Question

a) determine whether the following triangles are congruent. b) if they are, name the triangle congruence (pay attention to proper correspondence when naming the triangles) and then identify the theorem or postulate (sss, sas, asa, aas, hl) that supports your conclusion. c) be sure to show any additional congruence markings you used in your reasoning. d) if the triangles cannot be proven congruent, state “not possible.” then give the reason it is not possible. 1) congruence: △abd≅△__ sas reason: 2) congruence: △efg≅△ reason: 3) congruence: △emn≅△ reason: 4) congruence: △stu≅△ reason: 5) congruence: △yza≅△ reason: hl 6) congruence: △cde≅△__ reason:

Explanation:

Step1: Analyze triangle 1

In $\triangle ABD$ and $\triangle ACD$, $AB = AC$ (marked as 5), $\angle BAD=\angle CAD$ (common - angle), $AD = AD$ (common - side). By the Side - Angle - Side (SAS) congruence postulate, $\triangle ABD\cong\triangle ACD$.

Step2: Analyze triangle 2

In $\triangle EFG$ and $\triangle HJK$, we have $EF = HJ$ (marked), $\angle E=\angle H$ (right - angles), $FG = JK$ (marked). By the Side - Angle - Side (SAS) congruence postulate, $\triangle EFG\cong\triangle HJK$.

Step3: Analyze triangle 3

In $\triangle EMN$ and $\triangle PQR$, $EM = PQ$ (marked), $\angle M=\angle Q$ (marked), $MN = QR$ (marked). By the Side - Angle - Side (SAS) congruence postulate, $\triangle EMN\cong\triangle PQR$.

Step4: Analyze triangle 4

In $\triangle STU$ and $\triangle WVX$, $ST = WV$, $SU = WX$, $TU = VX$. By the Side - Side - Side (SSS) congruence postulate, $\triangle STU\cong\triangle WVX$.

Step5: Analyze triangle 5

In right - triangles $\triangle YZA$ and $\triangle YBA$, $YZ = YB$ (hypotenuse) and $YA$ is common (leg). By the Hypotenuse - Leg (HL) congruence theorem, $\triangle YZA\cong\triangle YBA$.

Step6: Analyze triangle 6

In right - triangles $\triangle CDE$ and $\triangle GFH$, $CD = GF$, $DE = FH$, and they are right - triangles. By the Hypotenuse - Leg (HL) congruence theorem, $\triangle CDE\cong\triangle GFH$.

Answer:

1. Congruence: $\triangle ABD\cong\triangle ACD$; Reason: SAS
2. Congruence: $\triangle EFG\cong\triangle HJK$; Reason: SAS
3. Congruence: $\triangle EMN\cong\triangle PQR$; Reason: SAS
4. Congruence: $\triangle STU\cong\triangle WVX$; Reason: SSS
5. Congruence: $\triangle YZA\cong\triangle YBA$; Reason: HL
6. Congruence: $\triangle CDE\cong\triangle GFH$; Reason: HL