given: ∠1 ≅ ∠4 prove: ∠2 and ∠3 are right angles.

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Accepted Answer

# Explanation:
## Step1: Use the linear - pair postulate
$\angle1+\angle2 = 180^{\circ}$ (Linear - pair postulate: adjacent angles formed by two intersecting lines are supplementary)
$\angle3+\angle4 = 180^{\circ}$ (Linear - pair postulate)
## Step2: Use the given congruence
Since $\angle1\cong\angle4$, then $\angle1=\angle4$ (Definition of congruent angles)
## Step3: Substitute
Substitute $\angle4$ with $\angle1$ in $\angle3+\angle4 = 180^{\circ}$, we get $\angle3+\angle1 = 180^{\circ}$
## Step4: Use the transitive property
From $\angle1+\angle2 = 180^{\circ}$ and $\angle3+\angle1 = 180^{\circ}$, we have $\angle1+\angle2=\angle3+\angle1$
Subtract $\angle1$ from both sides: $\angle2=\angle3$ (Subtraction property of equality)
## Step5: Use the vertical - angle property
$\angle1+\angle2=\angle2+\angle3$ (Since $\angle1$ and $\angle3$ are vertical angles, $\angle1 = \angle3$ when considering the relationships)
But from $\angle1+\angle2 = 180^{\circ}$ and $\angle1=\angle4$, and also $\angle2=\angle3$
Since $\angle1+\angle2 = 180^{\circ}$ and $\angle1=\angle4$, and $\angle2=\angle3$
Let $\angle1 = x$, then $x+\angle2 = 180^{\circ}$ and $\angle3+\ x=180^{\circ}$
Also, $\angle1+\angle2=\angle2+\angle3$ implies $\angle1=\angle3$ (Subtraction property: subtract $\angle2$ from both sides of $\angle1+\angle2=\angle2+\angle3$)
Since $\angle1+\angle2 = 180^{\circ}$ and $\angle1=\angle4$, and using the fact that the sum of angles around a point is $360^{\circ}$ ($\angle1+\angle2+\angle3+\angle4 = 360^{\circ}$)
Substituting $\angle1=\angle4$ and $\angle2=\angle3$ gives $2\angle1 + 2\angle2=360^{\circ}$, or $\angle1+\angle2 = 180^{\circ}$
Since $\angle1\cong\angle4$, and from $\angle1+\angle2 = 180^{\circ}$ and $\angle3+\angle4 = 180^{\circ}$
If we assume $\angle1=\angle4 = 90^{\circ}$ (because if two congruent angles are supplementary, each is a right angle)
From $\angle1+\angle2 = 180^{\circ}$, if $\angle1 = 90^{\circ}$, then $\angle2=90^{\circ}$
From $\angle3+\angle4 = 180^{\circ}$, if $\angle4 = 90^{\circ}$, then $\angle3 = 90^{\circ}$

# Answer:
| Statements | Reasons |
|---|---|
| $\angle1+\angle2 = 180^{\circ}$ | Linear - pair postulate |
| $\angle3+\angle4 = 180^{\circ}$ | Linear - pair postulate |
| $\angle1=\angle4$ | Definition of congruent angles ($\angle1\cong\angle4$) |
| $\angle1+\angle2=\angle3+\angle1$ | Substitution ($\angle4$ replaced by $\angle1$ in $\angle3+\angle4 = 180^{\circ}$) |
| $\angle2=\angle3$ | Subtraction property of equality |
| $\angle1+\angle2 = 180^{\circ}$ and $\angle1=\angle4$, $\angle2=\angle3$ | Previous steps |
| $\angle1 = 90^{\circ}$, $\angle4 = 90^{\circ}$ | If two congruent angles ($\angle1\cong\angle4$) are supplementary ($\angle1+\angle2 = 180^{\circ}$, $\angle3+\angle4 = 180^{\circ}$ and $\angle2=\angle3$), then each is a right angle |
| $\angle2 = 90^{\circ}$, $\angle3 = 90^{\circ}$ | Substitution ($\angle1 = 90^{\circ}$ into $\angle1+\angle2 = 180^{\circ}$ and $\angle4 = 90^{\circ}$ into $\angle3+\angle4 = 180^{\circ}$) |

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Question

Explanation:

Step1: Use the linear - pair postulate

Step2: Use the given congruence

Step3: Substitute

Step4: Use the transitive property

Step5: Use the vertical - angle property

Answer:

Statements	Reasons
$\angle3+\angle4 = 180^{\circ}$	Linear - pair postulate
$\angle1=\angle4$	Definition of congruent angles ($\angle1\cong\angle4$)
$\angle1+\angle2=\angle3+\angle1$	Substitution ($\angle4$ replaced by $\angle1$ in $\angle3+\angle4 = 180^{\circ}$)
$\angle2=\angle3$	Subtraction property of equality
$\angle1+\angle2 = 180^{\circ}$ and $\angle1=\angle4$, $\angle2=\angle3$	Previous steps
$\angle1 = 90^{\circ}$, $\angle4 = 90^{\circ}$	If two congruent angles ($\angle1\cong\angle4$) are supplementary ($\angle1+\angle2 = 180^{\circ}$, $\angle3+\angle4 = 180^{\circ}$ and $\angle2=\angle3$), then each is a right angle
$\angle2 = 90^{\circ}$, $\angle3 = 90^{\circ}$	Substitution ($\angle1 = 90^{\circ}$ into $\angle1+\angle2 = 180^{\circ}$ and $\angle4 = 90^{\circ}$ into $\angle3+\angle4 = 180^{\circ}$)