Question

geometry a standard 2.1 - practice essential standard #2 - score: /20 congruent triangles - form c a. state the corresponding parts that are congruent in each diagram. justify each statement b. determine if there is sufficient information to state the two triangles are congruent. if yes, justify your answer with an appropriate theorem. c. write a triangle congruent statement. 1. 2. 3. 4.

Explanation:

Step1: Analyze figure 1

In $\triangle ABC$ and $\triangle DCB$, we have $\angle A=\angle D = 90^{\circ}$, $BC = CB$ (common - side), and the markings show $AB = DC$.

Step2: Determine congruence for figure 1

By the Hypotenuse - Leg (HL) theorem (since we have right - triangles, a common hypotenuse and one equal leg), $\triangle ABC\cong\triangle DCB$. The corresponding parts: $\angle A\cong\angle D$, $\angle ABC\cong\angle DCB$, $\angle ACB\cong\angle DBC$, $AB\cong DC$, $AC\cong DB$, $BC\cong CB$. The congruent statement is $\triangle ABC\cong\triangle DCB$.

Step3: Analyze figure 2

We need to identify congruent parts. If we assume vertical angles at the intersection point $I$, say $\angle GIH=\angle JIK$. But we need more information like side - length equalities or other angle equalities to determine congruence. Without more info, we can't say the triangles are congruent.

Step4: Analyze figure 3

In $\triangle PQR$ and $\triangle VST$, we have $\angle R=\angle T = 90^{\circ}$, $\angle P=\angle V$ and $PR = VT$.

Step5: Determine congruence for figure 3

By the Angle - Angle - Side (AAS) theorem, $\triangle PQR\cong\triangle VST$. The corresponding parts: $\angle P\cong\angle V$, $\angle Q\cong\angle S$, $\angle R\cong\angle T$, $PQ\cong VS$, $QR\cong ST$, $PR\cong VT$. The congruent statement is $\triangle PQR\cong\triangle VST$.

Step6: Analyze figure 4

We need to identify congruent parts. If we assume the markings show equal sides and equal angles. For example, if $\angle O=\angle R$, $OP = RT$ and $OQ=RS$.

Step7: Determine congruence for figure 4

By the Side - Angle - Side (SAS) theorem, $\triangle OPQ\cong\triangle RTS$. The corresponding parts: $\angle O\cong\angle R$, $\angle P\cong\angle T$, $\angle Q\cong\angle S$, $OP\cong RT$, $OQ\cong RS$, $PQ\cong TS$. The congruent statement is $\triangle OPQ\cong\triangle RTS$.

Answer:

1.

• a. Corresponding parts: $\angle A\cong\angle D$, $\angle ABC\cong\angle DCB$, $\angle ACB\cong\angle DBC$, $AB\cong DC$, $AC\cong DB$, $BC\cong CB$.
• b. Yes, by HL theorem.
• c. $\triangle ABC\cong\triangle DCB$.

2.

• a. Need more information to identify all corresponding parts.
• b. No, not enough information.
• c. Cannot be determined.

3.

• a. Corresponding parts: $\angle P\cong\angle V$, $\angle Q\cong\angle S$, $\angle R\cong\angle T$, $PQ\cong VS$, $QR\cong ST$, $PR\cong VT$.
• b. Yes, by AAS theorem.
• c. $\triangle PQR\cong\triangle VST$.

4.

• a. Corresponding parts: $\angle O\cong\angle R$, $\angle P\cong\angle T$, $\angle Q\cong\angle S$, $OP\cong RT$, $OQ\cong RS$, $PQ\cong TS$.
• b. Yes, by SAS theorem.
• c. $\triangle OPQ\cong\triangle RTS$.