Question

given: ∠bcd is a right angle; ∠acb≅∠cad; a is in the interior of ∠bcd
prove: ∠cad is complementary to ∠acd
statements

1. ∠bcd is a right angle.
2. ∠acb≅∠cad
3. a is in the interior of ∠bcd.
4. m∠bcd = 90°
5. m∠acb + m∠acd = m∠bcd
6. m∠acb + m∠acd = 90°
7. ∠acb is complementary to ∠acd
8. ∠cad is complementary to ∠acd

reasons

1. def. right angle
2. given

3.
4.
5.
6.
7.
8.

Explanation:

Step1: Define right - angle

Since $\angle BCD$ is a right angle, by the definition of a right angle, $m\angle BCD = 90^{\circ}$.

Step2: Use angle - addition postulate

If $A$ is in the interior of $\angle BCD$, then by the angle - addition postulate, $m\angle ACB+m\angle ACD=m\angle BCD$.

Step3: Substitute the value of $\angle BCD$

Substitute $m\angle BCD = 90^{\circ}$ into the equation $m\angle ACB+m\angle ACD=m\angle BCD$, we get $m\angle ACB + m\angle ACD=90^{\circ}$.

Step4: Define complementary angles

Since $m\angle ACB + m\angle ACD = 90^{\circ}$, by the definition of complementary angles, $\angle ACB$ is complementary to $\angle ACD$.

Step5: Use angle congruence

Given that $\angle ACB\cong\angle CAD$, then $m\angle ACB=m\angle CAD$. Substitute $m\angle ACB$ with $m\angle CAD$ in the equation $m\angle ACB + m\angle ACD=90^{\circ}$, we can conclude that $\angle CAD$ is complementary to $\angle ACD$.

Answer:

1. Def. right angle
2. Given
3. Given
4. Def. right angle
5. Angle - addition postulate
6. Substitution property of equality
7. Def. complementary angles
8. Substitution property of equality (using $\angle ACB\cong\angle CAD$)