Geometry worksheet featuring five fill-in-the-blank proofs focused on proving triangle congruence using given angles and sides.

Here are the completed fill-in-the-blank proofs for each problem. I have worked through the logic for each one to ensure the statements and reasons match the geometry rules.

Problem 5

Logic Check:
* We have two angles given ($\angle I \cong \angle K$ and $\angle IHJ \cong \angle KJH$).
* The side connecting these angles is $\overline{HJ}$.
* Statement 3 identifies this shared side.
* Since we have Angle-Angle-Side (AAS) or Angle-Side-Angle (ASA) depending on the order, let's look closer. The side $\overline{HJ}$ is *between* $\angle IHJ$ and $\angle HJI$? No, $\overline{HJ}$ is adjacent to $\angle IHJ$ and opposite $\angle I$. Wait, looking at the diagram:
* $\angle I$ and $\angle K$ are corresponding.
* $\angle IHJ$ and $\angle KJH$ are corresponding.
* Side $\overline{HJ}$ is shared.
* Actually, usually, if you have two angles and a non-included side, it's AAS. If you have two angles and the included side, it's ASA.
* Let's check the position. In $\Delta HJK$, the side is $HJ$. The angles given are $\angle K$ and $\angle KJH$. The side $HJ$ is adjacent to $\angle KJH$ but opposite $\angle K$. So it is a non-included side. This implies AAS.
* *Correction*: Let's look at Statement 2 again: $\angle IHJ \cong \angle KJH$. These are the angles at vertices H and J inside the triangles.
* Triangle 1: $\Delta HJK$. Angles: $\angle K$, $\angle KJH$. Side: $\overline{HJ}$.
* Triangle 2: $\Delta JHI$. Angles: $\angle I$, $\angle IHJ$. Side: $\overline{HJ}$.
* The side $\overline{HJ}$ is not between the two given angles in either triangle (the side between $\angle I$ and $\angle IHJ$ would be $\overline{HI}$). Therefore, the postulate is AAS (Angle-Angle-Side).

Filled Blanks:
3. Reflexive Property of Congruence (This justifies that a segment is congruent to itself).
4. AAS Congruence Postulate (Angle-Angle-Side).

---

Problem 6

Logic Check:
* Given: $\angle MLN \cong \angle ONL$.
* Given: $\angle OLN \cong \angle$ \_\_\_\_. Looking at the diagram, the other pair of marked angles are at vertices N and L, specifically $\angle MNL$ and $\angle OL N$? No, the marks are on $\angle MNL$ and $\angle OLN$? Let's look at the arcs.
* Arc at L is split into $\angle MLN$ and $\angle OLN$? No, the diagonal is LN.
* Markings show $\angle MLN$ (top left part) and $\angle ONL$ (bottom right part) are congruent.
* Markings show $\angle MNL$ (top right part) and $\angle OLN$ (bottom left part) are congruent.
* So, Statement 2 is $\angle OLN \cong \angle MNL$.
* Statement 3 uses Reflexive Property. The shared side is $\overline{LN}$. So, $\overline{LN} \cong \overline{NL}$.
* Statement 4 concludes congruence. We have Angle ($\angle MLN \cong \angle ONL$), Side ($\overline{LN}$), Angle ($\angle MNL \cong \angle OLN$). The side is *between* the angles. This is ASA.

Filled Blanks:
2. MNL (Completing the angle name based on markings).
3. $\overline{LN} \cong \overline{NL}$ (Identifying the shared side).
4. ASA Congruence Postulate (Angle-Side-Angle).

---

Problem 7

Logic Check:
* Given: $\overline{PQ} \cong \overline{QS}$.
* Given: There are tick marks on $\overline{PT}$ and $\overline{SR}$. So Statement 2 is $\overline{PT} \cong \overline{SR}$.
* Statement 3: $\angle PQT \cong \angle RQS$. These are vertical angles. Reason: Vertical Angles are Congruent.
* Statement 4: Conclusion. We have Side ($\overline{PQ}$), Angle ($\angle Q$), Side ($\overline{PT}$? No, $\overline{PT}$ is not connected to Q).
* Let's re-read the diagram carefully.
* Tick marks are on $\overline{PQ}$ and $\overline{QS}$ (Given in Stmt 1).
* Tick marks are on $\overline{QT}$ and $\overline{QR}$? Or $\overline{PT}$ and $\overline{RS}$?
* Looking at Problem 7 image: The single tick is on PQ and QS. The double tick is on QT and QR.
* So, Statement 2 should be about the sides with double ticks: $\overline{QT} \cong \overline{QR}$.
* Statement 3 is the vertical angle at Q.
* This creates Side-Angle-Side (SAS).

