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4. given: ∠n and ∠q are right angles; \\(\\overline{no} \\cong \\overli…

Question

  1. given: ∠n and ∠q are right angles; \\(\overline{no} \cong \overline{pq}\\) prove: \\(\triangle onp \cong \triangle pqo\\) \
statementsreasons\
------\
1. ∠n and ∠q are right angles1.\
2. \\(\triangle onp\\) and \\(\triangle pqo\\) are \\(\underline{\quad\quad\quad\quad\quad}\\) triangles2. def. of right triangle\
33. reflexive prop.\
4. \\(\overline{no} \cong \overline{pq}\\)4.\
5.5.\

\

  1. given: \\(\overline{st} \parallel \overline{ru}\\); \\(\overline{sr} \parallel \overline{tu}\\) prove: \\(\triangle srt \cong \triangle utr\\) \
statementsreasons\
------\
1. \\(\overline{st} \parallel \overline{ru}\\)1.\
2.2. if lines \\(\parallel\\), alt. int. \\(\angle s \cong\\)\
3. \\(\overline{sr} \parallel \overline{tu}\\)3.\
4. \\(\angle srt \cong \angle utr\\)4.\
5.5.\
6.6.\

\

  1. 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\\) \
statementsreasons\
------\
1. ∠w and ∠y are right angles1.\
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\
5.5.\

Explanation:

Response
Problem 4: Prove \( \triangle ONP \cong \triangle PQO \)
Step 1: Identify Right Angles

Statement 1: \( \angle N \) and \( \angle Q \) are right angles.
Reason 1: Given.

Step 2: Classify Triangles

Statement 2: \( \triangle ONP \) and \( \triangle PQO \) are right triangles.
Reason 2: Def. of right triangle (a triangle with one right angle is a right triangle).

Step 3: Reflexive Property

Statement 3: \( \overline{OP} \cong \overline{PO} \).
Reason 3: Reflexive Prop. (a segment is congruent to itself).

Step 4: Given Side Congruence

Statement 4: \( \overline{NO} \cong \overline{PQ} \).
Reason 4: Given.

Step 5: Prove Congruence (HL)

Statement 5: \( \triangle ONP \cong \triangle PQO \).
Reason 5: HL (Hypotenuse-Leg) Congruence Theorem (for right triangles, if hypotenuse and one leg are congruent, triangles are congruent).

Problem 5: Prove \( \triangle SRT \cong \triangle UTR \)
Step 1: Given Parallel Lines

Statement 1: \( \overline{ST} \parallel \overline{RU} \).
Reason 1: Given.

Step 2: Alternate Interior Angles

Statement 2: \( \angle STR \cong \angle URT \).
Reason 2: If lines \( \parallel \), alt. int. \( \angle s \cong \).

Step 3: Given Parallel Lines

Statement 3: \( \overline{SR} \parallel \overline{TU} \).
Reason 3: Given.

Step 4: Alternate Interior Angles

Statement 4: \( \angle SRT \cong \angle UTR \).
Reason 4: If lines \( \parallel \), alt. int. \( \angle s \cong \).

Step 5: Common Side

Statement 5: \( \overline{RT} \cong \overline{TR} \).
Reason 5: Reflexive Prop. (a segment is congruent to itself).

Step 6: Prove Congruence (ASA)

Statement 6: \( \triangle SRT \cong \triangle UTR \).
Reason 6: ASA (Angle-Side-Angle) Congruence Postulate (two angles and included side congruent).

Problem 6: Prove \( \triangle VWX \cong \triangle ZYX \)
Step 1: Identify Right Angles

Statement 1: \( \angle W \) and \( \angle Y \) are right angles.
Reason 1: Given.

Step 2: Classify Triangles

Statement 2: \( \triangle VWX \) and \( \triangle ZYX \) are right triangles.
Reason 2: Def. of right triangle.

Step 3: Given Information

Statement 3: \( \overline{VX} \cong \overline{ZX} \); \( X \) is the midpoint of \( \overline{WY} \).
Reason 3: Given.

Step 4: Midpoint Definition

Statement 4: \( \overline{WX} \cong \overline{YX} \).
Reason 4: Def. of midpoint (midpoint divides segment into two congruent segments).

