Question

definition of angle bisector definition of complementary angles definition of supplementary angles definition of perpendicular definition of right angle angle addition postulate vertical angles theorem complement theorem linear pair (supplementary) theorem congruent complements theorem congruent supplements theorem 29. given: ∠1 and ∠2 are complementary; ∠1 ≅ ∠4 prove: ∠3 and ∠4 are complementary statements reasons 1. ∠1 and ∠2 are complementary 1. given 2. m∠1 + m∠2 = 90° 2. def of right angle 3. ∠1 ≅ ∠4 3. given 4. m∠1 = m∠4 4. 5. ∠2 ≅ ∠3 5. 6. m∠2 = m∠3 6. 7. m∠4 + m∠3 = 90° 7. def of right angle 8. ∠3 and ∠4 are complementary 8.

Explanation:

Step1: Use definition of congruent angles

If $\angle1\cong\angle4$, then $m\angle1 = m\angle4$ (Definition of congruent angles).

Step2: Identify vertical - angles

$\angle2$ and $\angle3$ are vertical angles. Since vertical angles are congruent, $\angle2\cong\angle3$ (Vertical Angles Theorem).

Step3: Use definition of congruent angles for $\angle2$ and $\angle3$

If $\angle2\cong\angle3$, then $m\angle2 = m\angle3$ (Definition of congruent angles).

Step4: Substitute angles

We know that $m\angle1 + m\angle2=90^{\circ}$. Substituting $m\angle1$ with $m\angle4$ and $m\angle2$ with $m\angle3$ (from previous steps), we get $m\angle4 + m\angle3 = 90^{\circ}$.

Step5: Use definition of complementary angles

If $m\angle4 + m\angle3 = 90^{\circ}$, then $\angle3$ and $\angle4$ are complementary (Definition of complementary angles).

Answer:

1. Definition of congruent angles
2. Vertical Angles Theorem
3. Definition of congruent angles
4. Definition of complementary angles