Question

given: mo ⊥ no, ∠1 and ∠2 are adjacent angles. prove: m∠1 + m∠2 = 90°. a. statements: 1. mo ⊥ no; 2. ∠1 and ∠2 are vertical angles; 3. ∠mon is a right - angle; 4. m∠mon = 90°; 5. m∠1 + m∠2 = ∠mon; 6. m∠1 + m∠2 = 90°. reasons: 1. given; 2. given; 3. perpendicular lines intersect to form right angles; 4. definition of a right angle; 5. betweenness of rays; 6. transitive property or substitution property. b. statements: 1. mo ⊥ no; 2. m∠1 + m∠2 = 90°; 3. ∠mon is a right - angle; 4. m∠mon = 90°; 5. m∠1 + m∠2 = ∠mon; 6. m∠1 + m∠2 = 90°. reasons: 1. given; 2. given; 3. perpendicular lines intersect to form right angles; 4. definition of a right angle; 5. betweenness of rays; 6. transitive property or substitution property. c. statements: 1. mo ⊥ no; 2. ∠1 and ∠2 are adjacent angles; 3. ∠mon is a right - angle; 4. m∠mon = 90°; 5. m∠1 + m∠2 = ∠mon; 6. m∠1 + m∠2 = 90°. reasons: 1. given; 2. given; 3. perpendicular lines intersect to form right angles; 4. definition of a right angle; 5. betweenness of rays; 6. transitive property or substitution property. d. statements: 1. mo ⊥ no; 2. ∠1 and ∠2 are adjacent angles; 3. ∠mon is a right - angle; 4. m∠mon = 90°; 5. m∠1 + m∠2 = ∠mon; 6. m∠1 + m∠2 = 90°. reasons: 1. given; 2. given; 3. perpendicular lines intersect to form right angles; 4. transitive property; 5. distributive property; 6. substitution property. e. statements: 1. mo ⊥ no; 2. ∠mon is a right - angle; 3. ∠1 and ∠2 are adjacent angles; 4. m∠mon = 90°; 5. m∠1 + m∠2 = ∠mon; 6. m∠1 + m∠2 = 90°. reasons: 1. given; 2. given; 3. perpendicular lines intersect to form right angles; 4. definition of a right angle; 5. betweenness of rays; 6. transitive property or substitution property.

Explanation:

Step1: Analyze given information

We are given that $MO\perp NO$ and $\angle1$ and $\angle2$ are adjacent angles.

Step2: Determine angle formed by perpendicular lines

Since $MO\perp NO$, by the property that perpendicular lines intersect to form right - angles, $\angle MON$ is a right angle.

Step3: Define measure of right angle

By the definition of a right angle, $m\angle MON = 90^{\circ}$.

Step4: Use betweenness of rays

By the betweenness of rays, $m\angle1 + m\angle2=\angle MON$.

Step5: Apply substitution

Using the transitive or substitution property, since $m\angle MON = 90^{\circ}$ and $m\angle1 + m\angle2=\angle MON$, we have $m\angle1 + m\angle2 = 90^{\circ}$.

Answer:

C. Statements: 1. $MO\perp NO$ (Given); 2. $\angle1$ and $\angle2$ are adjacent angles (Given); 3. $\angle MON$ is a right angle (Perpendicular lines intersect to form right angles); 4. $m\angle MON = 90^{\circ}$ (Definition of a right angle); 5. $m\angle1 + m\angle2=\angle MON$ (Betweenness of Rays); 6. $m\angle1 + m\angle2 = 90^{\circ}$ (Transitive Property or Substitution Property)