Filled Blanks:
2. $\overline{QT} \cong \overline{QR}$ (Based on the double tick marks in the diagram).
3. Vertical Angles are Congruent.
4. SAS Congruence Postulate (Side-Angle-Side).

---

Problem 8

Logic Check:
* Given: $\overline{UV} \cong \overline{UX}$.
* Given: $\angle VWU \cong \angle XWU$. (Note: The square symbol indicates right angles, so they are congruent).
* Statement 3: Reflexive Property. The shared side is $\overline{UW}$. So, $\overline{UW} \cong \overline{UW}$.
* Statement 4: $\angle V \cong \angle X$. Why? If we already have enough for congruence in Step 5, maybe this step is intermediate?
* Let's look at Step 5: $\Delta UVW \cong \Delta UXW$.
* To get to Step 5, we can use AAS or ASA.
* We have Side ($\overline{UV} \cong \overline{UX}$), Angle ($\angle W$ is 90 for both).
* If we use HL (Hypotenuse-Leg), we need right triangles. We have right angles. Hypotenuses are $\overline{UV}$ and $\overline{UX}$ (Given). Leg is $\overline{UW}$ (Reflexive). So HL works.
* However, Statement 4 asks for $\angle V \cong \angle X$. This suggests the proof might be aiming for AAS.
* If we assume the triangles are isosceles because $\overline{UV} \cong \overline{UX}$, then base angles $\angle V \cong \angle X$. Reason: Isosceles Triangle Theorem.
* Then Step 5 uses AAS ($\angle V, \angle W, \text{side } UW$? No, side is not included. Side $UV$ is opposite $\angle W$? No. Side $UV$ is adjacent to $\angle V$ and opposite $\angle W$? No.
* Let's stick to the most direct path often taught in these worksheets.
* Path A: HL Theorem. Steps: 1. Hypotenuse given. 2. Right angles given. 3. Leg shared. 4. Conclude. But there is a Step 4 asking for angles.
* Path B: AAS. 1. Side given. 2. Angle given. 3. Side shared. 4. Angle derived?
* Let's look at the structure again.
* 1. $\overline{UV} \cong \overline{UX}$ (Side)
* 2. $\angle VWU \cong \angle XWU$ (Angle)
* 3. $\overline{UW} \cong \overline{UW}$ (Side)
* This is SSA, which isn't a rule, UNLESS it's HL for right triangles.
* But Statement 4 is $\angle V \cong \angle X$.
* If Statement 4 is true, then we have $\angle V \cong \angle X$, $\angle W \cong \angle W$, and side $\overline{UV} \cong \overline{UX}$. This is AAS.
* How do we justify Statement 4? Since $\overline{UV} \cong \overline{UX}$, $\Delta UVX$ is isosceles. Therefore, base angles are congruent.
* Reason for 4: Isosceles Triangle Theorem.
* Reason for 5: AAS Congruence Postulate.

* *Alternative Interpretation*: Sometimes worksheets are simpler. Maybe Statement 4 isn't needed for the final proof but is a property? No, it's part of the flow.
* Let's check if SAS is possible. We don't know $\overline{VW} \cong \overline{XW}$.
* Let's check if ASA is possible. We don't know $\angle VUW \cong \angle XUW$.
* So, relying on the Isosceles Triangle Theorem for Step 4 and AAS for Step 5 is the most logical mathematical path that fills all blanks.

Filled Blanks:
3. $\overline{UW} \cong \overline{UW}$
4. Isosceles Triangle Theorem (Because $\overline{UV} \cong \overline{UX}$, the base angles $\angle V$ and $\angle X$ are congruent).
5. AAS Congruence Postulate (Using $\angle V \cong \angle X$, $\angle W \cong \angle W$, and side $\overline{UV} \cong \overline{UX}$).

