QUESTION IMAGE
Question
- given: ∠n and ∠q are right angles; (overline{no} cong overline{pq}) prove: (\triangle onp cong \triangle pqo)
| statements | reasons |
|---|---|
| 2. (\triangle onp) and (\triangle pqo) are ____ triangles | 2. def of right triangle |
| 3. | 3. reflexive prop. |
| 4. (overline{no} cong overline{pq}) | 4. |
| 5. | 5. |
- given: (overline{st} parallel overline{ru}); (overline{sr} parallel overline{tu}) prove: (\triangle srt cong \triangle utr)
| statements | reasons |
|---|---|
| 2. | 2. if lines (parallel), alt. int. ∠s ≅ |
| 3. (overline{sr} parallel overline{tu}) | 3. |
| 4. ∠srt ≅ ∠utr | 4. |
| 5. | 5. |
| 6. (\triangle srt cong \triangle utr) | 6. |
- given: ∠w and ∠y are right angles; (overline{vx} cong overline{zx}); x is the midpoint of (overline{wy}) prove: (\triangle vwx cong \triangle zyx)
| statements | reasons |
|---|---|
| 2. | 2. def. of right triangle |
| 3. (overline{vx} cong overline{zx}); x is the midpoint of (overline{wy}) | 3. |
| 4. | 4. def. of midpoint |
| 5. (\triangle vwx cong \triangle zyx) | 5. |
Problem 4 (Proving $\triangle ONP \cong \triangle PQO$)
Step 1: Identify Right Triangles
From the given right angles, $\triangle ONP$ and $\triangle PQO$ are right triangles (by definition of a right triangle, a triangle with one right angle).
Statement 2: $\triangle ONP$ and $\triangle PQO$ are right triangles.
Step 2: Reflexive Property
The reflexive property states a segment is congruent to itself. Here, $OP = PO$ (common hypotenuse).
Statement 3: $\boldsymbol{OP = PO}$ (Reflexive Prop.).
Step 3: Given Side Congruence
Statement 4: $\overline{NO} \cong \overline{PQ}$ (Given).
Step 4: HL Congruence
For right triangles, if the hypotenuse and one leg are congruent, the triangles are congruent (HL theorem). Here, hypotenuse $OP = PO$, leg $NO = PQ$.
Statement 5: $\boldsymbol{\triangle ONP \cong \triangle PQO}$ (HL Congruence Theorem).
Problem 5 (Proving $\triangle SRT \cong \triangle UTR$)
Step 1: Given Parallel Lines
Statement 1: $\overline{ST} \parallel \overline{RU}$ (Given).
Step 2: Alternate Interior Angles
If two lines are parallel, alternate interior angles are congruent. So, $\angle STR \cong \angle URT$ (since $ST \parallel RU$ and transversal $RT$).
Statement 2: $\boldsymbol{\angle STR \cong \angle URT}$ (If lines $\parallel$, alt. int. $\angle$s $\cong$).
Step 3: Given Parallel Lines
Statement 3: $\overline{SR} \parallel \overline{TU}$ (Given).
Step 4: Alternate Interior Angles
If $\overline{SR} \parallel \overline{TU}$ (transversal $RT$), then $\angle SRT \cong \angle UTR$ (alt. int. $\angle$s).
Statement 4: $\boldsymbol{\angle SRT \cong \angle UTR}$ (If lines $\parallel$, alt. int. $\angle$s $\cong$).
Step 5: Common Side
The segment $RT$ is common to both triangles, so $RT = TR$ (reflexive property).
Statement 5: $\boldsymbol{RT = TR}$ (Reflexive Prop.).
Step 6: ASA Congruence
Two angles and the included side: $\angle SRT \cong \angle UTR$, $RT = TR$, $\angle STR \cong \angle URT$. Thus, $\triangle SRT \cong \triangle UTR$ (ASA).
Statement 6: $\boldsymbol{\triangle SRT \cong \triangle UTR}$ (ASA Congruence Theorem).
Problem 6 (Proving $\triangle VWX \cong \triangle ZYX$)
Step 1: Identify Right Triangles
$\angle W$ and $\angle Y$ are right angles, so $\triangle VWX$ and $\triangle ZYX$ are right triangles (by definition of a right triangle).
Statement 2: $\boldsymbol{\triangle VWX}$ and $\boldsymbol{\triangle ZYX}$ are right triangles (Def. of right triangle).
Step 2: Given Information
Statement 3: $\overline{VX} \cong \overline{ZX}$; $X$ is the midpoint of $\overline{WY}$ (Given).
Step 3: Midpoint Definition
If $X$ is the midpoint of $\overline{WY}$, then $WX = XY$ (def. of midpoint: divides segment into two congruent parts).
