Using Triangle Congruence Theorems

Given: bisects ∠BAC; AB = AC Which congruence theorem can be used to prove △ABR ≅ △ACR?

A. AAS

Complete the paragraph proof. We are given AB ≅ AE and BC ≅ DE. This means ABE is an isosceles triangle. Base angles in an isosceles triangle are congruent based on the isosceles triangle theorem, so ∠ABE ≅ ∠AEB. We can then determine △ABC ≅ △AED by . Because of CPCTC, segment AC is congruent to segment . Triangle ACD is an isosceles triangle based on the definition of isosceles triangle. Therefore, based on the isosceles triangle theorem, ∠ACD ≅ ∠ADC.

1. SAS 2. AD

Given: ∠GHD and ∠EDH are right; GH ≅ ED. Which relationship in the diagram is true?

A. △GHD ≅ △EDH by SAS

Which congruency theorem can be used to prove that △ABD ≅ △DCA?

C. SAS

Mikal is proving that AE ≅ CE . Which reason does the ♣ represent in Mikal's proof?

D. AAS

Consider the diagram. The congruence theorem that can be used to prove △LON ≅ △LMN is

A. SSS.

Rowena is proving that AD ≅ EB. Which statement does the ♣ represent in her proof?

A. ΔACD ≅ ΔECB

Complete the paragraph proof: It is given that ∠TUW ≅ ∠SRW and RS ≅ TU. Because ∠RWS and ∠UWT are vertical angles and vertical angles are congruent, ∠RWS ≅ ∠UWT. Then, by AAS, △TUW ≅ △SRW. Because CPCTC, SW ≅ TW and WU ≅ RW. Because of the definition of congruence, SW = TW and WU = RW. If we add those equations together, SW + WU = TW + RW. Because of segment addition, SW + WU = SU and TW + RW = TR. Then by substitution, SU = TR. If segments are equal, then they are congruent, so SU ≅ TR. Because of , △TRS ≅ △SUT, and because of , ∠RST ≅ ∠UTS.

1.SAS 2.CPCTC

Given: ∠BCD is right; BC ≅ DC; DF ≅ BF; FA ≅ FE. Which relationships in the diagram are true? Check all that apply.

2. △CBF ≅ △CDF by SSS 3. △BFA ≅ △DFE by SAS 5. △CBE ≅ △CDA by HL

Which congruence theorem can be used to prove △WXZ ≅ △YZX?

A. AAS

Two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle. Which congruence theorem can be used to prove that the triangles are congruent?

B. AAS

Which congruency theorem can be used to prove that △GHL ≅ △KHJ?

B. ASA

In the figure below, WU ≅ VT. The congruency theorem can be used to prove that △WUT ≅ △VTU.

B. HL

Two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle. Which congruence theorem can be used to prove that the triangles are congruent?

C. SAS

Consider the diagram. Which congruence theorem can be used to prove △ABR ≅ △RCA?

A. HL

Which congruence theorem can be used to prove △BDA ≅ △BDC?

A. HL

Analyze the diagram below. Which statements regarding the diagram are correct? Check all that apply.

A. ST ≅ ST by the reflexive property. B. ∠RWS ≅ ∠UWT because they are vertical angles. C. △RWS ≅ △UWT by AAS. E. ∠WTU ≅ ∠WSR because CPCTC.

Line segments AD and BE intersect at C, and triangles ABC and DEC are formed. They have the following characteristics: ∠ACB and ∠DCE are vertical angles ∠B ≅ ∠E BC ≅ EC Which congruence theorem can be used to prove △ABC ≅ △DEC?

B. ASA