*(Note: If your class hasn't covered the Isosceles Triangle Theorem yet, they might accept "HL Theorem" for #5 and leave #4 as a separate observation, but typically every line leads to the next. Given the specific request for $\angle V \cong \angle X$, the Isosceles reasoning is required).*

---

Problem 9

Logic Check:
* Goal: Prove $\Delta YZA \cong \Delta CBA$.
* Diagram shows:
* $\angle Y$ and $\angle C$ have single arcs.
* $\overline{YA}$ and $\overline{CA}$ have single ticks? No, $\overline{YZ}$ and $\overline{CB}$? Let's look closely.
* Ticks are on $\overline{YA}$ and $\overline{CA}$? Or $\overline{ZA}$ and $\overline{BA}$?
* Actually, usually in these "bowtie" problems:
* Vertical angles at A are always congruent.
* We need two more pieces of info.
* Statement 1: $\angle Y \cong \angle C$. Reason: Given (implied by markings or text, usually "Given" if not stated, but here it's line 1). Let's assume the reason is "Given".
* Statement 2: Needs to be another given from the diagram. The tick marks are on $\overline{YA}$ and $\overline{CA}$? Or $\overline{YZ}$ and $\overline{BC}$?
* Looking at the ticks: There is a tick on $\overline{YA}$ and a tick on $\overline{CA}$? No, the tick is on the segment connecting Y to A? And C to A?
* Let's look at the letters. Y-Z-A and C-B-A.
* Ticks appear to be on $\overline{YA}$ and $\overline{CA}$? Or $\overline{ZA}$ and $\overline{BA}$?
* Let's assume the ticks are on $\overline{YA}$ and $\overline{CA}$ based on typical problem structures where the side adjacent to the known angle is given.
* Wait, looking at Problem 9 diagram again: The tick mark is on $\overline{YA}$ and $\overline{CA}$? No, it looks like it's on $\overline{YZ}$ and $\overline{CB}$?
* Actually, let's look at Statement 3: "Vertical Angles are $\cong$". This refers to $\angle YAZ \cong \angle CAB$.
* So we have Angle ($\angle Y \cong \angle C$) and Angle (Vertical).
* We need a Side.
* Statement 2 must provide the Side.
* The tick marks in the diagram are on $\overline{YA}$ and $\overline{CA}$? Or $\overline{ZA}$ and $\overline{BA}$?
* Let's look at the orientation. Y is top left, C is bottom right. Z is top right, B is bottom left.
* The tick is on the segment from Y to A? And C to A?
* If the tick is on $\overline{YA}$ and $\overline{CA}$, then Statement 2 is $\overline{YA} \cong \overline{CA}$.
* Then we have Angle ($\angle Y$), Side ($\overline{YA}$), Angle ($\angle A$). This is ASA.
* If the tick is on $\overline{ZA}$ and $\overline{BA}$, we have Angle ($\angle Y$), Angle ($\angle A$), Side ($\overline{ZA}$). This is AAS.
* Visually, the tick mark is on the segment connecting the outer vertex to the center. It looks like $\overline{YA}$ and $\overline{CA}$ have ticks? Or $\overline{ZA}$ and $\overline{BA}$?
* Let's look at the single tick on the left side (Y to A?) and single tick on the right side (C to A?).
* Actually, looking really closely at crop 6, the tick is on $\overline{YA}$ and $\overline{CA}$ is NOT marked. The tick is on $\overline{YZ}$? No.
* Let's look at the tick on the segment $Y-A$? And $C-A$?
* Wait, there is a tick on $\overline{YA}$ and a tick on $\overline{CA}$? No, the tick is on $\overline{ZA}$ and $\overline{BA}$?
* Let's assume the standard case: The tick marks are on $\overline{YA}$ and $\overline{CA}$ is unlikely if they are far apart.
* Let's look at the tick marks on $\overline{YZ}$ and $\overline{CB}$? No.
* Okay, let's look at the tick marks on $\overline{YA}$ and $\overline{CA}$?
* Actually, in many such problems, if $\angle Y \cong \angle C$ and Vertical Angles, and we need congruence, the side is usually the one *not* between the angles (AAS) or between (ASA).
* Let's look at the tick mark location in the image provided. It is on the segment connecting Y to A? And C to A?
* Let's assume Statement 2 is $\overline{YA} \cong \overline{CA}$ (Given).
* Then Reason 1 is Given.
* Reason 4 is ASA Congruence Postulate.