Statement 4: $\boldsymbol{WX = XY}$ (Def. of midpoint).
Step 4: HL Congruence
For right triangles, hypotenuse $VX = ZX$ (given) and leg $WX = XY$ (from midpoint). Thus, $\triangle VWX \cong \triangle ZYX$ (HL).
Statement 5: $\boldsymbol{\triangle VWX \cong \triangle ZYX}$ (HL Congruence Theorem).
Final Answers (Filled Proofs)
Problem 4:
| Statements | Reasons |
|---|---|
| 2. $\triangle ONP$ and $\triangle PQO$ are right triangles | 2. Def of right triangle |
| 3. $OP = PO$ | 3. Reflexive Prop. |
| 4. $\overline{NO} \cong \overline{PQ}$ | 4. Given |
| 5. $\triangle ONP \cong \triangle PQO$ | 5. HL Congruence Theorem |
Problem 5:
| Statements | Reasons |
|---|---|
| 2. $\angle STR \cong \angle URT$ | 2. If lines $\parallel$, alt. int. $\angle$s $\cong$ |
| 3. $\overline{SR} \parallel \overline{TU}$ | 3. Given |
| 4. $\angle SRT \cong \angle UTR$ | 4. If lines $\parallel$, alt. int. $\angle$s $\cong$ |
| 5. $RT = TR$ | 5. Reflexive Prop. |
| 6. $\triangle SRT \cong \triangle UTR$ | 6. ASA Congruence Theorem |
Problem 6:
| Statements | Reasons |
|---|---|
| 2. $\triangle VWX$ and $\triangle ZYX$ are right triangles | 2. Def of right triangle |
| 3. $\overline{VX} \cong \overline{ZX}$; $X$ is the midpoint of $\overline{WY}$ | 3. Given |
| 4. $WX = XY$ | 4. Def of midpoint |
| 5. $\triangle VWX \cong \triangle ZYX$ | 5. HL Congruence Theorem |
(Note: Replace $\boldsymbol{}$ with filled statements in the original table.)
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Step 1: Identify Right Triangles
$\angle W$ and $\angle Y$ are right angles, so $\triangle VWX$ and $\triangle ZYX$ are right triangles (by definition of a right triangle).
Statement 2: $\boldsymbol{\triangle VWX}$ and $\boldsymbol{\triangle ZYX}$ are right triangles (Def. of right triangle).
Step 2: Given Information
Statement 3: $\overline{VX} \cong \overline{ZX}$; $X$ is the midpoint of $\overline{WY}$ (Given).
Step 3: Midpoint Definition
If $X$ is the midpoint of $\overline{WY}$, then $WX = XY$ (def. of midpoint: divides segment into two congruent parts).
Statement 4: $\boldsymbol{WX = XY}$ (Def. of midpoint).
Step 4: HL Congruence
For right triangles, hypotenuse $VX = ZX$ (given) and leg $WX = XY$ (from midpoint). Thus, $\triangle VWX \cong \triangle ZYX$ (HL).
Statement 5: $\boldsymbol{\triangle VWX \cong \triangle ZYX}$ (HL Congruence Theorem).
Final Answers (Filled Proofs)
Problem 4:
| Statements | Reasons |
|---|---|
| 2. $\triangle ONP$ and $\triangle PQO$ are right triangles | 2. Def of right triangle |
| 3. $OP = PO$ | 3. Reflexive Prop. |
| 4. $\overline{NO} \cong \overline{PQ}$ | 4. Given |
| 5. $\triangle ONP \cong \triangle PQO$ | 5. HL Congruence Theorem |
Problem 5:
| Statements | Reasons |
|---|---|
| 2. $\angle STR \cong \angle URT$ | 2. If lines $\parallel$, alt. int. $\angle$s $\cong$ |
| 3. $\overline{SR} \parallel \overline{TU}$ | 3. Given |
| 4. $\angle SRT \cong \angle UTR$ | 4. If lines $\parallel$, alt. int. $\angle$s $\cong$ |
| 5. $RT = TR$ | 5. Reflexive Prop. |
| 6. $\triangle SRT \cong \triangle UTR$ | 6. ASA Congruence Theorem |
Problem 6:
| Statements | Reasons |
|---|---|
| 2. $\triangle VWX$ and $\triangle ZYX$ are right triangles | 2. Def of right triangle |
| 3. $\overline{VX} \cong \overline{ZX}$; $X$ is the midpoint of $\overline{WY}$ | 3. Given |
| 4. $WX = XY$ | 4. Def of midpoint |
| 5. $\triangle VWX \cong \triangle ZYX$ | 5. HL Congruence Theorem |
(Note: Replace $\boldsymbol{}$ with filled statements in the original table.)