Step 5: Prove Congruence (HL)

Statement 5: \( \triangle VWX \cong \triangle ZYX \).
Reason 5: HL (Hypotenuse-Leg) Congruence Theorem (for right triangles, if hypotenuse and one leg are congruent, triangles are congruent).

Final Answers (Summary of Congruence)
  1. \( \boldsymbol{\triangle ONP \cong \triangle PQO} \) (by HL)
  2. \( \boldsymbol{\triangle SRT \cong \triangle UTR} \) (by ASA)
  3. \( \boldsymbol{\triangle VWX \cong \triangle ZYX} \) (by HL)

Answer:

Problem 4: Prove \( \triangle ONP \cong \triangle PQO \)
Step 1: Identify Right Angles

Statement 1: \( \angle N \) and \( \angle Q \) are right angles.
Reason 1: Given.

Step 2: Classify Triangles

Statement 2: \( \triangle ONP \) and \( \triangle PQO \) are right triangles.
Reason 2: Def. of right triangle (a triangle with one right angle is a right triangle).

Step 3: Reflexive Property

Statement 3: \( \overline{OP} \cong \overline{PO} \).
Reason 3: Reflexive Prop. (a segment is congruent to itself).

Step 4: Given Side Congruence

Statement 4: \( \overline{NO} \cong \overline{PQ} \).
Reason 4: Given.

Step 5: Prove Congruence (HL)

Statement 5: \( \triangle ONP \cong \triangle PQO \).
Reason 5: HL (Hypotenuse-Leg) Congruence Theorem (for right triangles, if hypotenuse and one leg are congruent, triangles are congruent).

Problem 5: Prove \( \triangle SRT \cong \triangle UTR \)
Step 1: Given Parallel Lines

Statement 1: \( \overline{ST} \parallel \overline{RU} \).
Reason 1: Given.

Step 2: Alternate Interior Angles

Statement 2: \( \angle STR \cong \angle URT \).
Reason 2: If lines \( \parallel \), alt. int. \( \angle s \cong \).

Step 3: Given Parallel Lines

Statement 3: \( \overline{SR} \parallel \overline{TU} \).
Reason 3: Given.

Step 4: Alternate Interior Angles

Statement 4: \( \angle SRT \cong \angle UTR \).
Reason 4: If lines \( \parallel \), alt. int. \( \angle s \cong \).

Step 5: Common Side

Statement 5: \( \overline{RT} \cong \overline{TR} \).
Reason 5: Reflexive Prop. (a segment is congruent to itself).

Step 6: Prove Congruence (ASA)

Statement 6: \( \triangle SRT \cong \triangle UTR \).
Reason 6: ASA (Angle-Side-Angle) Congruence Postulate (two angles and included side congruent).

Problem 6: Prove \( \triangle VWX \cong \triangle ZYX \)
Step 1: Identify Right Angles

Statement 1: \( \angle W \) and \( \angle Y \) are right angles.
Reason 1: Given.

Step 2: Classify Triangles

Statement 2: \( \triangle VWX \) and \( \triangle ZYX \) are right triangles.
Reason 2: Def. of right triangle.

Step 3: Given Information

Statement 3: \( \overline{VX} \cong \overline{ZX} \); \( X \) is the midpoint of \( \overline{WY} \).
Reason 3: Given.

Step 4: Midpoint Definition

Statement 4: \( \overline{WX} \cong \overline{YX} \).
Reason 4: Def. of midpoint (midpoint divides segment into two congruent segments).

Step 5: Prove Congruence (HL)

Statement 5: \( \triangle VWX \cong \triangle ZYX \).
Reason 5: HL (Hypotenuse-Leg) Congruence Theorem (for right triangles, if hypotenuse and one leg are congruent, triangles are congruent).

Final Answers (Summary of Congruence)
  1. \( \boldsymbol{\triangle ONP \cong \triangle PQO} \) (by HL)
  2. \( \boldsymbol{\triangle SRT \cong \triangle UTR} \) (by ASA)
  3. \( \boldsymbol{\triangle VWX \cong \triangle ZYX} \) (by HL)