Question

viii. given: <fec is a right angle
prove: <1 and <4 are complementary
statements

1. <fec is a right angle
2. m<fec = 90
3. m<fec = m<1 + m<2
4. m<1 + m<2 = 90
5. <2 ≅ <4
6. m<2 = m<4
7. m<1 + m<4 = 90
8. <1 and <4 are complementary

reasons
1.
2.
3.
4.
5.
6.
7.
8.

Explanation:

Step1: Given information

Given

Step2: Definition of right - angle

Definition of a right - angle ($m\angle FEC = 90^{\circ}$ if $\angle FEC$ is a right - angle)

Step3: Angle - addition postulate

Angle - addition postulate (if point $E$ lies in the interior of $\angle FEC$, then $m\angle FEC=m\angle1 + m\angle2$)

Step4: Substitution property

Substitution property (substitute $m\angle FEC = 90$ into $m\angle FEC=m\angle1 + m\angle2$)

Step5: Vertical angles are congruent

Vertical angles are congruent ($\angle2$ and $\angle4$ are vertical angles)

Step6: Definition of congruent angles

Definition of congruent angles (if $\angle2\cong\angle4$, then $m\angle2 = m\angle4$)

Step7: Substitution property

Substitution property (substitute $m\angle2=m\angle4$ into $m\angle1 + m\angle2 = 90$)

Step8: Definition of complementary angles

Definition of complementary angles (if $m\angle1 + m\angle4=90$, then $\angle1$ and $\angle4$ are complementary)

Answer:

1. Given
2. Definition of a right - angle
3. Angle - addition postulate
4. Substitution property
5. Vertical angles are congruent
6. Definition of congruent angles
7. Substitution property
8. Definition of complementary angles