Question

example 3
proof write a two - column proof.
6 given: \\( \angle 2 \cong \angle 4 \\)
prove: \\( \angle 1 \cong \angle 3 \\)

Explanation:

Step1: Use the definition of complementary angles

We know that if two angles are complementary, their sum is \(90^{\circ}\). So, \(\angle1+\angle2 = 90^{\circ}\) (by the definition of complementary angles, as they form a right - angle) and \(\angle3+\angle4=90^{\circ}\) (by the definition of complementary angles, as they form a right - angle).

Step2: Use the substitution property

Since \(\angle2\cong\angle4\) (given), we can substitute \(\angle2\) with \(\angle4\) in the equation \(\angle1+\angle2 = 90^{\circ}\). So, \(\angle1+\angle4 = 90^{\circ}\).

Step3: Compare the two equations

We have \(\angle1+\angle4 = 90^{\circ}\) and \(\angle3+\angle4=90^{\circ}\). By the subtraction property of equality (subtract \(\angle4\) from both sides of each equation), we get \(\angle1=90^{\circ}-\angle4\) and \(\angle3 = 90^{\circ}-\angle4\).

Answer:

Statements	Reasons
2. \(\angle2\cong\angle4\)	Given
3. \(\angle1+\angle4=90^{\circ}\)	Substitution property (substitute \(\angle2\) with \(\angle4\) in \(\angle1+\angle2 = 90^{\circ}\))
4. \(\angle1\cong\angle3\)	If \(a + c=b + c\), then \(a = b\) (subtraction property: subtract \(\angle4\) from \(\angle1+\angle4\) and \(\angle3+\angle4\))