Problem 1 :
In the diagram given below, prove that ΔPQW ≅ ΔTSW using two column proof.
Problem 2 :
In the diagram given below, prove that ΔAEB ≅ ΔDEC using two column proof.
Problem 3 :
In the diagram given below, prove that ΔABD ≅ ΔEBC using two column proof.
Problem 4 :
In the diagram given below, prove that ΔEFG ≅ ΔJHG using two column proof.
Problem 5 :
In the diagram given below, prove that ΔABC ≅ ΔFGH.
Problem 6 :
Check whether two triangles ABC and CDE are congruent.
Problem 7 :
Check whether two triangles PQR and RST are congruent.
Problem 8 :
Check whether two triangles ABD and ACD are congruent.
1. Answer :
Statements PQ ≅ ST PW ≅ TW QW ≅ SW ΔPQW ≅ ΔTSW |
Reasons Given Given Given SSS Congruence Postulate |
2. Answer :
Statements AE ≅ DE, BE ≅ CE ∠1 ≅ ∠2 ΔAEB ≅ ΔDEC |
Reasons Given Vertical Angles Theorem SAS Congruence Postulate |
3. Answer :
Statements BD ≅ BC AD || EC ∠D ≅ ∠C ∠ABD ≅ ∠EBC ΔABD ≅ ΔEBC |
Reasons Given Given Alternate Interior Angles Theorem Vertical Angles Theorem ASA Congruence Postulate |
4. Answer :
Statements FE ≅ JH ∠E ≅ ∠J ∠EGF ≅ ∠JGH ΔEFG ≅ ΔJHG |
Reasons Given Given Vertical Angles Theorem AAS Congruence Postulate |
5. Answer :
Because AB = 5 in triangle ABC and FG = 5 in triangle FGH,
AB ≅ FG
Because AC = 3 in triangle ABC and FH = 3 in triangle FGH,
AC ≅ FH
Use the distance formula to find the lengths of BC and GH.
Length of BC :
BC = √[(x2 - x1)2 + (y2 - y1)2]
Here (x1, x1) = B(-7, 0) and (x2, x2) = C(-4, 5).
BC = √[(-4 + 7)2 + (5 - 0)2]
= √[32 + 52]
= √[9 + 25]
= √34
Length of GH :
GH = √[(x2 - x1)2 + (y2 - y1)2]
Here (x1, x1) = G(1, 2) and (x2, x2) = H(6, 5).
GH = √[(6 - 1)2 + (5 - 2)2]
= √[52 + 32]
= √[25 + 9]
= √34
Conclusion :
Because BC = √34 and GH = √34,
BC ≅ GH
All the three pairs of corresponding sides are congruent. By SSS congruence postulate,
ΔABC ≅ ΔFGH
6. Answer :
(i) Triangle ABC and triangle CDE are right triangles. Because they both have a right angle.
(i) AC = CE (Leg)
(ii) BC = CD (Leg)
Hence, the two triangles ABC and CDE are congruent by Leg-Leg theorem.
7. Answer :
(i) Triangle PQR and triangle RST are right triangles. Because they both have a right angle.
(ii) QR = RS (Given)
(iii) ∠PRQ = ∠SRT (Vertical Angles)
Hence, the two triangles PQR and RST are congruent by Leg-Acute (LA) Angle theorem.
8. Answer :
(i) Triangle ABD and triangle ACD are right triangles. Because they both have a right angle.
(i) AB = AC (Hypotenuse)
(ii) AD = AD (Common side, Leg)
Hence, the two triangles ABD and ACD are congruent by Hypotenuse-Leg (HL) theorem.
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