Question

analytic geometry
fill in the blank proofs:
problem 5:
statement reason

1. ∠i≅∠k 1. given
2. ∠jih≅∠kjh 2. given
3. jh≅jh 3.
4. △ihj≅△kjh 4.

problem 6:
statement reason

1. ∠mln≅∠onl 1. given
2. ∠oln≅∠ 2. given
3. ln≅ln 3. reflexive property
4. △lno≅△nlm 4.

problem 7:
statement reason

1. pq = qs 1. given
2. 2. given
3. ∠pqt≅∠rqs 3.
4. △pqt≅△sqr 4.

problem 8:
statement reason

1. uv≅ux 1. given
2. ∠vwu≅∠xwu 2. given
3. 3. reflexive property
4. ∠v≅∠x 4.
5. △uvw≅△uxw 5.

problem 9:
statement reason

1. ∠y≅∠c 1.
2. 2. given
3. 3. vertical angles are
4. △yz a≅△cba 4.

Explanation:

Problem 5

Step1: Identify the congruence - related reason

We know that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent. Here, since $\angle I\cong\angle K$, $\angle JIH\cong\angle JKH$ and $HI = HK$, the reason for $\triangle JIH\cong\triangle JKH$ is ASA (Angle - Side - Angle) Postulate.

Problem 6

Step1: Complete the angle equality

Since $\angle MLN\cong\angle ONL$ and $\angle LON\cong\angle LMN$ (by the given information and the reflexive property of the common side $LN$), the missing angle in the congruence - related statement is $\angle LMN$. The reason for $\triangle LNO\cong\triangle NLM$ is AAS (Angle - Angle - Side) Theorem (two angles and a non - included side are congruent).

Problem 7

Step1: Identify the congruence reason

Given $PQ = QS$, $\angle PQT=\angle RQS$ (vertically opposite angles are equal) and $\angle P = \angle S$ (from the given equal angles), the reason for $\triangle PQT\cong\triangle SQR$ is ASA (Angle - Side - Angle) Postulate.

Problem 8

Step1: Identify the congruence reason

Given $UV\cong UX$, $\angle VWU\cong\angle XWU$ and $WU$ is common (reflexive property), the reason for $\triangle UWY\cong\triangle UWX$ is SAS (Side - Angle - Side) Postulate.

Problem 9

Step1: Identify the given side equality

Since $\angle Y\cong\angle C$ and $\angle YAZ\cong\angle CAB$ (vertical angles are equal), for $\triangle YZA\cong\triangle CBA$, we need $YZ = CB$. The reason for the congruence is AAS (Angle - Angle - Side) Theorem if we assume the side - equality condition is met.

Answer:

Problem 5: ASA Postulate
Problem 6: $\angle LMN$, AAS Theorem
Problem 7: ASA Postulate
Problem 8: SAS Postulate
Problem 9: $YZ = CB$, AAS Theorem