* *Alternative*: If the ticks are on $\overline{ZA}$ and $\overline{BA}$, then Reason 4 is AAS.
* Let's look at the visual weight. The tick is on the left leg of the top triangle and the right leg of the bottom triangle? That would be $\overline{YA}$ and $\overline{CA}$? No, $\overline{YA}$ and $\overline{CA}$ are corresponding parts if we map Y to C.
* Let's go with $\overline{YA} \cong \overline{CA}$ as the most likely intended given side, making it ASA. If the tick was on the other pair, it would be AAS. Without higher resolution, ASA is a very common starting point. However, looking at the tick on the segment $Y...$ and $C...$, it seems to be on the segments extending from the vertical angles.
* Let's write the answer assuming the side adjacent to the given angle is the one marked, leading to ASA. If the side opposite is marked, it's AAS.
* Actually, looking at Problem 9 again: The tick is on $\overline{YA}$ and $\overline{CA}$?
* Let's provide the most robust answer:
1. Reason: Given
2. Statement: $\overline{YA} \cong \overline{CA}$ (Assuming these are the marked sides).
3. Statement: $\angle YAZ \cong \angle CAB$ (Vertical angles).
4. Reason: ASA Congruence Postulate.

* *Self-Correction*: What if the ticks are on $\overline{YZ}$ and $\overline{CB}$? Then it's AAS.
* Let's look at the tick marks in Problem 7. They were clearly on the "legs" touching the vertex. In Problem 9, the ticks are on the segments $YA$ and $CA$? Or $ZA$ and $BA$?
* Let's look at the letter ordering: $\Delta YZA \cong \Delta CBA$.
* Y corresponds to C. Z corresponds to B. A corresponds to A.
* If $\angle Y \cong \angle C$, and we have vertical angles at A.
* If the tick is on $\overline{YA}$ and $\overline{CA}$, then Side is between Angle Y and Angle A? Yes. So ASA.
* If the tick is on $\overline{ZA}$ and $\overline{BA}$, then Side is opposite Angle Y/C? No, adjacent to A. So AAS.
* Visually, the tick mark is on the segment closer to the "left" and "right" extremes?
* I will provide the solution for ASA as it is the most standard "fill in the blank" pattern when the first angle is given, but I will note that if the marked sides are the other pair, it is AAS. Given the ambiguity, I'll stick to the visual cue that the tick is on $\overline{YA}$ and $\overline{CA}$ (the sides connecting the named angle to the center).

* *Wait*, looking at the very last crop, the tick is on $\overline{YA}$ and $\overline{CA}$? No, the tick is on $\overline{YZ}$? No.
* Let's look at the tick on the segment $Y-A$? It looks like it.
* Okay, I will proceed with ASA.

Filled Blanks:
1. Given
2. $\overline{YA} \cong \overline{CA}$ (Or $\overline{ZA} \cong \overline{BA}$ depending on exact marking, but YA/CA is consistent with ASA). *Note: Based on standard diagrams, if the tick is on the side adjacent to the given angle and the vertical angle, it's ASA.*
3. $\angle YAZ \cong \angle CAB$
4. ASA Congruence Postulate

Final Answer:

Problem 5
3. Reflexive Property of Congruence
4. AAS Congruence Postulate

Problem 6
2. MNL
3. $\overline{LN} \cong \overline{NL}$
4. ASA Congruence Postulate

Problem 7
2. $\overline{QT} \cong \overline{QR}$
3. Vertical Angles are Congruent
4. SAS Congruence Postulate

Problem 8
3. $\overline{UW} \cong \overline{UW}$
4. Isosceles Triangle Theorem
5. AAS Congruence Postulate

Problem 9
1. Given
2. $\overline{YA} \cong \overline{CA}$ *(Note: Assumes the tick marks are on these sides. If they are on ZA and BA, change Reason 4 to AAS)*
3. $\angle YAZ \cong \angle CAB$
4. ASA Congruence Postulate