Triangles
Practice Questions

AAA is similarity criterion, not congruence criterion.

Angles opposite to equal sides are equal: ∠C = ∠B = 50°.

Corresponding parts of congruent triangles (CPCT) are equal: AB = PQ.

Sides opposite to equal angles are equal: PQ = QR = 4 cm.

Both are fundamental isosceles triangle theorems, but R is the converse of A.

Both A and R are true and R explains A.

In △ABC and △ABD: AC=AD, ∠CAB=∠DAB, AB=AB. By SAS, △ABC ≅ △ABD.

In △AOD and △BOC: ∠OAD=∠OBC=90°, ∠AOD=∠BOC (vert opp), AD=BC. By AAS, △AOD ≅ △BOC ⇒ OA = OB.

In △APB and △AQB: ∠APB=∠AQB=90°, ∠PAB=∠QAB, AB=AB. By AAS, △APB ≅ △AQB.

BE = CD ⇒ BD = EC. In △ABD and △ACE: AB=AC, ∠B=∠C, BD=EC. By SAS, △ABD ≅ △ACE ⇒ AD = AE.

∠EPA + ∠DPE = ∠DPB + ∠DPE ⇒ ∠APD = ∠BPE. In △DAP and △EBP: ∠A=∠B, AP=BP, ∠APD=∠BPE. By ASA, △DAP ≅ △EBP.

1. AM=BM, ∠AMC=∠BMD, CM=DM ⇒ △AMC ≅ △BMD by SAS. 2. AC || BD ⇒ ∠ACB + ∠DBC = 180° ⇒ ∠DBC = 90°.

In △ABE and △ACF: ∠AEB=∠AFC=90°, ∠A=∠A, AB=AC. By AAS, △ABE ≅ △ACF ⇒ BE = CF.

In △ABE and △ACF: ∠AEB=∠AFC=90°, ∠A=∠A, BE=CF. By AAS, △ABE ≅ △ACF ⇒ AB = AC.

AB = AC ⇒ ∠ABC = ∠ACB. DB = DC ⇒ ∠DBC = ∠DCB. Adding: ∠ABC + ∠DBC = ∠ACB + ∠DCB ⇒ ∠ABD = ∠ACD.

BD = 1/2 BC, QM = 1/2 QR ⇒ BD = QM. △ABD ≅ △PQM by SSS ⇒ ∠B = ∠Q. △ABC ≅ △PQR by SAS.

Hypotenuse = 13m, Height = 5m, Right angle = 90°. By RHS congruence criterion, the two ramp triangles are congruent.

In △AOB and △DOC: AO=DO, ∠AOB=∠DOC, ∠OAB=∠ODC (alt int). By ASA, △AOB ≅ △DOC ⇒ OB = OC.

∠ABC = ∠ACB and ∠ADC = ∠ACD. In △BCD: ∠B + ∠D + ∠BCD = 180° ⇒ 2 ∠BCD = 180° ⇒ ∠BCD = 90°.

In △ABD and △ACD: AD=AD, ∠ADB=∠ADC=90°, BD=CD. By SAS, △ABD ≅ △ACD ⇒ AB = AC.