS =1 The probability of choosing enough information =(ASA+SSS+SAS)/n=(3+1+3)/120=7/120 Exercise 34 Given that C(1,0) and D(5,4) . To find the midpoint of two points ( 𝑅𝑅 1 , 𝑦𝑦 1 ) and ( 𝑅𝑅 2 , 𝑦𝑦 2 ) = ( 𝑅𝑅 1 + 𝑅𝑅 2 )/2, ( 𝑦𝑦 1 + 𝑦𝑦 2 )/2 X-coordinate of the midpoint =(1+5)/2=6/2=3 Y-coordinate of the midpoint =(0+4)/2=4/2=2 Hence, midpoint =(3,2) Exercise 35 Given that C(-2,3) and D(4,-1) To find the midpoint of two points ( 𝑅𝑅 1 , 𝑦𝑦 1 ) and ( 𝑅𝑅 2 , 𝑦𝑦 2 ) = ( 𝑅𝑅 1 + 𝑅𝑅 2 )/2, ( 𝑦𝑦 1 + 𝑦𝑦 2 )/2 X-coordinate of the midpoint =(-2+4)/2 = 2/2=1 Y-coordinate of the midpoint =(3+(-1))/2 =2/2=1 Hence, midpoint =(1,1) Exercise 36 Given that C(-5,-7) and D(2,-4) To find the midpoint of two points ( 𝑅𝑅 1 , 𝑦𝑦 1 ) and ( 𝑅𝑅 2 , 𝑦𝑦 2 ) = ( 𝑅𝑅 1 + 𝑅𝑅 2 )/2, ( 𝑦𝑦 1 + 𝑦𝑦 2 )/2 X-coordinate of the midpoint =(-5+2)/2 = (-3)/2=-1.5 Y-coordinate of the midpoint =(-7+(-4))/2 =(-11)/2=-5.5 Hence, midpoint =(-1.5,-5.5) Exercise 37 1) Draw a line. 2) Label a point D on the line segment. 3) Draw a arc with centre A and label the intersection points B and C. 4) Using the same compass setting, draw an arc with centre D. 5) Label the intersection point E. 6) Draw an arc of radius BC with centre E. Label the intersection point of the two arcs F. 7) Join D and F. ∠ DEF is the required construction.

So, the image of the construction,